From fdcbb0260cf6dc4ee3e5347a6408656cd7a3e094 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Thu, 4 Jan 2024 08:00:44 +0000 Subject: [PATCH] build based on b820ec0 --- dev/.documenter-siteinfo.json | 2 +- dev/HarmonicOscillator/index.html | 4 ++-- dev/HydrogenAtom/index.html | 4 ++-- dev/InfinitePotentialWell/index.html | 4 ++-- dev/MorsePotential/index.html | 4 ++-- dev/index.html | 14 ++++++++------ dev/jmd/HarmonicOscillator.jmd | 2 +- dev/jmd/HydrogenAtom.jmd | 2 +- dev/jmd/InfinitePotentialWell.jmd | 2 +- dev/jmd/MorsePotential.jmd | 2 +- dev/search_index.js | 2 +- 11 files changed, 22 insertions(+), 20 deletions(-) diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 6b596ac..3069f69 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-12-14T23:52:04","documenter_version":"1.2.1"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-01-04T08:00:41","documenter_version":"1.2.1"}} \ No newline at end of file diff --git a/dev/HarmonicOscillator/index.html b/dev/HarmonicOscillator/index.html index 4f6f47a..1fd2ec4 100644 --- a/dev/HarmonicOscillator/index.html +++ b/dev/HarmonicOscillator/index.html @@ -22,7 +22,7 @@ H_{8}(x) &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}, \\ H_{9}(x) &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}, \\ &\vdots - \end{aligned}\]

Reference

Usage & Examples

Install Antique.jl for the first run and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.

using Antique
+  \end{aligned}\]
Reference
Usage & Examples
Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.
using Antique
 HO = antique(:HarmonicOscillator, k=1.0, m=1.0, ℏ=1.0)
Parameters:
julia> HO.k
 1.0
 
@@ -443,4 +443,4 @@
 5.0 |  9 |   21.242645786248 |   21.242617504750 ✔
 Test Summary:              | Pass  Total
 ∫ψₙ*Hψₙdx = <ψₙ|H|ψₙ> = Eₙ |   40     40
-

+
diff --git a/dev/HydrogenAtom/index.html b/dev/HydrogenAtom/index.html index 0e00b9e..9321b37 100644 --- a/dev/HydrogenAtom/index.html +++ b/dev/HydrogenAtom/index.html @@ -49,7 +49,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots. - \end{aligned}\]

References

Usage & Examples

Install Antique.jl for the first run and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.

using Antique
+  \end{aligned}\]
References
Usage & Examples
Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.
using Antique
 H = antique(:HydrogenAtom, Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)
Parameters:
julia> H.Z
 1
 
@@ -1304,4 +1304,4 @@
  3 |  3 |  2 |  2 |  2 |  2 |    1.000000000000 |    1.000300628566 ✔
 Test Summary:                       | Pass  Total
 <ψₙ₁ₗ₁ₘ₁|ψₙ₂ₗ₂ₘ₂> = δₙ₁ₙ₂δₗ₁ₗ₂δₘ₁ₘ₂ |  196    196
-

+
diff --git a/dev/InfinitePotentialWell/index.html b/dev/InfinitePotentialWell/index.html index 8ad9701..989394b 100644 --- a/dev/InfinitePotentialWell/index.html +++ b/dev/InfinitePotentialWell/index.html @@ -5,7 +5,7 @@ \infty & x \lt 0, L \lt x \\ 0 & 0 \leq x \leq L \end{array} - \right.\]

Eigen Values

E(; n=0, L=L, m=m, ℏ=ℏ)

\[ E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\]

Eigen Functions

ψ(x; n=0, L=L)

\[ \psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]

Proofs

Usage & Examples

Install Antique.jl for the first run and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.

using Antique
+  \right.\]
Eigen Values
E(; n=0, L=L, m=m, ℏ=ℏ)
\[  E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\]
Eigen Functions
ψ(x; n=0, L=L)
\[   \psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]
Proofs
Usage & Examples
Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.
using Antique
 IPW = antique(:InfinitePotentialWell, L=1.0, m=1.0, ℏ=1.0)
Parameters:
julia> IPW.L
 1.0
 
@@ -384,4 +384,4 @@
 7.0 |  1 |    0.201420496383 |    0.201420497981 ✔
 Test Summary:                              | Pass  Total
 <ψₙ|p²|ψₙ> = ∫ψₙ*(-ℏ²d²/dx²)ψₙdx = π²ℏ²/L² |    4      4
-

+
diff --git a/dev/MorsePotential/index.html b/dev/MorsePotential/index.html index dc77fad..45d46e9 100644 --- a/dev/MorsePotential/index.html +++ b/dev/MorsePotential/index.html @@ -21,7 +21,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots - \end{aligned}\]

References

Usage & Examples

Install Antique.jl for the first run and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.

Warning

Run using Pkg; Pkg.add("SpecialFunctions") if the following returns an error.

# Parameters for H₂⁺
+  \end{aligned}\]
References
Usage & Examples
Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.
Warning
Run using Pkg; Pkg.add("SpecialFunctions") if the following returns an error.
# Parameters for H₂⁺
 # https://doi.org/10.1002/slct.202102509
 # https://doi.org/10.5281/zenodo.5047817
 # https://physics.nist.gov/cgi-bin/cuu/Value?mpsme
@@ -845,4 +845,4 @@
 0.1 |  9 |   -0.026741858566 |   -0.026742018376 ✔
 Test Summary:              | Pass  Total
 <ψₙ|H|ψₙ> = ∫ψₙ*Hψₙdx = Eₙ |   40     40
-

+
diff --git a/dev/index.html b/dev/index.html index 6291579..7926729 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,9 +1,11 @@ -Home · Antique.jl
GitHub

Antique.jl

Self-contained, Well-Tested, Well-Documented Analytical Solutions of Quantum Mechanical Equations.

Install

To install this package, run the following code in your Jupyter Notebook:

using Pkg; Pkg.add("Antique")

Usage

To use this package, run the following code before each use:

using Antique

The function antique(model, parameters...) returns a module. Each module has E(), ψ(x) and some other functions.

Examples

The energy of $1\mathrm{S}$ state in $\mathrm{H}$:

julia> H = antique(:HydrogenAtom, Z=1)
-julia> H.E(n=1)
--0.5

The energy of $1\mathrm{S}$ state in $\mathrm{He}^+$:

julia> He⁺ = antique(:HydrogenAtom, Z=2)
-julia> He⁺.E(n=1)
--2.0

Supported Models

+Home · Antique.jl
GitHub

Antique.jl

Self-contained, Well-Tested, Well-Documented Analytical Solutions of Quantum Mechanical Equations.

Install

To install this package, run the following code in your Jupyter Notebook:

using Pkg; Pkg.add("Antique")

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E, ψ, V and some other functions. Here are examples in hydrogen-like atom. The analytical notation of energy (eigen value of the Hamiltonian) is written as

\[E_n = -\frac{Z^2}{2n^2} E_\mathrm{h}.\]

Hydrogen atom has symbol $\mathrm{H}$ and atomic number 1 ($Z=1$). Therefore the ground state ($n=1$) energy is $-\frac{1}{2} E_\mathrm{h}$.

using Antique
+H = antique(:HydrogenAtom, Z=1)
+H.E(n=1)
+# output> -0.5

Helium cation has symbol $\mathrm{He}^+$ and atomic number 2 ($Z=2$). Therefore the ground state ($n=1$) energy is $-2 E_\mathrm{h}$.

using Antique
+He⁺ = antique(:HydrogenAtom, Z=2)
+He⁺.E(n=1)
+# output> -2.0

There are more examples on each model page.

Supported Models

InfinitePotentialWell @@ -28,4 +30,4 @@ :HydrogenAtom
-

Future Works

List of quantum-mechanical systems with analytical solutions

Acknowledgment

This package was named by @KB-satou and @ultimatile.

+

Future Works

List of quantum-mechanical systems with analytical solutions

Acknowledgment

This package was named by @KB-satou and @ultimatile.

diff --git a/dev/jmd/HarmonicOscillator.jmd b/dev/jmd/HarmonicOscillator.jmd index 9fa2c62..65da077 100644 --- a/dev/jmd/HarmonicOscillator.jmd +++ b/dev/jmd/HarmonicOscillator.jmd @@ -88,7 +88,7 @@ Examples: ## Usage & Examples -[Install Antique.jl](@ref Install) for the first run and run `using Antique` before each use. The function `antique(model, parameters...)` returns a module that has `E()`, `ψ(x)`, `V(x)` and some other functions. In this system, the model name is specified by `:HarmonicOscillator` and several parameters `k`, `m` and `ℏ` are set as optional arguments. +[Install Antique.jl](@ref Install) for the first use and run `using Antique` before each use. The function `antique(model, parameters...)` returns a module that has `E()`, `ψ(x)`, `V(x)` and some other functions. In this system, the model name is specified by `:HarmonicOscillator` and several parameters `k`, `m` and `ℏ` are set as optional arguments. ```julia; cache = :all; results = "hidden" using Antique diff --git a/dev/jmd/HydrogenAtom.jmd b/dev/jmd/HydrogenAtom.jmd index 9f34c54..d78510d 100644 --- a/dev/jmd/HydrogenAtom.jmd +++ b/dev/jmd/HydrogenAtom.jmd @@ -151,7 +151,7 @@ Examples: ## Usage & Examples -[Install Antique.jl](@ref Install) for the first run and run `using Antique` before each use. The function `antique(model, parameters...)` returns a module that has `E()`, `ψ(r)`, `V(r)` and some other functions. In this system, the model name is specified by `:HydrogenAtom` and several parameters `Z`, `Eₕ`, `mₑ`, `a₀` and `ℏ` are set as optional arguments. +[Install Antique.jl](@ref Install) for the first use and run `using Antique` before each use. The function `antique(model, parameters...)` returns a module that has `E()`, `ψ(r)`, `V(r)` and some other functions. In this system, the model name is specified by `:HydrogenAtom` and several parameters `Z`, `Eₕ`, `mₑ`, `a₀` and `ℏ` are set as optional arguments. ```julia; cache = :all; results = "hidden" using Antique diff --git a/dev/jmd/InfinitePotentialWell.jmd b/dev/jmd/InfinitePotentialWell.jmd index 7376ce8..d23a5f6 100644 --- a/dev/jmd/InfinitePotentialWell.jmd +++ b/dev/jmd/InfinitePotentialWell.jmd @@ -50,7 +50,7 @@ The infinite potential well (particle in a box) is the simplest model for quantu ## Usage & Examples -[Install Antique.jl](@ref Install) for the first run and run `using Antique` before each use. The function `antique(model, parameters...)` returns a module that has `E()`, `ψ(x)`, `V(x)` and some other functions. In this system, the model name is specified by `:InfinitePotentialWell` and several parameters `L`, `m` and `ℏ` are set as optional arguments. +[Install Antique.jl](@ref Install) for the first use and run `using Antique` before each use. The function `antique(model, parameters...)` returns a module that has `E()`, `ψ(x)`, `V(x)` and some other functions. In this system, the model name is specified by `:InfinitePotentialWell` and several parameters `L`, `m` and `ℏ` are set as optional arguments. ```julia; cache = :all; results = "hidden" using Antique diff --git a/dev/jmd/MorsePotential.jmd b/dev/jmd/MorsePotential.jmd index ddecc9f..a09d10b 100644 --- a/dev/jmd/MorsePotential.jmd +++ b/dev/jmd/MorsePotential.jmd @@ -81,7 +81,7 @@ Examples: ## Usage & Examples -[Install Antique.jl](@ref Install) for the first run and run `using Antique` before each use. The function `antique(model, parameters...)` returns a module that has `E()`, `ψ(r)`, `V(r)` and some other functions. In this system, the model name is specified by `:MorsePotential` and several parameters `rₑ`, `Dₑ`, `k`, `µ` and `ℏ` are set as optional arguments. +[Install Antique.jl](@ref Install) for the first use and run `using Antique` before each use. The function `antique(model, parameters...)` returns a module that has `E()`, `ψ(r)`, `V(r)` and some other functions. In this system, the model name is specified by `:MorsePotential` and several parameters `rₑ`, `Dₑ`, `k`, `µ` and `ℏ` are set as optional arguments. !!! warning diff --git a/dev/search_index.js b/dev/search_index.js index 306e7f2..bfa0bef 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"CurrentModule = Antique","category":"page"},{"location":"InfinitePotentialWell/#Infinite-Potential-Well-(Particle-in-a-Box)","page":"Infinite Potential Well","title":"Infinite Potential Well (Particle in a Box)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.","category":"page"},{"location":"InfinitePotentialWell/#Definitions","page":"Infinite Potential Well","title":"Definitions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L is the length of the 1D-box, m is the mass of particle.","category":"page"},{"location":"InfinitePotentialWell/#Schrödinger-Equation","page":"Infinite Potential Well","title":"Schrödinger Equation","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"InfinitePotentialWell/#Hamiltonian","page":"Infinite Potential Well","title":"Hamiltonian","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH = frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"InfinitePotentialWell/#Potential","page":"Infinite Potential Well","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"V(x; L=L)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" V(x) =\n left\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\n right","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"E(; n=0, L=L, m=m, ℏ=ℏ)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" E_n = frachbar^2 n^2 pi^22 m L^2","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Functions","page":"Infinite Potential Well","title":"Eigen Functions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"ψ(x; n=0, L=L)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" psi_n(x) = sqrtfrac2L sin fracnpi xL","category":"page"},{"location":"InfinitePotentialWell/#Proofs","page":"Infinite Potential Well","title":"Proofs","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen Functions & Eigen Values\nNormalization","category":"page"},{"location":"InfinitePotentialWell/#Usage-and-Examples","page":"Infinite Potential Well","title":"Usage & Examples","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Install Antique.jl for the first run and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Antique\nIPW = antique(:InfinitePotentialWell, L=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.L\n1.0\n\njulia> IPW.m\n1.0\n\njulia> IPW.ℏ\n1.0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen values:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.E(n=1)\n4.934802200544679\n\njulia> IPW.E(n=2)\n19.739208802178716","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Plots\nplot(xlim=(0,1), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> IPW.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> IPW.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> IPW.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> IPW.ψ(x, n=4), label=\"n=4\", lw=2)\nplot!(x -> IPW.ψ(x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L = 1\nusing Plots\nplot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 1:5\n # energy\n plot!([0,L], fill(IPW.E(n=n,L=L),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(0:0.01:L, x->IPW.E(n=n,L=L)+5*IPW.ψ(x,n=n,L=L), lc=n, lw=2, label=\"\")\nend\n# potential\nplot!([0,0,L,L], [140,0,0,140], lc=:black, lw=2, label=\"\")","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/#Testing","page":"Infinite Potential Well","title":"Testing","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"InfinitePotentialWell/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Infinite Potential Well","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"int_0^L psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 10 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 10 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 10 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 4 | 10 | 0.000000000000 | 0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 10 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 10 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 7 | 10 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000000 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 8 | 10 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\n 9 | 10 | 0.000000000000 | 0.000000000000 ✔\n10 | 1 | 0.000000000000 | 0.000000000000 ✔\n10 | 2 | 0.000000000000 | 0.000000000000 ✔\n10 | 3 | 0.000000000000 | 0.000000000000 ✔\n10 | 4 | 0.000000000000 | 0.000000000000 ✔\n10 | 5 | 0.000000000000 | 0.000000000000 ✔\n10 | 6 | 0.000000000000 | -0.000000000000 ✔\n10 | 7 | 0.000000000000 | -0.000000000000 ✔\n10 | 8 | 0.000000000000 | 0.000000000000 ✔\n10 | 9 | 0.000000000000 | 0.000000000000 ✔\n10 | 10 | 1.000000000000 | 1.000000000000 ✔\nTest Summary: | Pass Total\n<ψᵢ|ψⱼ> = ∫ψₙ*ψₙdx = δᵢⱼ | 100 100","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values-2","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n E_n\n = int_0^L psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int_0^L psi^ast_n(x) left 0 - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | m | ℏ | n | analytical | numerical \n--- | --- | --- | -- | ----------------- | ----------------- \n0.1 | 0.1 | 0.1 | 1 | 49.348021579139 | 49.348022005447 ✔\n0.1 | 0.1 | 0.1 | 2 | 197.392081461942 | 197.392088021787 ✔\n0.1 | 0.1 | 0.1 | 3 | 444.132165131018 | 444.132198049021 ✔\n0.1 | 0.1 | 0.1 | 4 | 789.568248175979 | 789.568352087149 ✔\n0.1 | 0.1 | 0.1 | 5 | 1233.700296336187 | 1233.700550136170 ✔\n0.1 | 0.1 | 0.1 | 6 | 1776.528266243334 | 1776.528792196084 ✔\n0.1 | 0.1 | 0.1 | 7 | 2418.052103857080 | 2418.053078266893 ✔\n0.1 | 0.1 | 0.1 | 8 | 3158.271745875927 | 3158.273408348594 ✔\n0.1 | 0.1 | 0.1 | 9 | 3997.187119264267 | 3997.189782441189 ✔\n0.1 | 0.1 | 0.1 | 10 | 4934.798141994514 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 1 | 4934.802157913905 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 2 | 19739.208146194214 | 19739.208802178713 ✔\n0.1 | 0.1 | 1.0 | 3 | 44413.216513101754 | 44413.219804902103 ✔\n0.1 | 0.1 | 1.0 | 4 | 78956.824817597895 | 78956.835208714852 ✔\n0.1 | 0.1 | 1.0 | 5 | 123370.029633618717 | 123370.055013616948 ✔\n0.1 | 0.1 | 1.0 | 6 | 177652.826624333364 | 177652.879219608410 ✔\n0.1 | 0.1 | 1.0 | 7 | 241805.210385707964 | 241805.307826689212 ✔\n0.1 | 0.1 | 1.0 | 8 | 315827.174587592541 | 315827.340834859409 ✔\n0.1 | 0.1 | 1.0 | 9 | 399718.711926426622 | 399718.978244118916 ✔\n0.1 | 0.1 | 1.0 | 10 | 493479.814199451241 | 493480.220054467791 ✔\n0.1 | 1.0 | 0.1 | 1 | 4.934802157914 | 4.934802200545 ✔\n0.1 | 1.0 | 0.1 | 2 | 19.739208146194 | 19.739208802179 ✔\n0.1 | 1.0 | 0.1 | 3 | 44.413216513102 | 44.413219804902 ✔\n0.1 | 1.0 | 0.1 | 4 | 78.956824817598 | 78.956835208715 ✔\n0.1 | 1.0 | 0.1 | 5 | 123.370029633619 | 123.370055013617 ✔\n0.1 | 1.0 | 0.1 | 6 | 177.652826624333 | 177.652879219608 ✔\n0.1 | 1.0 | 0.1 | 7 | 241.805210385708 | 241.805307826689 ✔\n0.1 | 1.0 | 0.1 | 8 | 315.827174587593 | 315.827340834859 ✔\n0.1 | 1.0 | 0.1 | 9 | 399.718711926427 | 399.718978244119 ✔\n0.1 | 1.0 | 0.1 | 10 | 493.479814199451 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 1 | 493.480215791390 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 2 | 1973.920814619422 | 1973.920880217871 ✔\n0.1 | 1.0 | 1.0 | 3 | 4441.321651310176 | 4441.321980490210 ✔\n0.1 | 1.0 | 1.0 | 4 | 7895.682481759791 | 7895.683520871485 ✔\n0.1 | 1.0 | 1.0 | 5 | 12337.002963361872 | 12337.005501361695 ✔\n0.1 | 1.0 | 1.0 | 6 | 17765.282662433339 | 17765.287921960840 ✔\n0.1 | 1.0 | 1.0 | 7 | 24180.521038570794 | 24180.530782668924 ✔\n0.1 | 1.0 | 1.0 | 8 | 31582.717458759253 | 31582.734083485939 ✔\n0.1 | 1.0 | 1.0 | 9 | 39971.871192642662 | 39971.897824411892 ✔\n0.1 | 1.0 | 1.0 | 10 | 49347.981419945128 | 49348.022005446779 ✔\n1.0 | 0.1 | 0.1 | 1 | 0.493480215948 | 0.493480220054 ✔\n1.0 | 0.1 | 0.1 | 2 | 1.973920815419 | 1.973920880218 ✔\n1.0 | 0.1 | 0.1 | 3 | 4.441321651944 | 4.441321980490 ✔\n1.0 | 0.1 | 0.1 | 4 | 7.895682481265 | 7.895683520871 ✔\n1.0 | 0.1 | 0.1 | 5 | 12.337002965030 | 12.337005501362 ✔\n1.0 | 0.1 | 0.1 | 6 | 17.765282661715 | 17.765287921961 ✔\n1.0 | 0.1 | 0.1 | 7 | 24.180521036064 | 24.180530782669 ✔\n1.0 | 0.1 | 0.1 | 8 | 31.582717460023 | 31.582734083486 ✔\n1.0 | 0.1 | 0.1 | 9 | 39.971871195191 | 39.971897824412 ✔\n1.0 | 0.1 | 0.1 | 10 | 49.347981417827 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 1 | 49.348021594816 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 2 | 197.392081541864 | 197.392088021787 ✔\n1.0 | 0.1 | 1.0 | 3 | 444.132165194438 | 444.132198049021 ✔\n1.0 | 0.1 | 1.0 | 4 | 789.568248126463 | 789.568352087149 ✔\n1.0 | 0.1 | 1.0 | 5 | 1233.700296503016 | 1233.700550136170 ✔\n1.0 | 0.1 | 1.0 | 6 | 1776.528266171473 | 1776.528792196084 ✔\n1.0 | 0.1 | 1.0 | 7 | 2418.052103606433 | 2418.053078266892 ✔\n1.0 | 0.1 | 1.0 | 8 | 3158.271746002275 | 3158.273408348594 ✔\n1.0 | 0.1 | 1.0 | 9 | 3997.187119519121 | 3997.189782441190 ✔\n1.0 | 0.1 | 1.0 | 10 | 4934.798141782662 | 4934.802200544679 ✔\n1.0 | 1.0 | 0.1 | 1 | 0.049348021595 | 0.049348022005 ✔\n1.0 | 1.0 | 0.1 | 2 | 0.197392081542 | 0.197392088022 ✔\n1.0 | 1.0 | 0.1 | 3 | 0.444132165194 | 0.444132198049 ✔\n1.0 | 1.0 | 0.1 | 4 | 0.789568248126 | 0.789568352087 ✔\n1.0 | 1.0 | 0.1 | 5 | 1.233700296503 | 1.233700550136 ✔\n1.0 | 1.0 | 0.1 | 6 | 1.776528266171 | 1.776528792196 ✔\n1.0 | 1.0 | 0.1 | 7 | 2.418052103606 | 2.418053078267 ✔\n1.0 | 1.0 | 0.1 | 8 | 3.158271746002 | 3.158273408349 ✔\n1.0 | 1.0 | 0.1 | 9 | 3.997187119519 | 3.997189782441 ✔\n1.0 | 1.0 | 0.1 | 10 | 4.934798141783 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 1 | 4.934802159482 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 2 | 19.739208154186 | 19.739208802179 ✔\n1.0 | 1.0 | 1.0 | 3 | 44.413216519444 | 44.413219804902 ✔\n1.0 | 1.0 | 1.0 | 4 | 78.956824812646 | 78.956835208715 ✔\n1.0 | 1.0 | 1.0 | 5 | 123.370029650302 | 123.370055013617 ✔\n1.0 | 1.0 | 1.0 | 6 | 177.652826617147 | 177.652879219608 ✔\n1.0 | 1.0 | 1.0 | 7 | 241.805210360643 | 241.805307826689 ✔\n1.0 | 1.0 | 1.0 | 8 | 315.827174600228 | 315.827340834859 ✔\n1.0 | 1.0 | 1.0 | 9 | 399.718711951912 | 399.718978244119 ✔\n1.0 | 1.0 | 1.0 | 10 | 493.479814178266 | 493.480220054468 ✔\nTest Summary: | Pass Total\n<ψₙ|H|ψₙ> = ∫ψₙ*Tψₙdx = Eₙ | 80 80","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x","page":"Infinite Potential Well","title":"Expected Value of x","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x rangle_n=1\n= int_0^L psi_1^ast(x) hatx psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"for only n=1.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.050000000000 | 0.050000000000 ✔\n0.5 | 1 | 0.250000000000 | 0.250000000000 ✔\n1.0 | 1 | 0.500000000000 | 0.500000000000 ✔\n7.0 | 1 | 3.500000000000 | 3.500000000000 ✔\nTest Summary: | Pass Total\n<ψₙ|x|ψₙ> = L/2 | 4 4","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x2","page":"Infinite Potential Well","title":"Expected Value of x^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x^2 rangle_n=1\n= int_0^L psi_1^ast(x) hatx^2 psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.002826727415 | 0.002826727415 ✔\n0.5 | 1 | 0.070668185378 | 0.070668185378 ✔\n1.0 | 1 | 0.282672741512 | 0.282672741512 ✔\n7.0 | 1 | 13.850964334096 | 13.850964334096 ✔\nTest Summary: | Pass Total\n<ψₙ|x²|ψₙ> = 2L²/π³(π³/6-π/4) | 4 4","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p","page":"Infinite Potential Well","title":"Expected Value of p","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p rangle_n=1\n= int_0^L psi_1^ast(x) hatp psi_1(x) mathrmdx\n= 0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n langle p rangle_n=1\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -ihbarfracmathrmdmathrmd x right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -ihbar fracpsi(x+Delta x) - psi(x-Delta x)2Delta x right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2fracmathrmd psi(x)mathrmdx Delta x\n + Oleft(Delta x^3right)\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n \n 2fracmathrmd psi(x)mathrmdx Delta x\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n - Oleft(Delta x^3right)\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n - fracOleft(Delta x^3right)2Delta x\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n psi(x+Delta x)\n =\n psi(x)\n + fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\n \n psi(x-Delta x)\n =\n psi(x)\n - fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.000000000003 | 0.000000000000 ✔\n0.5 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.0 | 1 | 0.000000000000 | 0.000000000000 ✔\n7.0 | 1 | 0.000000000000 | 0.000000000000 ✔\nTest Summary: | Pass Total\n<ψₙ|p|ψₙ> = ∫ψₙ*(-iℏd/dx)ψₙdx = 0 | 4 4","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p2","page":"Infinite Potential Well","title":"Expected Value of p^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p^2 rangle\n= int_0^L psi_1^ast(x) hatp^2 psi_1(x) mathrmdx\n= fracpi^2hbar^2L^2","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n langle p^2 rangle\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -hbar^2fracmathrmd^2mathrmdx^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -hbar^2 fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 986.960431582781 | 986.960440108936 ✔\n0.5 | 1 | 39.478417274195 | 39.478417604357 ✔\n1.0 | 1 | 9.869604318963 | 9.869604401089 ✔\n7.0 | 1 | 0.201420496383 | 0.201420497981 ✔\nTest Summary: | Pass Total\n<ψₙ|p²|ψₙ> = ∫ψₙ*(-ℏ²d²/dx²)ψₙdx = π²ℏ²/L² | 4 4\n","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"CurrentModule = Antique","category":"page"},{"location":"MorsePotential/#Morse-Potential","page":"Morse Potential","title":"Morse Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.","category":"page"},{"location":"MorsePotential/#Definitions","page":"Morse Potential","title":"Definitions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"xi = 2lambdamathrme^-a(r-r_e), omega = sqrtkµ, k = 2D_mathrmea^2, lambda = fracsqrt2mD_mathrmeahbar, chi = frachbaromega4D_mathrme, N_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight) are used. The domains of the potential and the wave functions are 0leq r lt infty.","category":"page"},{"location":"MorsePotential/#Schrödinger-Equation","page":"Morse Potential","title":"Schrödinger Equation","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatHpsi(r) = E psi(r)","category":"page"},{"location":"MorsePotential/#Hamiltonian","page":"Morse Potential","title":"Hamiltonian","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"MorsePotential/#Potential","page":"Morse Potential","title":"Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"V(r; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)))","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" V(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)","category":"page"},{"location":"MorsePotential/#Eigen-Values","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"E(n; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)), µ=µ, ω=sqrt(k/µ), χ=ℏ*ω/(4*Dₑ), ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" E_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2","category":"page"},{"location":"MorsePotential/#Eigen-Functions","page":"Morse Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"ψ(n, r; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)), µ=µ, ω=sqrt(k/µ), χ=ℏ*ω/(4*Dₑ), ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" psi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials","page":"Morse Potential","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"L(x; n=0, α=0)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k left(beginarrayl n+alpha n-k endarrayright) fracx^kk \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Examples:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n L_0^(0)(x) = 1 \n L_1^(0)(x) = 1 - x \n L_1^(1)(x) = 2 - x \n L_2^(0)(x) = 1 - 2 x + 12 x^2 \n L_2^(1)(x) = 3 - 3 x + 12 x^2 \n L_2^(2)(x) = 6 - 4 x + 12 x^2 \n L_3^(0)(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^(1)(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_3^(2)(x) = 10 - 10 x + 52 x^2 - 16 x^3 \n L_3^(3)(x) = 20 - 15 x + 3 x^2 - 16 x^3 \n L_4^(0)(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 124 x^4 \n L_4^(1)(x) = 5 - 10 x + 5 x^2 - 56 x^3 + 124 x^4 \n L_4^(2)(x) = 15 - 20 x + 152 x^2 - 1 x^3 + 124 x^4 \n L_4^(3)(x) = 35 - 35 x + 212 x^2 - 76 x^3 + 124 x^4 \n L_4^(4)(x) = 70 - 56 x + 14 x^2 - 43 x^3 + 124 x^4 \n vdots\n endaligned","category":"page"},{"location":"MorsePotential/#References","page":"Morse Potential","title":"References","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"P. M. Morse, Phys. Rev. 34, 57 (1929)\nJ. P. Dahl, M. Springborg, J. Chem. Phys. 88, 4535 (1988). (62), (63)\nW. K. Shao, Y. He, J. Pan, J. Nonlinear Sci. Appl., 9, 5, 3388 (2016). (1.6) \nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.12","category":"page"},{"location":"MorsePotential/#Usage-and-Examples","page":"Morse Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Install Antique.jl for the first run and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"warning: Warning\nRun using Pkg; Pkg.add(\"SpecialFunctions\") if the following returns an error.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"# Parameters for H₂⁺\n# https://doi.org/10.1002/slct.202102509\n# https://doi.org/10.5281/zenodo.5047817\n# https://physics.nist.gov/cgi-bin/cuu/Value?mpsme\nrₑ = 1.997193319969992120068298141276\nVₑ = -0.602634619106539878727562156289\nDₑ = - 0.5 - Vₑ\nk = 2*((-1.1026342144949464615+1/2.00) - Vₑ) / (2.00 - rₑ)^2\nµ = 1/(1/1836.15267343 + 1/1836.15267343)\nℏ = 1.0\n\nusing Antique\nMP = antique(:MorsePotential, rₑ=rₑ, Vₑ=Vₑ, Dₑ=Dₑ, k=k, µ =µ, ℏ =ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.rₑ\n1.997193319969992\n\njulia> MP.Dₑ\n0.10263461910653993\n\njulia> MP.k\n0.1027265041900817\n\njulia> MP.µ\n918.076336715\n\njulia> MP.ℏ\n1.0","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Eigen values:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.E(n=0)\n-0.09741377794418261\n\njulia> MP.E(n=1)\n-0.08738092406760907","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(0.1:0.01:15, r -> MP.V(r), lw=2, label=\"\", xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"r\", ylabel=\"V(r)\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Wave functions:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(xlim=(0,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> MP.ψ(x, n=0), label=\"n=0\", lw=2)\nplot!(x -> MP.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> MP.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> MP.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> MP.ψ(x, n=4), label=\"n=4\", lw=2)\nplot!(x -> MP.ψ(x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve, Energy levels, Comparison with harmonic oscillator:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"MP = antique(:MorsePotential)\nHO = antique(:HarmonicOscillator, k=MP.k, m=MP.μ)\nusing Plots\nplot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"\\$r\\$\", ylabel=\"\\$V(r), E_n\\$\", legend=:bottomright, size=(480,400), dpi=300)\nfor n in 0:MP.nₘₐₓ()\n # energy\n EM = MP.E(n=n)\n EH = HO.E(n=n) - MP.Dₑ\n plot!(0.1:0.01:15, r -> EH > HO.V(r-MP.rₑ) - MP.Dₑ ? EH : NaN, lc=\"#BC1C5F\", lw=1, label=\"\")\n plot!(0.1:0.01:15, r -> EM > MP.V(r) ? EM : NaN, lc=\"#578FC7\", lw=1, label=\"\")\nend\n# potential\nplot!(0.1:0.01:15, r -> HO.V(r-MP.rₑ) - MP.Dₑ, lc=\"#BC1C5F\", lw=2, label=\"Harmonic Oscillator\")\nplot!(0.1:0.01:15, r -> MP.V(r), lc=\"#578FC7\", lw=2, label=\"Morse Potential\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"where, the potential of harmonic oscillator is defined as V(r) simeq frac12 k (r - r_mathrme)^2 + V_0.","category":"page"},{"location":"MorsePotential/#Testing","page":"Morse Potential","title":"Testing","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Generalized Laguerre Polynomials L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=0 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_0^(0)(x)\n = e^x e^ - x\n = 1 \n = 1\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_1^(0)(x)\n = e^x fracmathrmdmathrmdx x e^ - x\n = 1 - x \n = 1 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_1^(1)(x)\n = frace^x fracmathrmdmathrmdx x^2 e^ - xx\n = 2 - x \n = 2 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(0)(x)\n = frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n = 1 - 2 x + frac12 x^2 \n = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(1)(x)\n = fracfrac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - xx\n = 3 - 3 x + frac12 x^2 \n = 3 - 3 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(2)(x)\n = fracfrac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - xx^2\n = 6 - 4 x + frac12 x^2 \n = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(0)(x)\n = frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = 1 - frac16 x^3 - 3 x + frac32 x^2 \n = 1 - frac16 x^3 - 3 x + frac32 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(1)(x)\n = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - xx\n = 4 - 6 x - frac16 x^3 + 2 x^2 \n = 4 - frac16 x^3 - 6 x + 2 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(2)(x)\n = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - xx^2\n = 10 - 10 x - frac16 x^3 + frac52 x^2 \n = 10 - frac16 x^3 - 10 x + frac52 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(3)(x)\n = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - xx^3\n = 20 - 15 x - frac16 x^3 + 3 x^2 \n = 20 - frac16 x^3 - 15 x + 3 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(0)(x)\n = frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4 \n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(1)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - xx\n = 5 - 10 x - frac56 x^3 + 5 x^2 + frac124 x^4 \n = 5 - frac56 x^3 - 10 x + 5 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(2)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - xx^2\n = 15 - 20 x - x^3 + frac152 x^2 + frac124 x^4 \n = 15 - x^3 - 20 x + frac152 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(3)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^7 e^ - xx^3\n = 35 - 35 x - frac76 x^3 + frac212 x^2 + frac124 x^4 \n = 35 - frac76 x^3 - 35 x + frac212 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=4 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(4)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^8 e^ - xx^4\n = 70 - 56 x - frac43 x^3 + 14 x^2 + frac124 x^4 \n = 70 - frac43 x^3 - 56 x + 14 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Test Summary: | Pass Total\nLₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ | 15 15","category":"page"},{"location":"MorsePotential/#Normalization-and-Orthogonality-of-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Normalization & Orthogonality of L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"int_0^infty L_i^(alpha)(x) L_j^(alpha)(x) x^alpha mathrme^-x mathrmdx = fracGamma(n+alpha+1)n delta_ij","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" α | i | j | analytical | numerical \n---- | -- | -- | ----------------- | ----------------- \n0.01 | 0 | 0 | 0.994325851192 | 0.994325852936 ✔\n0.01 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 1 | 1.004269109703 | 1.004269111483 ✔\n0.01 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 8 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 9 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 2 | 2 | 1.009290455252 | 1.009290456144 ✔\n0.01 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 2 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 9 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 3 | 3 | 1.012654756769 | 1.012654758579 ✔\n0.01 | 3 | 4 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 7 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 3 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 3 | 9 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 4 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 4 | 4 | 1.015186393661 | 1.015186394564 ✔\n0.01 | 4 | 5 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 4 | 6 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 4 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 9 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 5 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 5 | 5 | 1.017216766449 | 1.017216768275 ✔\n0.01 | 5 | 6 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 6 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 6 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 6 | 4 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 6 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 6 | 6 | 1.018912127726 | 1.018912128636 ✔\n0.01 | 6 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 6 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 6 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 7 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 7 | 3 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 7 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 6 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 7 | 1.020367716480 | 1.020367717392 ✔\n0.01 | 7 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 7 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 8 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 8 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 8 | 3 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 6 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 7 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 8 | 1.021643176126 | 1.021643177967 ✔\n0.01 | 8 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 9 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 9 | 2 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 9 | 3 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 9 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 9 | 5 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 6 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 7 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 9 | 1.022778335210 | 1.022778336127 ✔\n0.05 | 0 | 0 | 0.973504265563 | 0.973504267703 ✔\n0.05 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 0 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 0 | 8 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 0 | 9 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 1 | 1 | 1.022179478841 | 1.022179479980 ✔\n0.05 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 1 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 5 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 8 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 1 | 9 | 0.000000000000 | 0.000000000007 ✔\n0.05 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 2 | 2 | 1.047733965812 | 1.047733966390 ✔\n0.05 | 2 | 3 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 2 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 2 | 5 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 2 | 6 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 2 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 2 | 8 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 2 | 9 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 3 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 3 | 3 | 1.065196198575 | 1.065196199813 ✔\n0.05 | 3 | 4 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 3 | 5 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 3 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 8 | 0.000000000000 | 0.000000000016 ✔\n0.05 | 3 | 9 | 0.000000000000 | 0.000000000015 ✔\n0.05 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 4 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 4 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 4 | 3 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 4 | 4 | 1.078511151058 | 1.078511152326 ✔\n0.05 | 4 | 5 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 4 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 4 | 7 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 4 | 8 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 4 | 9 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 5 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 5 | 2 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 5 | 3 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 5 | 4 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 5 | 5 | 1.089296262568 | 1.089296263862 ✔\n0.05 | 5 | 6 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 5 | 7 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 5 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 5 | 9 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 6 | 2 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 6 | 3 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 6 | 4 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 6 | 5 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 6 | 6 | 1.098373731423 | 1.098373732739 ✔\n0.05 | 6 | 7 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 6 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 6 | 9 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 7 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 7 | 3 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 7 | 4 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 7 | 5 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 7 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 7 | 1.106219258076 | 1.106219258720 ✔\n0.05 | 7 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 9 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 8 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 8 | 1 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 8 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 8 | 3 | 0.000000000000 | 0.000000000016 ✔\n0.05 | 8 | 4 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 8 | 5 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 7 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 8 | 1.113133128439 | 1.113133129790 ✔\n0.05 | 8 | 9 | 0.000000000000 | 0.000000000074 ✔\n0.05 | 9 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 9 | 1 | 0.000000000000 | 0.000000000007 ✔\n0.05 | 9 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 9 | 3 | 0.000000000000 | 0.000000000015 ✔\n0.05 | 9 | 4 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 9 | 5 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 9 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 9 | 7 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 9 | 8 | 0.000000000000 | 0.000000000074 ✔\n0.05 | 9 | 9 | 1.119317201375 | 1.119317202034 ✔\n0.10 | 0 | 0 | 0.951350769867 | 0.951350771636 ✔\n0.10 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 1 | 1 | 1.046485846854 | 1.046485847852 ✔\n0.10 | 1 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 1 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 1 | 9 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 2 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 2 | 1.098810139196 | 1.098810140297 ✔\n0.10 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 2 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 2 | 7 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 8 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 9 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 3 | 3 | 1.135437143836 | 1.135437145012 ✔\n0.10 | 3 | 4 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 3 | 6 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 7 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 8 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 3 | 9 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 4 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 4 | 4 | 1.163823072432 | 1.163823073667 ✔\n0.10 | 4 | 5 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 4 | 6 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 7 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.10 | 4 | 9 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 5 | 4 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 5 | 5 | 1.187099533881 | 1.187099535166 ✔\n0.10 | 5 | 6 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 5 | 7 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 5 | 8 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 5 | 9 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 6 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 6 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 6 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 6 | 4 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 6 | 5 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 6 | 6 | 1.206884526112 | 1.206884527440 ✔\n0.10 | 6 | 7 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 8 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 9 | 0.000000000000 | 0.000000000065 ✔\n0.10 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 7 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 7 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 7 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 7 | 4 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 7 | 5 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 7 | 6 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 7 | 7 | 1.224125733628 | 1.224125734265 ✔\n0.10 | 7 | 8 | 0.000000000000 | 0.000000000066 ✔\n0.10 | 7 | 9 | 0.000000000000 | 0.000000000031 ✔\n0.10 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 8 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 8 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 8 | 3 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.10 | 8 | 5 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 8 | 6 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 8 | 7 | 0.000000000000 | 0.000000000066 ✔\n0.10 | 8 | 8 | 1.239427305298 | 1.239427306699 ✔\n0.10 | 8 | 9 | 0.000000000000 | 0.000000000067 ✔\n0.10 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 9 | 1 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 9 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 9 | 3 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 9 | 4 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 9 | 5 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 9 | 6 | 0.000000000000 | 0.000000000065 ✔\n0.10 | 9 | 7 | 0.000000000000 | 0.000000000031 ✔\n0.10 | 9 | 8 | 0.000000000000 | 0.000000000067 ✔\n0.10 | 9 | 9 | 1.253198719802 | 1.253198721234 ✔\n0.50 | 0 | 0 | 0.886226925453 | 0.886226925863 ✔\n0.50 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 1 | 1.329340388179 | 1.329340389103 ✔\n0.50 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 2 | 1.661675485224 | 1.661675485734 ✔\n0.50 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 3 | 1.938621399428 | 1.938621400123 ✔\n0.50 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 4 | 2.180949074356 | 2.180949075236 ✔\n0.50 | 4 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 9 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 5 | 2.399043981792 | 2.399043982856 ✔\n0.50 | 5 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 5 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 6 | 2.598964313608 | 2.598964314050 ✔\n0.50 | 6 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 6 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 6 | 9 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 7 | 7 | 2.784604621723 | 2.784604622230 ✔\n0.50 | 7 | 8 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 7 | 9 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 8 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 8 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 8 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 8 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 8 | 8 | 2.958642410581 | 2.958642412199 ✔\n0.50 | 8 | 9 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 9 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 9 | 4 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 9 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 9 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 9 | 7 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 8 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 9 | 3.123011433391 | 3.123011435194 ✔\n1.00 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n1.00 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 1 | 2.000000000000 | 2.000000000000 ✔\n1.00 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 2 | 3.000000000000 | 3.000000000000 ✔\n1.00 | 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 3 | 4.000000000000 | 4.000000000000 ✔\n1.00 | 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 4 | 5.000000000000 | 4.999999999999 ✔\n1.00 | 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 5 | 6.000000000000 | 6.000000000000 ✔\n1.00 | 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 6 | 7.000000000000 | 7.000000000000 ✔\n1.00 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 7 | 8.000000000000 | 8.000000000000 ✔\n1.00 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 8 | 9.000000000000 | 9.000000000001 ✔\n1.00 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 9 | 10.000000000000 | 10.000000000001 ✔\nTest Summary: | Pass Total\n∫Lᵢ⁽ᵅ⁾(x)Lⱼ⁽ᵅ⁾(x)xᵅexp(-x)dx = Γ(i+α+1)/i! δᵢⱼ | 500 500","category":"page"},{"location":"MorsePotential/#Normalization-and-Orthogonality-of-\\psi_n(r)","page":"Morse Potential","title":"Normalization & Orthogonality of psi_n(r)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"int_0^infty psi_i^ast(r) psi_j(r) mathrmdr = delta_ij","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000002 ✔\n 0 | 8 | 0.000000000000 | -0.000000000026 ✔\n 0 | 9 | 0.000000000000 | -0.000000000104 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | 0.000000000001 ✔\n 1 | 8 | 0.000000000000 | -0.000000000022 ✔\n 1 | 9 | 0.000000000000 | -0.000000000067 ✔\n 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000009 ✔\n 2 | 9 | 0.000000000000 | -0.000000000030 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0.000000000000 | -0.000000000001 ✔\n 3 | 8 | 0.000000000000 | -0.000000000002 ✔\n 3 | 9 | 0.000000000000 | -0.000000000006 ✔\n 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000001 ✔\n 4 | 9 | 0.000000000000 | 0.000000000001 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000001 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | -0.000000000001 ✔\n 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000002 ✔\n 6 | 9 | 0.000000000000 | -0.000000000003 ✔\n 7 | 0 | 0.000000000000 | 0.000000000002 ✔\n 7 | 1 | 0.000000000000 | 0.000000000001 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | -0.000000000001 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000001 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000004 ✔\n 8 | 0 | 0.000000000000 | -0.000000000026 ✔\n 8 | 1 | 0.000000000000 | -0.000000000022 ✔\n 8 | 2 | 0.000000000000 | -0.000000000009 ✔\n 8 | 3 | 0.000000000000 | -0.000000000002 ✔\n 8 | 4 | 0.000000000000 | -0.000000000001 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000002 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 0.999999999995 ✔\n 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 9 | 0 | 0.000000000000 | -0.000000000104 ✔\n 9 | 1 | 0.000000000000 | -0.000000000067 ✔\n 9 | 2 | 0.000000000000 | -0.000000000030 ✔\n 9 | 3 | 0.000000000000 | -0.000000000006 ✔\n 9 | 4 | 0.000000000000 | 0.000000000001 ✔\n 9 | 5 | 0.000000000000 | -0.000000000001 ✔\n 9 | 6 | 0.000000000000 | -0.000000000003 ✔\n 9 | 7 | 0.000000000000 | 0.000000000004 ✔\n 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000015 ✔\nTest Summary: | Pass Total\n<ψᵢ|ψⱼ> = δᵢⱼ | 100 100","category":"page"},{"location":"MorsePotential/#Eigen-Values-2","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n E_n\n = int psi^ast_n(r) hatH psi_n(r) mathrmdx \n = int psi^ast_n(r) left hatV + hatT right psi(r) mathrmdx \n = int psi^ast_n(r) left V(r) - frachbar^22m fracmathrmd^2mathrmd r^2 right psi(r) mathrmdx \n simeq int psi^ast_n(r) left V(r)psi(r) -frachbar^22m fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2 right mathrmdx\n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n 2psi(r)\n + fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n + Oleft(Delta r^4right)\n =\n psi(r+Delta r)\n + psi(r-Delta r)\n \n fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n =\n psi(r+Delta r)\n - 2psi(r)\n + psi(r-Delta r)\n - Oleft(Delta r^4right)\n \n fracmathrmd^2 psi(r)mathrmd r^2\n =\n fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n - fracOleft(Delta r^4right)Delta r^2\n \n fracmathrmd^2 psi(r)mathrmd r^2\n =\n fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n + Oleft(Delta r^2right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\npsi(r+Delta r)\n= psi(r)\n+ fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n+ frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\n\npsi(r-Delta r)\n= psi(r)\n- fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n- frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | -0.097482629904 | -0.097482629943 ✔\n0.1 | 1 | -0.087576629073 | -0.087576629208 ✔\n0.1 | 2 | -0.078201265005 | -0.078201265359 ✔\n0.1 | 3 | -0.069356537702 | -0.069356538266 ✔\n0.1 | 4 | -0.061042447162 | -0.061042448777 ✔\n0.1 | 5 | -0.053258993386 | -0.053258996131 ✔\n0.1 | 6 | -0.046006176374 | -0.046006177829 ✔\n0.1 | 7 | -0.039283996126 | -0.039283997743 ✔\n0.1 | 8 | -0.033092452642 | -0.033092467851 ✔\n0.1 | 9 | -0.027431545922 | -0.027431467792 ✔\n0.2 | 0 | -0.095387461081 | -0.095387461144 ✔\n0.2 | 1 | -0.081689100176 | -0.081689100427 ✔\n0.2 | 2 | -0.069052012799 | -0.069052013380 ✔\n0.2 | 3 | -0.057476198949 | -0.057476199867 ✔\n0.2 | 4 | -0.046961658628 | -0.046961660317 ✔\n0.2 | 5 | -0.037508391834 | -0.037508393202 ✔\n0.2 | 6 | -0.029116398568 | -0.029116400340 ✔\n0.2 | 7 | -0.021785678830 | -0.021785684062 ✔\n0.2 | 8 | -0.015516232619 | -0.015516237539 ✔\n0.2 | 9 | -0.010308059937 | -0.010308062755 ✔\n0.3 | 0 | -0.093795214605 | -0.093795214695 ✔\n0.3 | 1 | -0.077310338322 | -0.077310338694 ✔\n0.3 | 2 | -0.062417372330 | -0.062417373167 ✔\n0.3 | 3 | -0.049116316630 | -0.049116318029 ✔\n0.3 | 4 | -0.037407171221 | -0.037407173073 ✔\n0.3 | 5 | -0.027289936105 | -0.027289938027 ✔\n0.3 | 6 | -0.018764611280 | -0.018764613693 ✔\n0.3 | 7 | -0.011831196747 | -0.011831198102 ✔\n0.3 | 8 | -0.006489692505 | -0.006489694275 ✔\n0.3 | 9 | -0.002740098556 | -0.002740100893 ✔\n0.1 | 0 | -0.097413777944 | -0.097413777967 ✔\n0.1 | 1 | -0.087380924068 | -0.087380924205 ✔\n0.1 | 2 | -0.077893174789 | -0.077893175145 ✔\n0.1 | 3 | -0.068950530107 | -0.068950530660 ✔\n0.1 | 4 | -0.060552990023 | -0.060552989095 ✔\n0.1 | 5 | -0.052700554537 | -0.052700557255 ✔\n0.1 | 6 | -0.045393223648 | -0.045393222818 ✔\n0.1 | 7 | -0.038630997356 | -0.038631017157 ✔\n0.1 | 8 | -0.032413875662 | -0.032413886246 ✔\n0.1 | 9 | -0.026741858566 | -0.026742018376 ✔\nTest Summary: | Pass Total\n<ψₙ|H|ψₙ> = ∫ψₙ*Hψₙdx = Eₙ | 40 40\n","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"CurrentModule = Antique","category":"page"},{"location":"HydrogenAtom/#Hydrogen-Atom","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The hydrogen atom is the simplest 2-body Coulomb system.","category":"page"},{"location":"HydrogenAtom/#Definitions","page":"Hydrogen Atom","title":"Definitions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Z is the atomic number. The domains of the potential and the wave functions are 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.","category":"page"},{"location":"HydrogenAtom/#Schrödinger-Equation","page":"Hydrogen Atom","title":"Schrödinger Equation","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"HydrogenAtom/#Hamiltonian","page":"Hydrogen Atom","title":"Hamiltonian","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, mu=left(frac1m_mathrme+frac1m_mathrmpright)^-1 is the reduced mass of electron mathrme and proton mathrmp. mu = m_mathrme holds in the limit m_mathrmprightarrowinfty. ","category":"page"},{"location":"HydrogenAtom/#Potential","page":"Hydrogen Atom","title":"Potential","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"V(r; Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\n endaligned","category":"page"},{"location":"HydrogenAtom/#Eigen-Values","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"E(; n=1, Z=Z, Eₕ=Eₕ)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" E_n = -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2 = -fracZ^22n^2 E_mathrmh","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, E_mathrmh is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, E_mathrmh = 27211386245988(53)mathrmeV from here.","category":"page"},{"location":"HydrogenAtom/#Eigen-Functions","page":"Hydrogen Atom","title":"Eigen Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ψ(r, θ, φ; n=1, l=0, m=0, Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" psi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)","category":"page"},{"location":"HydrogenAtom/#Radial-Functions","page":"Hydrogen Atom","title":"Radial Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"R(r; n=1, l=0, Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" R_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_0right)^3 left(frac2Zrn a_0right)^l exp left(-fracZrn a_0right) L_n+l^2l+1 left(frac2Zrn a_0right)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Laguerre polynomials are defined as L_n(x) = frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right), and associated Laguerre polynomials are defined as L_n^k(x) = fracmathrmd^kmathrmdx^k L_n(x). Note that, replace 2n(n+l) with 2n(n+l)^3 if Laguerre polynomials are defined as L_n(x) = mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).","category":"page"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"L(x; n=0, k=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n L_n^k(x)\n = fracmathrmd^kmathrmdx^k L_n(x) \n = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n = (-1)^k L_n-k^(k)(x)\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Examples:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n L_0^0(x) = 1 \n L_1^0(x) = 1 - x \n L_1^1(x) = 1 \n L_2^0(x) = 1 - 2 x + 12 x^2 \n L_2^1(x) = 2 - x \n L_2^2(x) = 1 \n L_3^0(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^1(x) = 3 - 3 x + 12 x^2 \n L_3^2(x) = 3 - x \n L_3^3(x) = 1 \n L_4^0(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 512 x^4 \n L_4^1(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_4^2(x) = 6 - 4 x + 12 x^2 \n L_4^3(x) = 4 - x \n L_4^4(x) = 1 \n vdots\n endaligned","category":"page"},{"location":"HydrogenAtom/#Spherical-Harmonics","page":"Hydrogen Atom","title":"Spherical Harmonics","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Y(θ, φ; l=0, m=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" Y_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Note that some variants are connected by ","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"i^m+m sqrtfrac(l-m)(l+m) P_l^m = (-1)^fracm+m2 sqrtfrac(l-m)(l+m) P_l^m = (-1)^m sqrtfrac(l-m)(l+m) P_l^m","category":"page"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials","page":"Hydrogen Atom","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"P(x; n=0, m=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n P_n^m(x)\n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Legendre polynomials are defined as P_n(x) = frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right. Note that P_l^-m = (-1)^m frac(l-m)(l+m) P_l^m for m0. (It is not compatible with P_k^m(t) = (-1)^mleft( 1-t^2 right)^m2 fracmathrmd^m P_k(t)mathrmdt^m caused by (-1)^m.) The specific formulae are given below.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Examples:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n P_0^0(x) = 1 \n P_1^0(x) = x \n P_1^1(x) = left(+1right)sqrt1-x^2 \n P_2^0(x) = -12 + 32 x^2 \n P_2^1(x) = left(-3 xright)sqrt1-x^2 \n P_2^2(x) = 3 - 6 x \n P_3^0(x) = -32 x + 52 x^3 \n P_3^1(x) = left(32 - 152 x^2right)sqrt1-x^2 \n P_3^2(x) = 15 x - 30 x^2 \n P_3^3(x) = left(15 - 30 xright)sqrt1-x^2 \n P_4^0(x) = 38 - 154 x^2 + 358 x^4 \n P_4^1(x) = left(- 152 x + 352 x^3right)sqrt1-x^2 \n P_4^2(x) = -152 + 15 x + 1052 x^2 - 105 x^3 \n P_4^3(x) = left(105 x - 210 x^2right)sqrt1-x^2 \n P_4^4(x) = 105 - 420 x + 420 x^2 \n vdots\n endaligned","category":"page"},{"location":"HydrogenAtom/#References","page":"Hydrogen Atom","title":"References","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"cpprefjp, legendre, assoc_legendre, laguerre, assoc_laguerre\nThe Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.3 Table1, 18.5 Table1, 18.5.17, 18.3 Table1, 18.5 Table1, 18.5.12\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965), p.598 (c.1), p.598 (c.4), p.603 (d.13), p.603 (d.13)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968), p.79 (14.12), p.93 (16.19)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999), p.493 (B.72), p.494 Table, p.493 (B.72), p.483 (B.12), p.483 (B.12)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994), p.83 (4), p.83 (5), p.149 (21)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995), p.126 (4.28), p.96 Table3.1, p.126 (4.27), p.139 (4.88), p.140 Table4.4, p.139 (4.87), p.140 Table4.5\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997), p.195 Table6.1, p.196 (6.26), p.196 Table6.2, p.207 Table6.4\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008), p.234\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021), p.245 Problem 3.30.b, ","category":"page"},{"location":"HydrogenAtom/#Usage-and-Examples","page":"Hydrogen Atom","title":"Usage & Examples","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Install Antique.jl for the first run and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Antique\nH = antique(:HydrogenAtom, Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Parameters:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.Z\n1\n\njulia> H.Eₕ\n1.0\n\njulia> H.mₑ\n1.0\n\njulia> H.a₀\n1.0\n\njulia> H.ℏ\n1.0","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eigen values:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.E(n=1)\n-0.5\n\njulia> H.E(n=2)\n-0.125","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Wave length (n=2rightarrow1, the first line of the Lyman series):","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eₕ2nm⁻¹ = 2.1947463136320e-2 # https://physics.nist.gov/cgi-bin/cuu/CCValue?hrminv\nprintln(\"ΔE = \", H.E(n=2) - H.E(n=1), \" Eₕ\")\nprintln(\"λ = \", ((H.E(n=2)-H.E(n=1))*Eₕ2nm⁻¹)^-1, \" nm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ΔE = 0.375 Eₕ\nλ = 121.50227341098497 nm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Hyperfine Splitting:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"# constants: https://doi.org/10.1103/RevModPhys.93.025010\ne = 1.602176634e-19 # C https://physics.nist.gov/cgi-bin/cuu/Value?e\nh = 6.62607015e-34 # J Hz-1 https://physics.nist.gov/cgi-bin/cuu/Value?h\nc = 299792458 # m s-1 https://physics.nist.gov/cgi-bin/cuu/Value?c\na0 = 5.29177210903e-11 # m https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0\nµ0 = 1.25663706212e-6 # N A-2 https://physics.nist.gov/cgi-bin/cuu/Value?mu0\nµB = 9.2740100783e-24 # J T-1 https://physics.nist.gov/cgi-bin/cuu/Value?mub\nµN = 5.0507837461e-27 # J T-1 https://physics.nist.gov/cgi-bin/cuu/Value?mun\nge = 2.00231930436256 # https://physics.nist.gov/cgi-bin/cuu/Value?gem\ngp = 5.5856946893 # https://physics.nist.gov/cgi-bin/cuu/Value?gp\n\n# calculation: https://doi.org/10.1119/1.12733\nδ = abs(H.ψ(0,0,0))^2\nΔE = 2 / 3 * µ0 * µN * µB * gp * ge * δ * a0^(-3)\nprintln(\"1/π = \", 1/π)\nprintln(\"<δ(r)> = \", δ, \" a₀⁻³\")\nprintln(\"<δ(r)> = \", δ * a0^(-3), \" m⁻³\")\nprintln(\"ΔE = \", ΔE, \" J\")\nprintln(\"ν = ΔE/h = \", ΔE / h * 1e-6, \" MHz\")\nprintln(\"λ = hc/ΔE = \", h*c/ΔE*100, \" cm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"1/π = 0.3183098861837907\n<δ(r)> = 0.3183098861837908 a₀⁻³\n<δ(r)> = 2.1480615849063944e30 m⁻³\nΔE = 9.427622831641132e-25 J\nν = ΔE/h = 1422.8075794882932 MHz\nλ = hc/ΔE = 21.070485027063118 cm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h},~E_n/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nplot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve, Energy levels:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nfor n in 0:10\n plot!(0.0:0.01:15, r -> H.E(n=n) > H.V(r) ? H.E(n=n) : NaN, lc=n, lw=1, label=\"\") # energy level\nend\nplot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Radial functions:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\\$\", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=300)\nfor n in 1:3\n for l in 0:n-1\n plot!(0:0.01:20, r->r^2*H.R(r,n=n,l=l)^2, lc=n, lw=2, ls=[:solid,:dash,:dot,:dashdot,:dashdotdot][l+1], label=\"\\$n = $n, l=$l\\$\")\n end\nend\nplot!()","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/#Testing","page":"Hydrogen Atom","title":"Testing","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials-P_nm(x)","page":"Hydrogen Atom","title":"Associated Legendre Polynomials P_n^m(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n P_n^m(x)\n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=0 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_0^0(x)\n = 1\n = 1 \n = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_1^0(x)\n = frac12 fracmathrmdmathrmdx left( -1 + x^2 right)\n = x \n = x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_1^1(x)\n = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac12 fracmathrmdmathrmdx left( -1 + x^2 right)\n = left( 1 - x^2 right)^frac12 \n = left( 1 - x^2 right)^frac12\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_2^0(x)\n = frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n = frac-12 + frac32 x^2 \n = frac-12 + frac32 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_2^1(x)\n = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n = 3 left( 1 - x^2 right)^frac12 x \n = 3 left( 1 - x^2 right)^frac12 x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_2^2(x)\n = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n = 3 - 3 x^2 \n = 3 - 3 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_3^0(x)\n = frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n = - frac32 x + frac52 x^3 \n = - frac32 x + frac52 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_3^1(x)\n = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n = - frac32 left( 1 - x^2 right)^frac12 + frac152 left( 1 - x^2 right)^frac12 x^2 \n = - frac32 left( 1 - x^2 right)^frac12 + frac152 left( 1 - x^2 right)^frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_3^2(x)\n = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n = 15 x - 15 x^3 \n = 15 x - 15 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_3^3(x)\n = left( 1 - x^2 right)^frac32 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n = 15 left( 1 - x^2 right)^frac32 \n = 15 left( 1 - x^2 right)^frac32\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^0(x)\n = frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = frac38 - frac154 x^2 + frac358 x^4 \n = frac38 - frac154 x^2 + frac358 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^1(x)\n = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = - frac152 left( 1 - x^2 right)^frac12 x + frac352 left( 1 - x^2 right)^frac12 x^3 \n = - frac152 left( 1 - x^2 right)^frac12 x + frac352 left( 1 - x^2 right)^frac12 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^2(x)\n = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = frac-152 + 60 x^2 - frac1052 x^4 \n = frac-152 + 60 x^2 - frac1052 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^3(x)\n = left( 1 - x^2 right)^frac32 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = 105 left( 1 - x^2 right)^frac32 x \n = 105 left( 1 - x^2 right)^frac32 x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=4 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^4(x)\n = left( 1 - x^2 right)^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = 105 left( 1 - x^2 right)^2 \n = 105 left( 1 - x^2 right)^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Test Summary: | Pass Total\nPₙᵐ(x) = √(1-x²)ᵐ dᵐ/dxᵐ Pₙ(x); Pₙ(x) = 1/(2ⁿn!) dⁿ/dxⁿ (x²-1)ⁿ | 15 15","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-P_nm(x)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of P_n^m(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_-1^1 P_i^m(x) P_j^m(x) mathrmdx = frac2(j+m)(2j+1)(j-m) delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" m | i | j | analytical | numerical \n-- | -- | -- | ----------------- | ----------------- \n 0 | 0 | 0 | 2.000000000000 | 2.000000000000 ✔\n 0 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 1 | 0.666666666667 | 0.666666666667 ✔\n 0 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 2 | 0.400000000000 | 0.400000000000 ✔\n 0 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 3 | 0.285714285714 | 0.285714285714 ✔\n 0 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 4 | 0.222222222222 | 0.222222222222 ✔\n 0 | 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 0 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n 0 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 5 | 0.181818181818 | 0.181818181818 ✔\n 0 | 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 6 | 0.153846153846 | 0.153846153846 ✔\n 0 | 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 7 | 0.133333333333 | 0.133333333333 ✔\n 0 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 8 | 0.117647058824 | 0.117647058824 ✔\n 0 | 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 0 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 9 | 9 | 0.105263157895 | 0.105263157895 ✔\n 1 | 1 | 1 | 1.333333333333 | 1.333333333333 ✔\n 1 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 2 | 2.400000000000 | 2.400000000000 ✔\n 1 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 3 | 3.428571428571 | 3.428571428571 ✔\n 1 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 4 | 4.444444444444 | 4.444444444444 ✔\n 1 | 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 5 | 5.454545454545 | 5.454545454545 ✔\n 1 | 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 6 | 6.461538461538 | 6.461538461538 ✔\n 1 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 7 | 7.466666666667 | 7.466666666667 ✔\n 1 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 8 | 8.470588235294 | 8.470588235294 ✔\n 1 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 9 | 9.473684210526 | 9.473684210526 ✔\n 2 | 2 | 2 | 9.600000000000 | 9.600000000000 ✔\n 2 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 3 | 34.285714285714 | 34.285714285714 ✔\n 2 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 4 | 80.000000000000 | 80.000000000000 ✔\n 2 | 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 5 | 152.727272727273 | 152.727272727273 ✔\n 2 | 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 6 | 258.461538461538 | 258.461538461538 ✔\n 2 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 7 | 403.200000000000 | 403.200000000000 ✔\n 2 | 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 8 | 8 | 592.941176470588 | 592.941176470588 ✔\n 2 | 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 9 | 833.684210526316 | 833.684210526316 ✔\n 3 | 3 | 3 | 205.714285714286 | 205.714285714286 ✔\n 3 | 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 9 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 3 | 4 | 4 | 1120.000000000000 | 1120.000000000000 ✔\n 3 | 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 3 | 5 | 5 | 3665.454545454545 | 3665.454545454545 ✔\n 3 | 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 6 | 9304.615384615385 | 9304.615384615387 ✔\n 3 | 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 9 | 0.000000000000 | -0.000000000002 ✔\n 3 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 7 | 20160.000000000000 | 20160.000000000004 ✔\n 3 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 9 | 0.000000000000 | -0.000000000003 ✔\n 3 | 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 8 | 39134.117647058825 | 39134.117647058825 ✔\n 3 | 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 6 | 0.000000000000 | -0.000000000002 ✔\n 3 | 9 | 7 | 0.000000000000 | -0.000000000003 ✔\n 3 | 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 9 | 70029.473684210534 | 70029.473684210505 ✔\n 4 | 4 | 4 | 8960.000000000000 | 8960.000000000002 ✔\n 4 | 4 | 5 | 0.000000000000 | -0.000000000002 ✔\n 4 | 4 | 6 | 0.000000000000 | -0.000000000001 ✔\n 4 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 8 | 0.000000000000 | 0.000000000007 ✔\n 4 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 4 | 0.000000000000 | -0.000000000002 ✔\n 4 | 5 | 5 | 65978.181818181823 | 65978.181818181838 ✔\n 4 | 5 | 6 | 0.000000000000 | -0.000000000001 ✔\n 4 | 5 | 7 | 0.000000000000 | -0.000000000058 ✔\n 4 | 5 | 8 | 0.000000000000 | -0.000000000002 ✔\n 4 | 5 | 9 | 0.000000000000 | -0.000000000007 ✔\n 4 | 6 | 4 | 0.000000000000 | -0.000000000001 ✔\n 4 | 6 | 5 | 0.000000000000 | -0.000000000001 ✔\n 4 | 6 | 6 | 279138.461538461561 | 279138.461538461503 ✔\n 4 | 6 | 7 | 0.000000000000 | -0.000000000018 ✔\n 4 | 6 | 8 | 0.000000000000 | 0.000000000055 ✔\n 4 | 6 | 9 | 0.000000000000 | 0.000000000029 ✔\n 4 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 5 | 0.000000000000 | -0.000000000058 ✔\n 4 | 7 | 6 | 0.000000000000 | -0.000000000018 ✔\n 4 | 7 | 7 | 887040.000000000000 | 887040.000000000000 ✔\n 4 | 7 | 8 | 0.000000000000 | 0.000000000031 ✔\n 4 | 7 | 9 | 0.000000000000 | 0.000000000104 ✔\n 4 | 8 | 4 | 0.000000000000 | 0.000000000007 ✔\n 4 | 8 | 5 | 0.000000000000 | -0.000000000002 ✔\n 4 | 8 | 6 | 0.000000000000 | 0.000000000055 ✔\n 4 | 8 | 7 | 0.000000000000 | 0.000000000031 ✔\n 4 | 8 | 8 | 2348047.058823529165 | 2348047.058823529631 ✔\n 4 | 8 | 9 | 0.000000000000 | -0.000000000015 ✔\n 4 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 5 | 0.000000000000 | -0.000000000007 ✔\n 4 | 9 | 6 | 0.000000000000 | 0.000000000029 ✔\n 4 | 9 | 7 | 0.000000000000 | 0.000000000104 ✔\n 4 | 9 | 8 | 0.000000000000 | -0.000000000015 ✔\n 4 | 9 | 9 | 5462298.947368421592 | 5462298.947368418798 ✔\n 5 | 5 | 5 | 659781.818181818235 | 659781.818181818351 ✔\n 5 | 5 | 6 | 0.000000000000 | -0.000000000002 ✔\n 5 | 5 | 7 | 0.000000000000 | 0.000000000233 ✔\n 5 | 5 | 8 | 0.000000000000 | 0.000000000567 ✔\n 5 | 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 5 | 0.000000000000 | -0.000000000002 ✔\n 5 | 6 | 6 | 6141046.153846153989 | 6141046.153846156783 ✔\n 5 | 6 | 7 | 0.000000000000 | 0.000000000250 ✔\n 5 | 6 | 8 | 0.000000000000 | 0.000000001630 ✔\n 5 | 6 | 9 | 0.000000000000 | 0.000000000931 ✔\n 5 | 7 | 5 | 0.000000000000 | 0.000000000233 ✔\n 5 | 7 | 6 | 0.000000000000 | 0.000000000250 ✔\n 5 | 7 | 7 | 31933440.000000000000 | 31933440.000000000000 ✔\n 5 | 7 | 8 | 0.000000000000 | 0.000000002503 ✔\n 5 | 7 | 9 | 0.000000000000 | 0.000000003725 ✔\n 5 | 8 | 5 | 0.000000000000 | 0.000000000567 ✔\n 5 | 8 | 6 | 0.000000000000 | 0.000000001630 ✔\n 5 | 8 | 7 | 0.000000000000 | 0.000000002503 ✔\n 5 | 8 | 8 | 122098447.058823525906 | 122098447.058823525906 ✔\n 5 | 8 | 9 | 0.000000000000 | -0.000000001397 ✔\n 5 | 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 6 | 0.000000000000 | 0.000000000931 ✔\n 5 | 9 | 7 | 0.000000000000 | 0.000000003725 ✔\n 5 | 9 | 8 | 0.000000000000 | -0.000000001397 ✔\n 5 | 9 | 9 | 382360926.315789461136 | 382360926.315789461136 ✔\nTest Summary: | Pass Total\n∫Pᵢᵐ(x)Pⱼᵐ(x)dx = 2(j+m)!/(2j+1)(j-m)! δᵢⱼ | 355 355","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-Y_{lm}(\\theta,\\varphi)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of Y_lm(thetavarphi)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_0^2pi\nint_0^pi\nY_lm(thetavarphi)^* Y_lm(thetavarphi) sin(theta)\nmathrmdtheta mathrmdvarphi\n= delta_ll delta_mm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"l₁ | l₂ | m₁ | m₂ | analytical | numerical \n-- | -- | -- | -- | ----------------- | ----------------- \n 0 | 0 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 0 | 2 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 0 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | -1 | -1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 1 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 1 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 1 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 1 | 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | -1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | -1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 0 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 2 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | -2 | -2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | -2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | -2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | -2 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | -1 | -1 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | -1 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 0 | -2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | -2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 2 | -2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 2 | 2 | 1.000000000000 | 1.000000000000 ✔\nTest Summary: | Pass Total\n∫Yₗ₁ₘ₁(θ,φ)Yₗ₂ₘ₂(θ,φ)sinθdθdφ = δₗ₁ₗ₂δₘ₁ₘ₂ | 81 81","category":"page"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials-L_n{k}(x)","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials L_n^k(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n L_n^k(x)\n = fracmathrmd^kmathrmdx^k L_n(x) \n = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n = (-1)^k L_n-k^(k)(x)\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=0 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_0^0(x)\n = e^x e^ - x\n = 1 \n = 1 \n = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_1^0(x)\n = e^x fracmathrmdmathrmdx x e^ - x\n = 1 - x \n = 1 - x \n = 1 - x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_1^1(x)\n = fracmathrmdmathrmdx e^x fracmathrmdmathrmdx x e^ - x\n = -1 \n = -1 \n = -1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_2^0(x)\n = frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n = 1 - 2 x + frac12 x^2 \n = 1 - 2 x + frac12 x^2 \n = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_2^1(x)\n = fracmathrmdmathrmdx frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n = -2 + x \n = -2 + x \n = -2 + x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_2^2(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n = 1 \n = 1 \n = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_3^0(x)\n = frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = 1 - frac16 x^3 - 3 x + frac32 x^2 \n = 1 - frac16 x^3 - 3 x + frac32 x^2 \n = 1 - frac16 x^3 - 3 x + frac32 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_3^1(x)\n = fracmathrmdmathrmdx frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = -3 + 3 x - frac12 x^2 \n = -3 + 3 x - frac12 x^2 \n = -3 + 3 x - frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_3^2(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = 3 - x \n = 3 - x \n = 3 - x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_3^3(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = -1 \n = -1 \n = -1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^0(x)\n = frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4 \n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4 \n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^1(x)\n = fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = -4 + frac16 x^3 + 6 x - 2 x^2 \n = -4 + frac16 x^3 + 6 x - 2 x^2 \n = -4 + frac16 x^3 + 6 x - 2 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^2(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = 6 - 4 x + frac12 x^2 \n = 6 - 4 x + frac12 x^2 \n = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^3(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = -4 + x \n = -4 + x \n = -4 + x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=4 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^4(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = 1 \n = 1 \n = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Test Summary: | Pass Total\nLₙᵏ(x) = dᵏ/dxᵏ Lₙ(x); Lₙ(x) = 1/(n!) eˣ dⁿ/dxⁿ e⁻ˣ xⁿ | 15 15","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-L_n{k}(x)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of L_n^k(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_0^infty mathrme^-x x^k L_i^k(x) L_j^k(x) mathrmdx = fraci(i-k) delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Replace n+k with n for the definition of Wolfram MathWorld.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" i | j | k | analytical | numerical \n-- | -- | -- | ----------------- | ----------------- \n 0 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 0 | 1.000000000000 | 1.000000000000 ✔\n 1 | 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | 1 | 2.000000000000 | 2.000000000000 ✔\n 2 | 2 | 2 | 2.000000000000 | 2.000000000000 ✔\n 2 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 0 | 1.000000000000 | 1.000000000000 ✔\n 3 | 3 | 1 | 3.000000000000 | 3.000000000000 ✔\n 3 | 3 | 2 | 6.000000000000 | 6.000000000000 ✔\n 3 | 3 | 3 | 6.000000000000 | 6.000000000000 ✔\n 3 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 0 | 1.000000000000 | 1.000000000000 ✔\n 4 | 4 | 1 | 4.000000000000 | 4.000000000000 ✔\n 4 | 4 | 2 | 12.000000000000 | 12.000000000000 ✔\n 4 | 4 | 3 | 24.000000000000 | 24.000000000000 ✔\n 4 | 4 | 4 | 24.000000000000 | 24.000000000000 ✔\n 4 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 5 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 0 | 1.000000000000 | 1.000000000000 ✔\n 5 | 5 | 1 | 5.000000000000 | 4.999999999999 ✔\n 5 | 5 | 2 | 20.000000000000 | 20.000000000000 ✔\n 5 | 5 | 3 | 60.000000000000 | 60.000000000000 ✔\n 5 | 5 | 4 | 120.000000000000 | 120.000000000000 ✔\n 5 | 5 | 5 | 120.000000000000 | 120.000000000000 ✔\n 5 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 6 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 2 | 0.000000000000 | -0.000000000000 ✔\n 6 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 0 | 1.000000000000 | 1.000000000000 ✔\n 6 | 6 | 1 | 6.000000000000 | 6.000000000000 ✔\n 6 | 6 | 2 | 30.000000000000 | 30.000000000000 ✔\n 6 | 6 | 3 | 120.000000000000 | 119.999999999978 ✔\n 6 | 6 | 4 | 360.000000000000 | 359.999999999996 ✔\n 6 | 6 | 5 | 720.000000000000 | 720.000000000000 ✔\n 6 | 6 | 6 | 720.000000000000 | 720.000000000000 ✔\n 6 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 2 | 0.000000000000 | -0.000000000000 ✔\n 6 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n 7 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 7 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n 7 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 7 | 3 | 3 | 0.000000000000 | -0.000000000000 ✔\n 7 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 6 | 0.000000000000 | 0.000000000000 ✔\n 7 | 7 | 0 | 1.000000000000 | 1.000000000000 ✔\n 7 | 7 | 1 | 7.000000000000 | 7.000000000000 ✔\n 7 | 7 | 2 | 42.000000000000 | 42.000000000000 ✔\n 7 | 7 | 3 | 210.000000000000 | 210.000000000000 ✔\n 7 | 7 | 4 | 840.000000000000 | 840.000000000000 ✔\n 7 | 7 | 5 | 2520.000000000000 | 2519.999999999775 ✔\n 7 | 7 | 6 | 5040.000000000000 | 5039.999999999985 ✔\n 7 | 7 | 7 | 5040.000000000000 | 5040.000000000000 ✔\nTest Summary: | Pass Total\n∫exp(-x)xᵏLᵢᵏ(x)Lⱼᵏ(x)dx = (2i+k)!/(i+k)! δᵢⱼ | 204 204","category":"page"},{"location":"HydrogenAtom/#Normalization-of-R_{nl}(r)","page":"Hydrogen Atom","title":"Normalization of R_nl(r)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int R_nl(r)^2 r^2 mathrmdr = 1","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" n | l | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 0 | 1.000000000000 | 1.000000000000 ✔\n 2 | 0 | 1.000000000000 | 1.000000000000 ✔\n 2 | 1 | 1.000000000000 | 1.000000000000 ✔\n 3 | 0 | 1.000000000000 | 1.000000000000 ✔\n 3 | 1 | 1.000000000000 | 0.999999999999 ✔\n 3 | 2 | 1.000000000000 | 1.000000000000 ✔\n 4 | 0 | 1.000000000000 | 1.000000000000 ✔\n 4 | 1 | 1.000000000000 | 1.000000000000 ✔\n 4 | 2 | 1.000000000000 | 1.000000000000 ✔\n 4 | 3 | 1.000000000000 | 1.000000000000 ✔\n 5 | 0 | 1.000000000000 | 1.000000000000 ✔\n 5 | 1 | 1.000000000000 | 1.000000000000 ✔\n 5 | 2 | 1.000000000000 | 1.000000000000 ✔\n 5 | 3 | 1.000000000000 | 1.000000000000 ✔\n 5 | 4 | 1.000000000000 | 1.000000000000 ✔\n 6 | 0 | 1.000000000000 | 1.000000000000 ✔\n 6 | 1 | 1.000000000000 | 1.000000000000 ✔\n 6 | 2 | 1.000000000000 | 1.000000000000 ✔\n 6 | 3 | 1.000000000000 | 1.000000000000 ✔\n 6 | 4 | 1.000000000000 | 1.000000000000 ✔\n 6 | 5 | 1.000000000000 | 1.000000000000 ✔\n 7 | 0 | 1.000000000000 | 1.000000000000 ✔\n 7 | 1 | 1.000000000000 | 1.000000000000 ✔\n 7 | 2 | 1.000000000000 | 1.000000000000 ✔\n 7 | 3 | 1.000000000000 | 1.000000000000 ✔\n 7 | 4 | 1.000000000000 | 1.000000000000 ✔\n 7 | 5 | 1.000000000000 | 1.000000000000 ✔\n 7 | 6 | 1.000000000000 | 1.000000000000 ✔\n 8 | 0 | 1.000000000000 | 1.000000000000 ✔\n 8 | 1 | 1.000000000000 | 1.000000000000 ✔\n 8 | 2 | 1.000000000000 | 1.000000000000 ✔\n 8 | 3 | 1.000000000000 | 1.000000000000 ✔\n 8 | 4 | 1.000000000000 | 1.000000000000 ✔\n 8 | 5 | 1.000000000000 | 1.000000000000 ✔\n 8 | 6 | 1.000000000000 | 1.000000000000 ✔\n 8 | 7 | 1.000000000000 | 1.000000000000 ✔\n 9 | 0 | 1.000000000000 | 1.000000000000 ✔\n 9 | 1 | 1.000000000000 | 1.000000000000 ✔\n 9 | 2 | 1.000000000000 | 1.000000000000 ✔\n 9 | 3 | 1.000000000000 | 1.000000000000 ✔\n 9 | 4 | 1.000000000000 | 1.000000000000 ✔\n 9 | 5 | 1.000000000000 | 1.000000000000 ✔\n 9 | 6 | 1.000000000000 | 1.000000000000 ✔\n 9 | 7 | 1.000000000000 | 1.000000000000 ✔\n 9 | 8 | 1.000000000000 | 1.000000000000 ✔\nTest Summary: | Pass Total\n∫|Rₙₗ(r)|²r²dr = δₙ₁ₙ₂δₗ₁ₗ₂ | 45 45","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-r","page":"Hydrogen Atom","title":"Expected Value of r","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle r rangle\n= int r R_n_1 l_1(r)^2 r^2 mathrmdr\n= fraca_mu2Z left 3n^2 - l(l+1) right \na_mu = a_0 fracm_mathrmemu \nfrac1mu = frac1m_mathrme + frac1m_mathrmp","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22\n Quan­tum Me­chan­ics for En­gi­neers by Leon van Dom­me­len","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" n | l | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 0 | 1.500000000000 | 1.500000000000 ✔\n 2 | 0 | 6.000000000000 | 6.000000000000 ✔\n 2 | 1 | 5.000000000000 | 5.000000000000 ✔\n 3 | 0 | 13.500000000000 | 13.500000000000 ✔\n 3 | 1 | 12.500000000000 | 12.500000000000 ✔\n 3 | 2 | 10.500000000000 | 10.500000000000 ✔\n 4 | 0 | 24.000000000000 | 23.999999999999 ✔\n 4 | 1 | 23.000000000000 | 22.999999999999 ✔\n 4 | 2 | 21.000000000000 | 21.000000000000 ✔\n 4 | 3 | 18.000000000000 | 18.000000000000 ✔\n 5 | 0 | 37.500000000000 | 37.500000000000 ✔\n 5 | 1 | 36.500000000000 | 36.500000000000 ✔\n 5 | 2 | 34.500000000000 | 34.500000000000 ✔\n 5 | 3 | 31.500000000000 | 31.500000000000 ✔\n 5 | 4 | 27.500000000000 | 27.499999999943 ✔\n 6 | 0 | 54.000000000000 | 54.000000000001 ✔\n 6 | 1 | 53.000000000000 | 53.000000000001 ✔\n 6 | 2 | 51.000000000000 | 51.000000000000 ✔\n 6 | 3 | 48.000000000000 | 48.000000000000 ✔\n 6 | 4 | 44.000000000000 | 44.000000000000 ✔\n 6 | 5 | 39.000000000000 | 39.000000000000 ✔\n 7 | 0 | 73.500000000000 | 73.500000000000 ✔\n 7 | 1 | 72.500000000000 | 72.500000000000 ✔\n 7 | 2 | 70.500000000000 | 70.500000000000 ✔\n 7 | 3 | 67.500000000000 | 67.500000000000 ✔\n 7 | 4 | 63.500000000000 | 63.500000000000 ✔\n 7 | 5 | 58.500000000000 | 58.500000000000 ✔\n 7 | 6 | 52.500000000000 | 52.499999999992 ✔\n 8 | 0 | 96.000000000000 | 96.000000000001 ✔\n 8 | 1 | 95.000000000000 | 94.999999999999 ✔\n 8 | 2 | 93.000000000000 | 93.000000000000 ✔\n 8 | 3 | 90.000000000000 | 90.000000000000 ✔\n 8 | 4 | 86.000000000000 | 86.000000000000 ✔\n 8 | 5 | 81.000000000000 | 81.000000000000 ✔\n 8 | 6 | 75.000000000000 | 75.000000000000 ✔\n 8 | 7 | 68.000000000000 | 68.000000000000 ✔\n 9 | 0 | 121.500000000000 | 121.500000000001 ✔\n 9 | 1 | 120.500000000000 | 120.500000000000 ✔\n 9 | 2 | 118.500000000000 | 118.500000000001 ✔\n 9 | 3 | 115.500000000000 | 115.500000000000 ✔\n 9 | 4 | 111.500000000000 | 111.499999999998 ✔\n 9 | 5 | 106.500000000000 | 106.499999999999 ✔\n 9 | 6 | 100.500000000000 | 100.500000000000 ✔\n 9 | 7 | 93.500000000000 | 93.500000000000 ✔\n 9 | 8 | 85.500000000000 | 85.500000000000 ✔\nTest Summary: | Pass Total\n∫r|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)/2Z × [3n²-l(l+1)]; 1/μ = 1/mₑ + 1/mₚ | 45 45","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-r2","page":"Hydrogen Atom","title":"Expected Value of r^2","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle r^2 rangle\n= int r^2 R_n_1 l_1(r)^2 r^2 mathrmdr\n= fraca_mu^22Z^2 n^2 left 5n^2 + 1 - 3l(l+1) right \na_mu = a_0 fracm_mathrmemu \nfrac1mu = frac1m_mathrme + frac1m_mathrmp","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22\n Quan­tum Me­chan­ics for En­gi­neers by Leon van Dom­me­len","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" n | l | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 0 | 3.000000000000 | 3.000000000000 ✔\n 2 | 0 | 42.000000000000 | 42.000000000000 ✔\n 2 | 1 | 30.000000000000 | 30.000000000000 ✔\n 3 | 0 | 207.000000000000 | 207.000000000000 ✔\n 3 | 1 | 180.000000000000 | 180.000000000000 ✔\n 3 | 2 | 126.000000000000 | 126.000000000000 ✔\n 4 | 0 | 648.000000000000 | 647.999999999903 ✔\n 4 | 1 | 600.000000000000 | 599.999999999936 ✔\n 4 | 2 | 504.000000000000 | 503.999999999975 ✔\n 4 | 3 | 360.000000000000 | 359.999999999996 ✔\n 5 | 0 | 1575.000000000000 | 1574.999999999999 ✔\n 5 | 1 | 1500.000000000000 | 1499.999999999998 ✔\n 5 | 2 | 1350.000000000000 | 1350.000000000000 ✔\n 5 | 3 | 1125.000000000000 | 1125.000000000003 ✔\n 5 | 4 | 825.000000000000 | 825.000000000000 ✔\n 6 | 0 | 3258.000000000000 | 3257.999999999997 ✔\n 6 | 1 | 3150.000000000000 | 3149.999999999992 ✔\n 6 | 2 | 2934.000000000000 | 2933.999999999998 ✔\n 6 | 3 | 2610.000000000000 | 2610.000000000033 ✔\n 6 | 4 | 2178.000000000000 | 2178.000000000008 ✔\n 6 | 5 | 1638.000000000000 | 1638.000000000000 ✔\n 7 | 0 | 6027.000000000000 | 6026.999999999992 ✔\n 7 | 1 | 5880.000000000000 | 5880.000000000003 ✔\n 7 | 2 | 5586.000000000000 | 5585.999999999990 ✔\n 7 | 3 | 5145.000000000000 | 5144.999999999992 ✔\n 7 | 4 | 4557.000000000000 | 4556.999999999997 ✔\n 7 | 5 | 3822.000000000000 | 3821.999999999999 ✔\n 7 | 6 | 2940.000000000000 | 2940.000000000001 ✔\n 8 | 0 | 10272.000000000000 | 10272.000000000029 ✔\n 8 | 1 | 10080.000000000000 | 10079.999999999995 ✔\n 8 | 2 | 9696.000000000000 | 9695.999999999993 ✔\n 8 | 3 | 9120.000000000000 | 9120.000000000011 ✔\n 8 | 4 | 8352.000000000000 | 8352.000000000002 ✔\n 8 | 5 | 7392.000000000000 | 7392.000000000010 ✔\n 8 | 6 | 6240.000000000000 | 6240.000000000000 ✔\n 8 | 7 | 4896.000000000000 | 4896.000000000008 ✔\n 9 | 0 | 16443.000000000000 | 16443.000000000102 ✔\n 9 | 1 | 16200.000000000000 | 16200.000000000040 ✔\n 9 | 2 | 15714.000000000000 | 15714.000000000149 ✔\n 9 | 3 | 14985.000000000000 | 14984.999999999918 ✔\n 9 | 4 | 14013.000000000000 | 14012.999999999545 ✔\n 9 | 5 | 12798.000000000000 | 12797.999999999807 ✔\n 9 | 6 | 11340.000000000000 | 11339.999999999945 ✔\n 9 | 7 | 9639.000000000000 | 9638.999999999991 ✔\n 9 | 8 | 7695.000000000000 | 7694.999999999998 ✔\nTest Summary: | Pass Total\n∫r²|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)²/2Z² × n²[5n²+1-3l(l+1)]; 1/μ = 1/mₑ + 1/mₚ | 45 45","category":"page"},{"location":"HydrogenAtom/#Virial-Theorem","page":"Hydrogen Atom","title":"Virial Theorem","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The virial theorem 2langle T rangle + langle V rangle = 0 and the definition of Hamiltonian langle H rangle = langle T rangle + langle V rangle derive langle H rangle = frac12 langle V rangle and langle H rangle = -langle T rangle.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"frac12 int psi_n^ast(x) V(x) psi_n(x) mathrmdx = E_n","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" n | analytical | numerical \n-- | ----------------- | ----------------- \n 1 | -1.000000000000 | -1.000000000000 ✔\n 2 | -0.250000000000 | -0.250000000000 ✔\n 3 | -0.111111111111 | -0.111111111111 ✔\n 4 | -0.062500000000 | -0.062500000000 ✔\n 5 | -0.040000000000 | -0.040000000000 ✔\n 6 | -0.027777777778 | -0.027777777778 ✔\n 7 | -0.020408163265 | -0.020408163265 ✔\n 8 | -0.015625000000 | -0.015625000000 ✔\n 9 | -0.012345679012 | -0.012345679012 ✔\n10 | -0.010000000000 | -0.010000000000 ✔\nTest Summary: | Pass Total\n<ψₙ|V|ψₙ> / 2 = Eₙ | 10 10","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-\\psi_n(r,\\theta,\\varphi)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of psi_n(rthetavarphi)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int psi_i^ast(rthetavarphi) psi_j(rthetavarphi) r^2 sin(theta) mathrmdr mathrmdtheta mathrmdvarphi = delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n₁ | n₂ | l₁ | l₂ | m₁ | m₂ | analytical | numerical \n-- | -- | -- | -- | -- | -- | ----------------- | ----------------- \n 1 | 1 | 0 | 0 | 0 | 0 | 1.000000000000 | 1.000000000252 ✔\n 1 | 2 | 0 | 0 | 0 | 0 | 0.000000000000 | -0.000000011223 ✔\n 1 | 2 | 0 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 0 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 0 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 0 | 0 | 0 | 0.000000000000 | -0.000000045661 ✔\n 1 | 3 | 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | -2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0 | 0 | 0 | 0 | 0.000000000000 | -0.000000011223 ✔\n 2 | 1 | 1 | 0 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 0 | 0 | 0 | 1.000000000000 | 1.000006970517 ✔\n 2 | 2 | 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 0 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | -1 | -1 | 1.000000000000 | 1.000002301351 ✔\n 2 | 2 | 1 | 1 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | 0 | 0 | 1.000000000000 | 1.000002301351 ✔\n 2 | 2 | 1 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 1 | 1 | 1 | 1.000000000000 | 1.000002301351 ✔\n 2 | 3 | 0 | 0 | 0 | 0 | 0.000000000000 | 0.000088519421 ✔\n 2 | 3 | 0 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 1 | -1 | -1 | 0.000000000000 | 0.000038730338 ✔\n 2 | 3 | 1 | 1 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 1 | 0 | 0 | 0.000000000000 | 0.000038730338 ✔\n 2 | 3 | 1 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 1 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 1 | 1 | 1 | 0.000000000000 | 0.000038730338 ✔\n 2 | 3 | 1 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | -1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | -1 | 2 | 0.000000000000 | -0.000000000272 ✔\n 2 | 3 | 1 | 2 | 0 | -2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | 1 | -2 | 0.000000000000 | 0.000000000272 ✔\n 2 | 3 | 1 | 2 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0 | 0 | 0 | 0 | 0.000000000000 | -0.000000045661 ✔\n 3 | 1 | 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 1 | 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 2 | 0 | -2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 2 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 2 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 1 | 2 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 1 | 2 | 0 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0 | 0 | 0 | 0 | 0.000000000000 | 0.000088519421 ✔\n 3 | 2 | 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 0 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 1 | 1 | -1 | -1 | 0.000000000000 | 0.000038730338 ✔\n 3 | 2 | 1 | 1 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 1 | 1 | 0 | 0 | 0.000000000000 | 0.000038730338 ✔\n 3 | 2 | 1 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 1 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 1 | 1 | 1 | 1 | 0.000000000000 | 0.000038730338 ✔\n 3 | 2 | 2 | 0 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 0 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | -2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | -2 | 1 | 0.000000000000 | 0.000000000272 ✔\n 3 | 2 | 2 | 1 | -1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | 2 | -1 | 0.000000000000 | -0.000000000272 ✔\n 3 | 2 | 2 | 1 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 0 | 0 | 0 | 1.000000000000 | 1.002052594504 ✔\n 3 | 3 | 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | -2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 0 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | -1 | -1 | 1.000000000000 | 1.001223346388 ✔\n 3 | 3 | 1 | 1 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | 0 | 0 | 1.000000000000 | 1.001223346388 ✔\n 3 | 3 | 1 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 1 | 1 | 1 | 1.000000000000 | 1.001223346388 ✔\n 3 | 3 | 1 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | -1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | -1 | 2 | 0.000000000000 | 0.000000000308 ✔\n 3 | 3 | 1 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | 1 | -2 | 0.000000000000 | -0.000000000308 ✔\n 3 | 3 | 1 | 2 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 0 | -2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 0 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 0 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | -2 | 1 | 0.000000000000 | -0.000000000308 ✔\n 3 | 3 | 2 | 1 | -1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | 2 | -1 | 0.000000000000 | 0.000000000308 ✔\n 3 | 3 | 2 | 1 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | -2 | -2 | 1.000000000000 | 1.000300628566 ✔\n 3 | 3 | 2 | 2 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | -2 | 2 | 0.000000000000 | 0.000000193779 ✔\n 3 | 3 | 2 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -1 | -1 | 1.000000000000 | 1.000300628559 ✔\n 3 | 3 | 2 | 2 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 0 | 0 | 1.000000000000 | 1.000300628572 ✔\n 3 | 3 | 2 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 1 | 1 | 1.000000000000 | 1.000300628559 ✔\n 3 | 3 | 2 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 2 | -2 | 0.000000000000 | 0.000000193779 ✔\n 3 | 3 | 2 | 2 | 2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 2 | 2 | 1.000000000000 | 1.000300628566 ✔\nTest Summary: | Pass Total\n<ψₙ₁ₗ₁ₘ₁|ψₙ₂ₗ₂ₘ₂> = δₙ₁ₙ₂δₗ₁ₗ₂δₘ₁ₘ₂ | 196 196\n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"CurrentModule = Antique","category":"page"},{"location":"HarmonicOscillator/#Harmonic-Oscillator","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The harmonic oscillator is the most frequently used model in quantum physics.","category":"page"},{"location":"HarmonicOscillator/#Definitions","page":"Harmonic Oscillator","title":"Definitions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"k is the spring constant. And omega = sqrtkm, xi = sqrtfracmomegahbarx, A_n = sqrtfrac1n 2^n sqrtfracmomegapihbar, H_n(x) = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 are used.","category":"page"},{"location":"HarmonicOscillator/#Schrödinger-Equation","page":"Harmonic Oscillator","title":"Schrödinger Equation","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatHpsi(x) = E psi(x)","category":"page"},{"location":"HarmonicOscillator/#Hamiltonian","page":"Harmonic Oscillator","title":"Hamiltonian","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n hatH\n = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x) \n = - frac12 hbaromega fracmathrmd^2mathrmdxi^2 + V(x)\n endaligned","category":"page"},{"location":"HarmonicOscillator/#Potential","page":"Harmonic Oscillator","title":"Potential","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"V(x; k=k, m=m, ω=sqrt(k/m))","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" V(x)\n = frac12 k x^2\n = frac12 m omega^2 x^2\n = frac12 hbar omega xi^2","category":"page"},{"location":"HarmonicOscillator/#Eigen-Values","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"E(n; k, m, ω=sqrt(k/m), ℏ=ℏ)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" E_n = hbar omega left( n + frac12 right)","category":"page"},{"location":"HarmonicOscillator/#Eigen-Functions","page":"Harmonic Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"ψ(n, x; k=k, m=m, ω=sqrt(k/m), ℏ=ℏ)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" psi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials","page":"Harmonic Oscillator","title":"Hermite Polynomials","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"H(x; n=0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Examples:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n H_0(x) = 1 \n H_1(x) = 2 x \n H_2(x) = -2 + 4 x^2 \n H_3(x) = -12 x + 8 x^3 \n H_4(x) = 12 - 48 x^2 + 16 x^4 \n H_5(x) = 120 x - 160 x^3 + 32 x^5 \n H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n H_7(x) = -1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n H_9(x) = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n vdots\n endaligned","category":"page"},{"location":"HarmonicOscillator/#Reference","page":"Harmonic Oscillator","title":"Reference","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"DLMF 18.5.18\ncpprefjp\nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965) p.595 (a.4), (a.6)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968) p.71 (13.12)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999) p.491 (B.59)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994) p.152 (7.22)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995) p.41 Table 2.1, p.43 (2.70)\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008) p.293 Table 9.1\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)","category":"page"},{"location":"HarmonicOscillator/#Usage-and-Examples","page":"Harmonic Oscillator","title":"Usage & Examples","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Install Antique.jl for the first run and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Antique\nHO = antique(:HarmonicOscillator, k=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.k\n1.0\n\njulia> HO.m\n1.0\n\njulia> HO.ℏ\n1.0","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Eigen values:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.E(n=0)\n0.5\n\njulia> HO.E(n=1)\n1.5","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(-5:0.1:5, x -> HO.V(x), lw=2, label=\"\", xlabel=\"x\", ylabel=\"V(x)\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> HO.ψ(x, n=0), label=\"n=0\", lw=2)\nplot!(x -> HO.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> HO.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> HO.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> HO.ψ(x, n=4), label=\"n=4\", lw=2)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times0.5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 0:4\n # energy\n hline!([HO.E(n=n)], lc=:black, ls=:dash, label=\"\")\n plot!([-sqrt(2*HO.k*HO.E(n=n)),sqrt(2*HO.k*HO.E(n=n))], fill(HO.E(n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(x -> HO.E(n=n) + 0.5*HO.ψ(x,n=n), lc=n+1, lw=2, label=\"\")\nend\n# potential\nplot!(x -> HO.V(x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/#Testing","page":"Harmonic Oscillator","title":"Testing","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials-H_n(x)","page":"Harmonic Oscillator","title":"Hermite Polynomials H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=0 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_0(x)\n = e^x^2 e^ - x^2\n = 1 \n = 1\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=1 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_1(x)\n = - e^x^2 fracmathrmd e^ - x^2mathrmdx\n = 2 x \n = 2 x\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=2 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_2(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = -2 + 4 x^2 \n = -2 + 4 x^2\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=3 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_3(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 8 x^3 - 12 x \n = - 12 x + 8 x^3\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=4 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_4(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 12 - 48 x^2 + 16 x^4 \n = 12 - 48 x^2 + 16 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=5 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_5(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 120 x - 160 x^3 + 32 x^5 \n = 120 x - 160 x^3 + 32 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=6 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_6(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = -120 + 720 x^2 + 64 x^6 - 480 x^4 \n = -120 + 720 x^2 + 64 x^6 - 480 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=7 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_7(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = - 1680 x + 3360 x^3 + 128 x^7 - 1344 x^5 \n = - 1680 x + 3360 x^3 + 128 x^7 - 1344 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=8 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_8(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 1680 + 256 x^8 - 13440 x^2 - 3584 x^6 + 13440 x^4 \n = 1680 + 256 x^8 - 13440 x^2 - 3584 x^6 + 13440 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=9 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_9(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 30240 x - 80640 x^3 + 512 x^9 - 9216 x^7 + 48384 x^5 \n = 30240 x - 80640 x^3 + 512 x^9 - 9216 x^7 + 48384 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Test Summary: | Pass Total\nHₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ... | 10 10","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-H_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int_-infty^infty H_j(x) H_i(x) mathrme^-x^2 mathrmdx = sqrtpi 2^j j delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.772453850906 | 1.772453850906 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000001 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 3.544907701811 | 3.544907701811 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000014 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 14.179630807244 | 14.179630807244 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000011 ✔\n 2 | 9 | 0.000000000000 | -0.000000000002 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 85.077784843465 | 85.077784843465 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000139 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 680.622278747718 | 680.622278747718 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000002 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000063 ✔\n 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 6806.222787477181 | 6806.222787477180 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | 0.000000000009 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000001339 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000002 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 81674.673449726179 | 81674.673449726135 ✔\n 6 | 7 | 0.000000000000 | 0.000000000004 ✔\n 6 | 8 | 0.000000000000 | 0.000000000397 ✔\n 6 | 9 | 0.000000000000 | -0.000000000087 ✔\n 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | 0.000000000009 ✔\n 7 | 6 | 0.000000000000 | 0.000000000004 ✔\n 7 | 7 | 1143445.428296166472 | 1143445.428296166705 ✔\n 7 | 8 | 0.000000000000 | -0.000000000007 ✔\n 7 | 9 | 0.000000000000 | 0.000000011649 ✔\n 8 | 0 | 0.000000000000 | -0.000000000001 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000011 ✔\n 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 8 | 4 | 0.000000000000 | -0.000000000063 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000397 ✔\n 8 | 7 | 0.000000000000 | -0.000000000007 ✔\n 8 | 8 | 18295126.852738663554 | 18295126.852738667279 ✔\n 8 | 9 | 0.000000000000 | 0.000000001630 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000014 ✔\n 9 | 2 | 0.000000000000 | -0.000000000002 ✔\n 9 | 3 | 0.000000000000 | 0.000000000139 ✔\n 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000001339 ✔\n 9 | 6 | 0.000000000000 | -0.000000000087 ✔\n 9 | 7 | 0.000000000000 | 0.000000011649 ✔\n 9 | 8 | 0.000000000000 | 0.000000001630 ✔\n 9 | 9 | 329312283.349295914173 | 329312283.349295675755 ✔\nTest Summary: | Pass Total\n∫Hⱼ(x)Hᵢ(x)exp(-x²)dx = √π2ʲj!δᵢⱼ | 100 100","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000000 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | 0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000000 ✔\n 8 | 0 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000000 ✔\n 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n 9 | 3 | 0.000000000000 | 0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | 0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\nTest Summary: | Pass Total\n<ψᵢ|ψⱼ> = δᵢⱼ | 100 100","category":"page"},{"location":"HarmonicOscillator/#Virial-Theorem","page":"Harmonic Oscillator","title":"Virial Theorem","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The virial theorem langle T rangle = langle V rangle and the definition of Hamiltonian langle H rangle = langle T rangle + langle V rangle derive langle H rangle = 2 langle V rangle = 2 langle T rangle.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"2 int psi_n^ast(x) V(x) psi_n(x) mathrmdx = E_n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | 0.500000000000 | 0.500000000000 ✔\n0.1 | 1 | 1.500000000000 | 1.500000000000 ✔\n0.1 | 2 | 2.500000000000 | 2.500000000000 ✔\n0.1 | 3 | 3.500000000000 | 3.500000000000 ✔\n0.1 | 4 | 4.500000000000 | 4.500000000000 ✔\n0.1 | 5 | 5.500000000000 | 5.500000000000 ✔\n0.1 | 6 | 6.500000000000 | 6.500000000000 ✔\n0.1 | 7 | 7.500000000000 | 7.500000000000 ✔\n0.1 | 8 | 8.500000000000 | 8.500000000000 ✔\n0.1 | 9 | 9.500000000000 | 9.500000000000 ✔\n0.5 | 0 | 0.500000000000 | 0.500000000000 ✔\n0.5 | 1 | 1.500000000000 | 1.500000000000 ✔\n0.5 | 2 | 2.500000000000 | 2.500000000000 ✔\n0.5 | 3 | 3.500000000000 | 3.500000000000 ✔\n0.5 | 4 | 4.500000000000 | 4.500000000000 ✔\n0.5 | 5 | 5.500000000000 | 5.500000000000 ✔\n0.5 | 6 | 6.500000000000 | 6.500000000000 ✔\n0.5 | 7 | 7.500000000000 | 7.500000000000 ✔\n0.5 | 8 | 8.500000000000 | 8.500000000000 ✔\n0.5 | 9 | 9.500000000000 | 9.500000000000 ✔\n1.0 | 0 | 0.500000000000 | 0.500000000000 ✔\n1.0 | 1 | 1.500000000000 | 1.500000000000 ✔\n1.0 | 2 | 2.500000000000 | 2.500000000000 ✔\n1.0 | 3 | 3.500000000000 | 3.500000000000 ✔\n1.0 | 4 | 4.500000000000 | 4.500000000000 ✔\n1.0 | 5 | 5.500000000000 | 5.500000000000 ✔\n1.0 | 6 | 6.500000000000 | 6.500000000000 ✔\n1.0 | 7 | 7.500000000000 | 7.500000000000 ✔\n1.0 | 8 | 8.500000000000 | 8.500000000000 ✔\n1.0 | 9 | 9.500000000000 | 9.500000000000 ✔\n5.0 | 0 | 0.500000000000 | 0.500000000000 ✔\n5.0 | 1 | 1.500000000000 | 1.500000000000 ✔\n5.0 | 2 | 2.500000000000 | 2.500000000000 ✔\n5.0 | 3 | 3.500000000000 | 3.500000000000 ✔\n5.0 | 4 | 4.500000000000 | 4.500000000000 ✔\n5.0 | 5 | 5.500000000000 | 5.500000000000 ✔\n5.0 | 6 | 6.500000000000 | 6.500000000000 ✔\n5.0 | 7 | 7.500000000000 | 7.500000000000 ✔\n5.0 | 8 | 8.500000000000 | 8.500000000000 ✔\n5.0 | 9 | 9.500000000000 | 9.500000000000 ✔\nTest Summary: | Pass Total\n2 × <ψₙ|V|ψₙ> = Eₙ | 40 40","category":"page"},{"location":"HarmonicOscillator/#Eigen-Values-2","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n E_n\n = int psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int psi^ast_n(x) left V(x) - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int psi^ast_n(x) left V(x)psi(x) -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | 0.158113883008 | 0.158113879883 ✔\n0.1 | 1 | 0.474341649025 | 0.474341633410 ✔\n0.1 | 2 | 0.790569415042 | 0.790569374409 ✔\n0.1 | 3 | 1.106797181059 | 1.106797102928 ✔\n0.1 | 4 | 1.423024947076 | 1.423024818987 ✔\n0.1 | 5 | 1.739252713093 | 1.739252522506 ✔\n0.1 | 6 | 2.055480479109 | 2.055480213500 ✔\n0.1 | 7 | 2.371708245126 | 2.371707891950 ✔\n0.1 | 8 | 2.687936011143 | 2.687935558100 ✔\n0.1 | 9 | 3.004163777160 | 3.004163211450 ✔\n0.5 | 0 | 0.353553390593 | 0.353553374944 ✔\n0.5 | 1 | 1.060660171780 | 1.060660093649 ✔\n0.5 | 2 | 1.767766952966 | 1.767766749878 ✔\n0.5 | 3 | 2.474873734153 | 2.474873343556 ✔\n0.5 | 4 | 3.181980515339 | 3.181979874817 ✔\n0.5 | 5 | 3.889087296526 | 3.889086343463 ✔\n0.5 | 6 | 4.596194077713 | 4.596192749665 ✔\n0.5 | 7 | 5.303300858899 | 5.303299093519 ✔\n0.5 | 8 | 6.010407640086 | 6.010405374197 ✔\n0.5 | 9 | 6.717514421272 | 6.717511593266 ✔\n1.0 | 0 | 0.500000000000 | 0.499999968773 ✔\n1.0 | 1 | 1.500000000000 | 1.499999843774 ✔\n1.0 | 2 | 2.500000000000 | 2.499999593764 ✔\n1.0 | 3 | 3.500000000000 | 3.499999218732 ✔\n1.0 | 4 | 4.500000000000 | 4.499998718747 ✔\n1.0 | 5 | 5.500000000000 | 5.499998093755 ✔\n1.0 | 6 | 6.500000000000 | 6.499997343602 ✔\n1.0 | 7 | 7.500000000000 | 7.499996468887 ✔\n1.0 | 8 | 8.500000000000 | 8.499995468843 ✔\n1.0 | 9 | 9.500000000000 | 9.499994343445 ✔\n5.0 | 0 | 1.118033988750 | 1.118033832523 ✔\n5.0 | 1 | 3.354101966250 | 3.354101184969 ✔\n5.0 | 2 | 5.590169943749 | 5.590167912524 ✔\n5.0 | 3 | 7.826237921249 | 7.826234014984 ✔\n5.0 | 4 | 10.062305898749 | 10.062299492494 ✔\n5.0 | 5 | 12.298373876249 | 12.298364344997 ✔\n5.0 | 6 | 14.534441853749 | 14.534428572309 ✔\n5.0 | 7 | 16.770509831248 | 16.770492175222 ✔\n5.0 | 8 | 19.006577808748 | 19.006555152416 ✔\n5.0 | 9 | 21.242645786248 | 21.242617504750 ✔\nTest Summary: | Pass Total\n∫ψₙ*Hψₙdx = <ψₙ|H|ψₙ> = Eₙ | 40 40\n","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = Antique","category":"page"},{"location":"#Antique.jl","page":"Home","title":"Antique.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Self-contained, Well-Tested, Well-Documented Analytical Solutions of Quantum Mechanical Equations.","category":"page"},{"location":"#Install","page":"Home","title":"Install","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install this package, run the following code in your Jupyter Notebook:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg; Pkg.add(\"Antique\")","category":"page"},{"location":"#Usage","page":"Home","title":"Usage","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To use this package, run the following code before each use:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Antique","category":"page"},{"location":"","page":"Home","title":"Home","text":"The function antique(model, parameters...) returns a module. Each module has E(), ψ(x) and some other functions.","category":"page"},{"location":"#Examples","page":"Home","title":"Examples","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The energy of 1mathrmS state in mathrmH:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> H = antique(:HydrogenAtom, Z=1)\njulia> H.E(n=1)\n-0.5","category":"page"},{"location":"","page":"Home","title":"Home","text":"The energy of 1mathrmS state in mathrmHe^+:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> He⁺ = antique(:HydrogenAtom, Z=2)\njulia> He⁺.E(n=1)\n-2.0","category":"page"},{"location":"#Supported-Models","page":"Home","title":"Supported Models","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
\n
\n \n \"InfinitePotentialWell\"/\n \n :InfinitePotentialWell\n
\n
\n \n \"HarmonicOscillator\"/\n \n :HarmonicOscillator\n
\n
\n \n \"MorsePotential\"/\n \n :MorsePotential\n
\n
\n \n \"HydrogenAtom\"/\n \n :HydrogenAtom\n
\n
","category":"page"},{"location":"#Future-Works","page":"Home","title":"Future Works","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"List of quantum-mechanical systems with analytical solutions","category":"page"},{"location":"#Acknowledgment","page":"Home","title":"Acknowledgment","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package was named by @KB-satou and @ultimatile.","category":"page"}] +[{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"CurrentModule = Antique","category":"page"},{"location":"InfinitePotentialWell/#Infinite-Potential-Well-(Particle-in-a-Box)","page":"Infinite Potential Well","title":"Infinite Potential Well (Particle in a Box)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.","category":"page"},{"location":"InfinitePotentialWell/#Definitions","page":"Infinite Potential Well","title":"Definitions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L is the length of the 1D-box, m is the mass of particle.","category":"page"},{"location":"InfinitePotentialWell/#Schrödinger-Equation","page":"Infinite Potential Well","title":"Schrödinger Equation","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"InfinitePotentialWell/#Hamiltonian","page":"Infinite Potential Well","title":"Hamiltonian","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH = frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"InfinitePotentialWell/#Potential","page":"Infinite Potential Well","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"V(x; L=L)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" V(x) =\n left\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\n right","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"E(; n=0, L=L, m=m, ℏ=ℏ)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" E_n = frachbar^2 n^2 pi^22 m L^2","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Functions","page":"Infinite Potential Well","title":"Eigen Functions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"ψ(x; n=0, L=L)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" psi_n(x) = sqrtfrac2L sin fracnpi xL","category":"page"},{"location":"InfinitePotentialWell/#Proofs","page":"Infinite Potential Well","title":"Proofs","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen Functions & Eigen Values\nNormalization","category":"page"},{"location":"InfinitePotentialWell/#Usage-and-Examples","page":"Infinite Potential Well","title":"Usage & Examples","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Antique\nIPW = antique(:InfinitePotentialWell, L=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.L\n1.0\n\njulia> IPW.m\n1.0\n\njulia> IPW.ℏ\n1.0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen values:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.E(n=1)\n4.934802200544679\n\njulia> IPW.E(n=2)\n19.739208802178716","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Plots\nplot(xlim=(0,1), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> IPW.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> IPW.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> IPW.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> IPW.ψ(x, n=4), label=\"n=4\", lw=2)\nplot!(x -> IPW.ψ(x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L = 1\nusing Plots\nplot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 1:5\n # energy\n plot!([0,L], fill(IPW.E(n=n,L=L),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(0:0.01:L, x->IPW.E(n=n,L=L)+5*IPW.ψ(x,n=n,L=L), lc=n, lw=2, label=\"\")\nend\n# potential\nplot!([0,0,L,L], [140,0,0,140], lc=:black, lw=2, label=\"\")","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/#Testing","page":"Infinite Potential Well","title":"Testing","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"InfinitePotentialWell/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Infinite Potential Well","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"int_0^L psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 10 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 10 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 10 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 4 | 10 | 0.000000000000 | 0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 10 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 10 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 7 | 10 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000000 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 8 | 10 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\n 9 | 10 | 0.000000000000 | 0.000000000000 ✔\n10 | 1 | 0.000000000000 | 0.000000000000 ✔\n10 | 2 | 0.000000000000 | 0.000000000000 ✔\n10 | 3 | 0.000000000000 | 0.000000000000 ✔\n10 | 4 | 0.000000000000 | 0.000000000000 ✔\n10 | 5 | 0.000000000000 | 0.000000000000 ✔\n10 | 6 | 0.000000000000 | -0.000000000000 ✔\n10 | 7 | 0.000000000000 | -0.000000000000 ✔\n10 | 8 | 0.000000000000 | 0.000000000000 ✔\n10 | 9 | 0.000000000000 | 0.000000000000 ✔\n10 | 10 | 1.000000000000 | 1.000000000000 ✔\nTest Summary: | Pass Total\n<ψᵢ|ψⱼ> = ∫ψₙ*ψₙdx = δᵢⱼ | 100 100","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values-2","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n E_n\n = int_0^L psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int_0^L psi^ast_n(x) left 0 - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | m | ℏ | n | analytical | numerical \n--- | --- | --- | -- | ----------------- | ----------------- \n0.1 | 0.1 | 0.1 | 1 | 49.348021579139 | 49.348022005447 ✔\n0.1 | 0.1 | 0.1 | 2 | 197.392081461942 | 197.392088021787 ✔\n0.1 | 0.1 | 0.1 | 3 | 444.132165131018 | 444.132198049021 ✔\n0.1 | 0.1 | 0.1 | 4 | 789.568248175979 | 789.568352087149 ✔\n0.1 | 0.1 | 0.1 | 5 | 1233.700296336187 | 1233.700550136170 ✔\n0.1 | 0.1 | 0.1 | 6 | 1776.528266243334 | 1776.528792196084 ✔\n0.1 | 0.1 | 0.1 | 7 | 2418.052103857080 | 2418.053078266893 ✔\n0.1 | 0.1 | 0.1 | 8 | 3158.271745875927 | 3158.273408348594 ✔\n0.1 | 0.1 | 0.1 | 9 | 3997.187119264267 | 3997.189782441189 ✔\n0.1 | 0.1 | 0.1 | 10 | 4934.798141994514 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 1 | 4934.802157913905 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 2 | 19739.208146194214 | 19739.208802178713 ✔\n0.1 | 0.1 | 1.0 | 3 | 44413.216513101754 | 44413.219804902103 ✔\n0.1 | 0.1 | 1.0 | 4 | 78956.824817597895 | 78956.835208714852 ✔\n0.1 | 0.1 | 1.0 | 5 | 123370.029633618717 | 123370.055013616948 ✔\n0.1 | 0.1 | 1.0 | 6 | 177652.826624333364 | 177652.879219608410 ✔\n0.1 | 0.1 | 1.0 | 7 | 241805.210385707964 | 241805.307826689212 ✔\n0.1 | 0.1 | 1.0 | 8 | 315827.174587592541 | 315827.340834859409 ✔\n0.1 | 0.1 | 1.0 | 9 | 399718.711926426622 | 399718.978244118916 ✔\n0.1 | 0.1 | 1.0 | 10 | 493479.814199451241 | 493480.220054467791 ✔\n0.1 | 1.0 | 0.1 | 1 | 4.934802157914 | 4.934802200545 ✔\n0.1 | 1.0 | 0.1 | 2 | 19.739208146194 | 19.739208802179 ✔\n0.1 | 1.0 | 0.1 | 3 | 44.413216513102 | 44.413219804902 ✔\n0.1 | 1.0 | 0.1 | 4 | 78.956824817598 | 78.956835208715 ✔\n0.1 | 1.0 | 0.1 | 5 | 123.370029633619 | 123.370055013617 ✔\n0.1 | 1.0 | 0.1 | 6 | 177.652826624333 | 177.652879219608 ✔\n0.1 | 1.0 | 0.1 | 7 | 241.805210385708 | 241.805307826689 ✔\n0.1 | 1.0 | 0.1 | 8 | 315.827174587593 | 315.827340834859 ✔\n0.1 | 1.0 | 0.1 | 9 | 399.718711926427 | 399.718978244119 ✔\n0.1 | 1.0 | 0.1 | 10 | 493.479814199451 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 1 | 493.480215791390 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 2 | 1973.920814619422 | 1973.920880217871 ✔\n0.1 | 1.0 | 1.0 | 3 | 4441.321651310176 | 4441.321980490210 ✔\n0.1 | 1.0 | 1.0 | 4 | 7895.682481759791 | 7895.683520871485 ✔\n0.1 | 1.0 | 1.0 | 5 | 12337.002963361872 | 12337.005501361695 ✔\n0.1 | 1.0 | 1.0 | 6 | 17765.282662433339 | 17765.287921960840 ✔\n0.1 | 1.0 | 1.0 | 7 | 24180.521038570794 | 24180.530782668924 ✔\n0.1 | 1.0 | 1.0 | 8 | 31582.717458759253 | 31582.734083485939 ✔\n0.1 | 1.0 | 1.0 | 9 | 39971.871192642662 | 39971.897824411892 ✔\n0.1 | 1.0 | 1.0 | 10 | 49347.981419945128 | 49348.022005446779 ✔\n1.0 | 0.1 | 0.1 | 1 | 0.493480215948 | 0.493480220054 ✔\n1.0 | 0.1 | 0.1 | 2 | 1.973920815419 | 1.973920880218 ✔\n1.0 | 0.1 | 0.1 | 3 | 4.441321651944 | 4.441321980490 ✔\n1.0 | 0.1 | 0.1 | 4 | 7.895682481265 | 7.895683520871 ✔\n1.0 | 0.1 | 0.1 | 5 | 12.337002965030 | 12.337005501362 ✔\n1.0 | 0.1 | 0.1 | 6 | 17.765282661715 | 17.765287921961 ✔\n1.0 | 0.1 | 0.1 | 7 | 24.180521036064 | 24.180530782669 ✔\n1.0 | 0.1 | 0.1 | 8 | 31.582717460023 | 31.582734083486 ✔\n1.0 | 0.1 | 0.1 | 9 | 39.971871195191 | 39.971897824412 ✔\n1.0 | 0.1 | 0.1 | 10 | 49.347981417827 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 1 | 49.348021594816 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 2 | 197.392081541864 | 197.392088021787 ✔\n1.0 | 0.1 | 1.0 | 3 | 444.132165194438 | 444.132198049021 ✔\n1.0 | 0.1 | 1.0 | 4 | 789.568248126463 | 789.568352087149 ✔\n1.0 | 0.1 | 1.0 | 5 | 1233.700296503016 | 1233.700550136170 ✔\n1.0 | 0.1 | 1.0 | 6 | 1776.528266171473 | 1776.528792196084 ✔\n1.0 | 0.1 | 1.0 | 7 | 2418.052103606433 | 2418.053078266892 ✔\n1.0 | 0.1 | 1.0 | 8 | 3158.271746002275 | 3158.273408348594 ✔\n1.0 | 0.1 | 1.0 | 9 | 3997.187119519121 | 3997.189782441190 ✔\n1.0 | 0.1 | 1.0 | 10 | 4934.798141782662 | 4934.802200544679 ✔\n1.0 | 1.0 | 0.1 | 1 | 0.049348021595 | 0.049348022005 ✔\n1.0 | 1.0 | 0.1 | 2 | 0.197392081542 | 0.197392088022 ✔\n1.0 | 1.0 | 0.1 | 3 | 0.444132165194 | 0.444132198049 ✔\n1.0 | 1.0 | 0.1 | 4 | 0.789568248126 | 0.789568352087 ✔\n1.0 | 1.0 | 0.1 | 5 | 1.233700296503 | 1.233700550136 ✔\n1.0 | 1.0 | 0.1 | 6 | 1.776528266171 | 1.776528792196 ✔\n1.0 | 1.0 | 0.1 | 7 | 2.418052103606 | 2.418053078267 ✔\n1.0 | 1.0 | 0.1 | 8 | 3.158271746002 | 3.158273408349 ✔\n1.0 | 1.0 | 0.1 | 9 | 3.997187119519 | 3.997189782441 ✔\n1.0 | 1.0 | 0.1 | 10 | 4.934798141783 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 1 | 4.934802159482 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 2 | 19.739208154186 | 19.739208802179 ✔\n1.0 | 1.0 | 1.0 | 3 | 44.413216519444 | 44.413219804902 ✔\n1.0 | 1.0 | 1.0 | 4 | 78.956824812646 | 78.956835208715 ✔\n1.0 | 1.0 | 1.0 | 5 | 123.370029650302 | 123.370055013617 ✔\n1.0 | 1.0 | 1.0 | 6 | 177.652826617147 | 177.652879219608 ✔\n1.0 | 1.0 | 1.0 | 7 | 241.805210360643 | 241.805307826689 ✔\n1.0 | 1.0 | 1.0 | 8 | 315.827174600228 | 315.827340834859 ✔\n1.0 | 1.0 | 1.0 | 9 | 399.718711951912 | 399.718978244119 ✔\n1.0 | 1.0 | 1.0 | 10 | 493.479814178266 | 493.480220054468 ✔\nTest Summary: | Pass Total\n<ψₙ|H|ψₙ> = ∫ψₙ*Tψₙdx = Eₙ | 80 80","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x","page":"Infinite Potential Well","title":"Expected Value of x","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x rangle_n=1\n= int_0^L psi_1^ast(x) hatx psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"for only n=1.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.050000000000 | 0.050000000000 ✔\n0.5 | 1 | 0.250000000000 | 0.250000000000 ✔\n1.0 | 1 | 0.500000000000 | 0.500000000000 ✔\n7.0 | 1 | 3.500000000000 | 3.500000000000 ✔\nTest Summary: | Pass Total\n<ψₙ|x|ψₙ> = L/2 | 4 4","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x2","page":"Infinite Potential Well","title":"Expected Value of x^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x^2 rangle_n=1\n= int_0^L psi_1^ast(x) hatx^2 psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.002826727415 | 0.002826727415 ✔\n0.5 | 1 | 0.070668185378 | 0.070668185378 ✔\n1.0 | 1 | 0.282672741512 | 0.282672741512 ✔\n7.0 | 1 | 13.850964334096 | 13.850964334096 ✔\nTest Summary: | Pass Total\n<ψₙ|x²|ψₙ> = 2L²/π³(π³/6-π/4) | 4 4","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p","page":"Infinite Potential Well","title":"Expected Value of p","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p rangle_n=1\n= int_0^L psi_1^ast(x) hatp psi_1(x) mathrmdx\n= 0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n langle p rangle_n=1\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -ihbarfracmathrmdmathrmd x right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -ihbar fracpsi(x+Delta x) - psi(x-Delta x)2Delta x right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2fracmathrmd psi(x)mathrmdx Delta x\n + Oleft(Delta x^3right)\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n \n 2fracmathrmd psi(x)mathrmdx Delta x\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n - Oleft(Delta x^3right)\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n - fracOleft(Delta x^3right)2Delta x\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n psi(x+Delta x)\n =\n psi(x)\n + fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\n \n psi(x-Delta x)\n =\n psi(x)\n - fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.000000000003 | 0.000000000000 ✔\n0.5 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.0 | 1 | 0.000000000000 | 0.000000000000 ✔\n7.0 | 1 | 0.000000000000 | 0.000000000000 ✔\nTest Summary: | Pass Total\n<ψₙ|p|ψₙ> = ∫ψₙ*(-iℏd/dx)ψₙdx = 0 | 4 4","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p2","page":"Infinite Potential Well","title":"Expected Value of p^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p^2 rangle\n= int_0^L psi_1^ast(x) hatp^2 psi_1(x) mathrmdx\n= fracpi^2hbar^2L^2","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n langle p^2 rangle\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -hbar^2fracmathrmd^2mathrmdx^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -hbar^2 fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 986.960431582781 | 986.960440108936 ✔\n0.5 | 1 | 39.478417274195 | 39.478417604357 ✔\n1.0 | 1 | 9.869604318963 | 9.869604401089 ✔\n7.0 | 1 | 0.201420496383 | 0.201420497981 ✔\nTest Summary: | Pass Total\n<ψₙ|p²|ψₙ> = ∫ψₙ*(-ℏ²d²/dx²)ψₙdx = π²ℏ²/L² | 4 4\n","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"CurrentModule = Antique","category":"page"},{"location":"MorsePotential/#Morse-Potential","page":"Morse Potential","title":"Morse Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.","category":"page"},{"location":"MorsePotential/#Definitions","page":"Morse Potential","title":"Definitions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"xi = 2lambdamathrme^-a(r-r_e), omega = sqrtkµ, k = 2D_mathrmea^2, lambda = fracsqrt2mD_mathrmeahbar, chi = frachbaromega4D_mathrme, N_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight) are used. The domains of the potential and the wave functions are 0leq r lt infty.","category":"page"},{"location":"MorsePotential/#Schrödinger-Equation","page":"Morse Potential","title":"Schrödinger Equation","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatHpsi(r) = E psi(r)","category":"page"},{"location":"MorsePotential/#Hamiltonian","page":"Morse Potential","title":"Hamiltonian","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"MorsePotential/#Potential","page":"Morse Potential","title":"Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"V(r; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)))","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" V(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)","category":"page"},{"location":"MorsePotential/#Eigen-Values","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"E(n; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)), µ=µ, ω=sqrt(k/µ), χ=ℏ*ω/(4*Dₑ), ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" E_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2","category":"page"},{"location":"MorsePotential/#Eigen-Functions","page":"Morse Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"ψ(n, r; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)), µ=µ, ω=sqrt(k/µ), χ=ℏ*ω/(4*Dₑ), ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" psi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials","page":"Morse Potential","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"L(x; n=0, α=0)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k left(beginarrayl n+alpha n-k endarrayright) fracx^kk \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Examples:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n L_0^(0)(x) = 1 \n L_1^(0)(x) = 1 - x \n L_1^(1)(x) = 2 - x \n L_2^(0)(x) = 1 - 2 x + 12 x^2 \n L_2^(1)(x) = 3 - 3 x + 12 x^2 \n L_2^(2)(x) = 6 - 4 x + 12 x^2 \n L_3^(0)(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^(1)(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_3^(2)(x) = 10 - 10 x + 52 x^2 - 16 x^3 \n L_3^(3)(x) = 20 - 15 x + 3 x^2 - 16 x^3 \n L_4^(0)(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 124 x^4 \n L_4^(1)(x) = 5 - 10 x + 5 x^2 - 56 x^3 + 124 x^4 \n L_4^(2)(x) = 15 - 20 x + 152 x^2 - 1 x^3 + 124 x^4 \n L_4^(3)(x) = 35 - 35 x + 212 x^2 - 76 x^3 + 124 x^4 \n L_4^(4)(x) = 70 - 56 x + 14 x^2 - 43 x^3 + 124 x^4 \n vdots\n endaligned","category":"page"},{"location":"MorsePotential/#References","page":"Morse Potential","title":"References","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"P. M. Morse, Phys. Rev. 34, 57 (1929)\nJ. P. Dahl, M. Springborg, J. Chem. Phys. 88, 4535 (1988). (62), (63)\nW. K. Shao, Y. He, J. Pan, J. Nonlinear Sci. Appl., 9, 5, 3388 (2016). (1.6) \nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.12","category":"page"},{"location":"MorsePotential/#Usage-and-Examples","page":"Morse Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"warning: Warning\nRun using Pkg; Pkg.add(\"SpecialFunctions\") if the following returns an error.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"# Parameters for H₂⁺\n# https://doi.org/10.1002/slct.202102509\n# https://doi.org/10.5281/zenodo.5047817\n# https://physics.nist.gov/cgi-bin/cuu/Value?mpsme\nrₑ = 1.997193319969992120068298141276\nVₑ = -0.602634619106539878727562156289\nDₑ = - 0.5 - Vₑ\nk = 2*((-1.1026342144949464615+1/2.00) - Vₑ) / (2.00 - rₑ)^2\nµ = 1/(1/1836.15267343 + 1/1836.15267343)\nℏ = 1.0\n\nusing Antique\nMP = antique(:MorsePotential, rₑ=rₑ, Vₑ=Vₑ, Dₑ=Dₑ, k=k, µ =µ, ℏ =ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.rₑ\n1.997193319969992\n\njulia> MP.Dₑ\n0.10263461910653993\n\njulia> MP.k\n0.1027265041900817\n\njulia> MP.µ\n918.076336715\n\njulia> MP.ℏ\n1.0","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Eigen values:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.E(n=0)\n-0.09741377794418261\n\njulia> MP.E(n=1)\n-0.08738092406760907","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(0.1:0.01:15, r -> MP.V(r), lw=2, label=\"\", xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"r\", ylabel=\"V(r)\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Wave functions:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(xlim=(0,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> MP.ψ(x, n=0), label=\"n=0\", lw=2)\nplot!(x -> MP.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> MP.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> MP.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> MP.ψ(x, n=4), label=\"n=4\", lw=2)\nplot!(x -> MP.ψ(x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve, Energy levels, Comparison with harmonic oscillator:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"MP = antique(:MorsePotential)\nHO = antique(:HarmonicOscillator, k=MP.k, m=MP.μ)\nusing Plots\nplot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"\\$r\\$\", ylabel=\"\\$V(r), E_n\\$\", legend=:bottomright, size=(480,400), dpi=300)\nfor n in 0:MP.nₘₐₓ()\n # energy\n EM = MP.E(n=n)\n EH = HO.E(n=n) - MP.Dₑ\n plot!(0.1:0.01:15, r -> EH > HO.V(r-MP.rₑ) - MP.Dₑ ? EH : NaN, lc=\"#BC1C5F\", lw=1, label=\"\")\n plot!(0.1:0.01:15, r -> EM > MP.V(r) ? EM : NaN, lc=\"#578FC7\", lw=1, label=\"\")\nend\n# potential\nplot!(0.1:0.01:15, r -> HO.V(r-MP.rₑ) - MP.Dₑ, lc=\"#BC1C5F\", lw=2, label=\"Harmonic Oscillator\")\nplot!(0.1:0.01:15, r -> MP.V(r), lc=\"#578FC7\", lw=2, label=\"Morse Potential\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"where, the potential of harmonic oscillator is defined as V(r) simeq frac12 k (r - r_mathrme)^2 + V_0.","category":"page"},{"location":"MorsePotential/#Testing","page":"Morse Potential","title":"Testing","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Generalized Laguerre Polynomials L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=0 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_0^(0)(x)\n = e^x e^ - x\n = 1 \n = 1\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_1^(0)(x)\n = e^x fracmathrmdmathrmdx x e^ - x\n = 1 - x \n = 1 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_1^(1)(x)\n = frace^x fracmathrmdmathrmdx x^2 e^ - xx\n = 2 - x \n = 2 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(0)(x)\n = frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n = 1 - 2 x + frac12 x^2 \n = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(1)(x)\n = fracfrac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - xx\n = 3 - 3 x + frac12 x^2 \n = 3 - 3 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(2)(x)\n = fracfrac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - xx^2\n = 6 - 4 x + frac12 x^2 \n = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(0)(x)\n = frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = 1 - frac16 x^3 - 3 x + frac32 x^2 \n = 1 - frac16 x^3 - 3 x + frac32 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(1)(x)\n = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - xx\n = 4 - 6 x - frac16 x^3 + 2 x^2 \n = 4 - frac16 x^3 - 6 x + 2 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(2)(x)\n = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - xx^2\n = 10 - 10 x - frac16 x^3 + frac52 x^2 \n = 10 - frac16 x^3 - 10 x + frac52 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(3)(x)\n = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - xx^3\n = 20 - 15 x - frac16 x^3 + 3 x^2 \n = 20 - frac16 x^3 - 15 x + 3 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(0)(x)\n = frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4 \n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(1)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - xx\n = 5 - 10 x - frac56 x^3 + 5 x^2 + frac124 x^4 \n = 5 - frac56 x^3 - 10 x + 5 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(2)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - xx^2\n = 15 - 20 x - x^3 + frac152 x^2 + frac124 x^4 \n = 15 - x^3 - 20 x + frac152 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(3)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^7 e^ - xx^3\n = 35 - 35 x - frac76 x^3 + frac212 x^2 + frac124 x^4 \n = 35 - frac76 x^3 - 35 x + frac212 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=4 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(4)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^8 e^ - xx^4\n = 70 - 56 x - frac43 x^3 + 14 x^2 + frac124 x^4 \n = 70 - frac43 x^3 - 56 x + 14 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Test Summary: | Pass Total\nLₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ | 15 15","category":"page"},{"location":"MorsePotential/#Normalization-and-Orthogonality-of-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Normalization & Orthogonality of L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"int_0^infty L_i^(alpha)(x) L_j^(alpha)(x) x^alpha mathrme^-x mathrmdx = fracGamma(n+alpha+1)n delta_ij","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" α | i | j | analytical | numerical \n---- | -- | -- | ----------------- | ----------------- \n0.01 | 0 | 0 | 0.994325851192 | 0.994325852936 ✔\n0.01 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 1 | 1.004269109703 | 1.004269111483 ✔\n0.01 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 8 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 9 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 2 | 2 | 1.009290455252 | 1.009290456144 ✔\n0.01 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 2 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 9 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 3 | 3 | 1.012654756769 | 1.012654758579 ✔\n0.01 | 3 | 4 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 7 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 3 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 3 | 9 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 4 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 4 | 4 | 1.015186393661 | 1.015186394564 ✔\n0.01 | 4 | 5 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 4 | 6 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 4 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 9 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 5 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 5 | 5 | 1.017216766449 | 1.017216768275 ✔\n0.01 | 5 | 6 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 6 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 6 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 6 | 4 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 6 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 6 | 6 | 1.018912127726 | 1.018912128636 ✔\n0.01 | 6 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 6 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 6 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 7 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 7 | 3 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 7 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 6 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 7 | 1.020367716480 | 1.020367717392 ✔\n0.01 | 7 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 7 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 8 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 8 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 8 | 3 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 6 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 7 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 8 | 1.021643176126 | 1.021643177967 ✔\n0.01 | 8 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 9 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 9 | 2 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 9 | 3 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 9 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 9 | 5 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 6 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 7 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 9 | 1.022778335210 | 1.022778336127 ✔\n0.05 | 0 | 0 | 0.973504265563 | 0.973504267703 ✔\n0.05 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 0 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 0 | 8 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 0 | 9 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 1 | 1 | 1.022179478841 | 1.022179479980 ✔\n0.05 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 1 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 5 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 8 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 1 | 9 | 0.000000000000 | 0.000000000007 ✔\n0.05 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 2 | 2 | 1.047733965812 | 1.047733966390 ✔\n0.05 | 2 | 3 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 2 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 2 | 5 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 2 | 6 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 2 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 2 | 8 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 2 | 9 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 3 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 3 | 3 | 1.065196198575 | 1.065196199813 ✔\n0.05 | 3 | 4 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 3 | 5 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 3 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 8 | 0.000000000000 | 0.000000000016 ✔\n0.05 | 3 | 9 | 0.000000000000 | 0.000000000015 ✔\n0.05 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 4 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 4 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 4 | 3 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 4 | 4 | 1.078511151058 | 1.078511152326 ✔\n0.05 | 4 | 5 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 4 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 4 | 7 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 4 | 8 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 4 | 9 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 5 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 5 | 2 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 5 | 3 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 5 | 4 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 5 | 5 | 1.089296262568 | 1.089296263862 ✔\n0.05 | 5 | 6 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 5 | 7 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 5 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 5 | 9 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 6 | 2 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 6 | 3 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 6 | 4 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 6 | 5 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 6 | 6 | 1.098373731423 | 1.098373732739 ✔\n0.05 | 6 | 7 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 6 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 6 | 9 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 7 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 7 | 3 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 7 | 4 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 7 | 5 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 7 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 7 | 1.106219258076 | 1.106219258720 ✔\n0.05 | 7 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 9 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 8 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 8 | 1 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 8 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 8 | 3 | 0.000000000000 | 0.000000000016 ✔\n0.05 | 8 | 4 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 8 | 5 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 7 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 8 | 1.113133128439 | 1.113133129790 ✔\n0.05 | 8 | 9 | 0.000000000000 | 0.000000000074 ✔\n0.05 | 9 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 9 | 1 | 0.000000000000 | 0.000000000007 ✔\n0.05 | 9 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 9 | 3 | 0.000000000000 | 0.000000000015 ✔\n0.05 | 9 | 4 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 9 | 5 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 9 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 9 | 7 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 9 | 8 | 0.000000000000 | 0.000000000074 ✔\n0.05 | 9 | 9 | 1.119317201375 | 1.119317202034 ✔\n0.10 | 0 | 0 | 0.951350769867 | 0.951350771636 ✔\n0.10 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 1 | 1 | 1.046485846854 | 1.046485847852 ✔\n0.10 | 1 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 1 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 1 | 9 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 2 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 2 | 1.098810139196 | 1.098810140297 ✔\n0.10 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 2 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 2 | 7 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 8 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 9 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 3 | 3 | 1.135437143836 | 1.135437145012 ✔\n0.10 | 3 | 4 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 3 | 6 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 7 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 8 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 3 | 9 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 4 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 4 | 4 | 1.163823072432 | 1.163823073667 ✔\n0.10 | 4 | 5 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 4 | 6 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 7 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.10 | 4 | 9 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 5 | 4 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 5 | 5 | 1.187099533881 | 1.187099535166 ✔\n0.10 | 5 | 6 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 5 | 7 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 5 | 8 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 5 | 9 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 6 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 6 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 6 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 6 | 4 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 6 | 5 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 6 | 6 | 1.206884526112 | 1.206884527440 ✔\n0.10 | 6 | 7 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 8 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 9 | 0.000000000000 | 0.000000000065 ✔\n0.10 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 7 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 7 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 7 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 7 | 4 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 7 | 5 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 7 | 6 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 7 | 7 | 1.224125733628 | 1.224125734265 ✔\n0.10 | 7 | 8 | 0.000000000000 | 0.000000000066 ✔\n0.10 | 7 | 9 | 0.000000000000 | 0.000000000031 ✔\n0.10 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 8 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 8 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 8 | 3 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.10 | 8 | 5 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 8 | 6 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 8 | 7 | 0.000000000000 | 0.000000000066 ✔\n0.10 | 8 | 8 | 1.239427305298 | 1.239427306699 ✔\n0.10 | 8 | 9 | 0.000000000000 | 0.000000000067 ✔\n0.10 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 9 | 1 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 9 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 9 | 3 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 9 | 4 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 9 | 5 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 9 | 6 | 0.000000000000 | 0.000000000065 ✔\n0.10 | 9 | 7 | 0.000000000000 | 0.000000000031 ✔\n0.10 | 9 | 8 | 0.000000000000 | 0.000000000067 ✔\n0.10 | 9 | 9 | 1.253198719802 | 1.253198721234 ✔\n0.50 | 0 | 0 | 0.886226925453 | 0.886226925863 ✔\n0.50 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 1 | 1.329340388179 | 1.329340389103 ✔\n0.50 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 2 | 1.661675485224 | 1.661675485734 ✔\n0.50 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 3 | 1.938621399428 | 1.938621400123 ✔\n0.50 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 4 | 2.180949074356 | 2.180949075236 ✔\n0.50 | 4 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 9 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 5 | 2.399043981792 | 2.399043982856 ✔\n0.50 | 5 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 5 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 6 | 2.598964313608 | 2.598964314050 ✔\n0.50 | 6 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 6 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 6 | 9 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 7 | 7 | 2.784604621723 | 2.784604622230 ✔\n0.50 | 7 | 8 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 7 | 9 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 8 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 8 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 8 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 8 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 8 | 8 | 2.958642410581 | 2.958642412199 ✔\n0.50 | 8 | 9 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 9 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 9 | 4 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 9 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 9 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 9 | 7 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 8 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 9 | 3.123011433391 | 3.123011435194 ✔\n1.00 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n1.00 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 1 | 2.000000000000 | 2.000000000000 ✔\n1.00 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 2 | 3.000000000000 | 3.000000000000 ✔\n1.00 | 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 3 | 4.000000000000 | 4.000000000000 ✔\n1.00 | 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 4 | 5.000000000000 | 4.999999999999 ✔\n1.00 | 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 5 | 6.000000000000 | 6.000000000000 ✔\n1.00 | 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 6 | 7.000000000000 | 7.000000000000 ✔\n1.00 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 7 | 8.000000000000 | 8.000000000000 ✔\n1.00 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 8 | 9.000000000000 | 9.000000000001 ✔\n1.00 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 9 | 10.000000000000 | 10.000000000001 ✔\nTest Summary: | Pass Total\n∫Lᵢ⁽ᵅ⁾(x)Lⱼ⁽ᵅ⁾(x)xᵅexp(-x)dx = Γ(i+α+1)/i! δᵢⱼ | 500 500","category":"page"},{"location":"MorsePotential/#Normalization-and-Orthogonality-of-\\psi_n(r)","page":"Morse Potential","title":"Normalization & Orthogonality of psi_n(r)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"int_0^infty psi_i^ast(r) psi_j(r) mathrmdr = delta_ij","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000002 ✔\n 0 | 8 | 0.000000000000 | -0.000000000026 ✔\n 0 | 9 | 0.000000000000 | -0.000000000104 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | 0.000000000001 ✔\n 1 | 8 | 0.000000000000 | -0.000000000022 ✔\n 1 | 9 | 0.000000000000 | -0.000000000067 ✔\n 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000009 ✔\n 2 | 9 | 0.000000000000 | -0.000000000030 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0.000000000000 | -0.000000000001 ✔\n 3 | 8 | 0.000000000000 | -0.000000000002 ✔\n 3 | 9 | 0.000000000000 | -0.000000000006 ✔\n 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000001 ✔\n 4 | 9 | 0.000000000000 | 0.000000000001 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000001 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | -0.000000000001 ✔\n 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000002 ✔\n 6 | 9 | 0.000000000000 | -0.000000000003 ✔\n 7 | 0 | 0.000000000000 | 0.000000000002 ✔\n 7 | 1 | 0.000000000000 | 0.000000000001 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | -0.000000000001 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000001 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000004 ✔\n 8 | 0 | 0.000000000000 | -0.000000000026 ✔\n 8 | 1 | 0.000000000000 | -0.000000000022 ✔\n 8 | 2 | 0.000000000000 | -0.000000000009 ✔\n 8 | 3 | 0.000000000000 | -0.000000000002 ✔\n 8 | 4 | 0.000000000000 | -0.000000000001 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000002 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 0.999999999995 ✔\n 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 9 | 0 | 0.000000000000 | -0.000000000104 ✔\n 9 | 1 | 0.000000000000 | -0.000000000067 ✔\n 9 | 2 | 0.000000000000 | -0.000000000030 ✔\n 9 | 3 | 0.000000000000 | -0.000000000006 ✔\n 9 | 4 | 0.000000000000 | 0.000000000001 ✔\n 9 | 5 | 0.000000000000 | -0.000000000001 ✔\n 9 | 6 | 0.000000000000 | -0.000000000003 ✔\n 9 | 7 | 0.000000000000 | 0.000000000004 ✔\n 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000015 ✔\nTest Summary: | Pass Total\n<ψᵢ|ψⱼ> = δᵢⱼ | 100 100","category":"page"},{"location":"MorsePotential/#Eigen-Values-2","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n E_n\n = int psi^ast_n(r) hatH psi_n(r) mathrmdx \n = int psi^ast_n(r) left hatV + hatT right psi(r) mathrmdx \n = int psi^ast_n(r) left V(r) - frachbar^22m fracmathrmd^2mathrmd r^2 right psi(r) mathrmdx \n simeq int psi^ast_n(r) left V(r)psi(r) -frachbar^22m fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2 right mathrmdx\n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n 2psi(r)\n + fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n + Oleft(Delta r^4right)\n =\n psi(r+Delta r)\n + psi(r-Delta r)\n \n fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n =\n psi(r+Delta r)\n - 2psi(r)\n + psi(r-Delta r)\n - Oleft(Delta r^4right)\n \n fracmathrmd^2 psi(r)mathrmd r^2\n =\n fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n - fracOleft(Delta r^4right)Delta r^2\n \n fracmathrmd^2 psi(r)mathrmd r^2\n =\n fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n + Oleft(Delta r^2right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\npsi(r+Delta r)\n= psi(r)\n+ fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n+ frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\n\npsi(r-Delta r)\n= psi(r)\n- fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n- frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | -0.097482629904 | -0.097482629943 ✔\n0.1 | 1 | -0.087576629073 | -0.087576629208 ✔\n0.1 | 2 | -0.078201265005 | -0.078201265359 ✔\n0.1 | 3 | -0.069356537702 | -0.069356538266 ✔\n0.1 | 4 | -0.061042447162 | -0.061042448777 ✔\n0.1 | 5 | -0.053258993386 | -0.053258996131 ✔\n0.1 | 6 | -0.046006176374 | -0.046006177829 ✔\n0.1 | 7 | -0.039283996126 | -0.039283997743 ✔\n0.1 | 8 | -0.033092452642 | -0.033092467851 ✔\n0.1 | 9 | -0.027431545922 | -0.027431467792 ✔\n0.2 | 0 | -0.095387461081 | -0.095387461144 ✔\n0.2 | 1 | -0.081689100176 | -0.081689100427 ✔\n0.2 | 2 | -0.069052012799 | -0.069052013380 ✔\n0.2 | 3 | -0.057476198949 | -0.057476199867 ✔\n0.2 | 4 | -0.046961658628 | -0.046961660317 ✔\n0.2 | 5 | -0.037508391834 | -0.037508393202 ✔\n0.2 | 6 | -0.029116398568 | -0.029116400340 ✔\n0.2 | 7 | -0.021785678830 | -0.021785684062 ✔\n0.2 | 8 | -0.015516232619 | -0.015516237539 ✔\n0.2 | 9 | -0.010308059937 | -0.010308062755 ✔\n0.3 | 0 | -0.093795214605 | -0.093795214695 ✔\n0.3 | 1 | -0.077310338322 | -0.077310338694 ✔\n0.3 | 2 | -0.062417372330 | -0.062417373167 ✔\n0.3 | 3 | -0.049116316630 | -0.049116318029 ✔\n0.3 | 4 | -0.037407171221 | -0.037407173073 ✔\n0.3 | 5 | -0.027289936105 | -0.027289938027 ✔\n0.3 | 6 | -0.018764611280 | -0.018764613693 ✔\n0.3 | 7 | -0.011831196747 | -0.011831198102 ✔\n0.3 | 8 | -0.006489692505 | -0.006489694275 ✔\n0.3 | 9 | -0.002740098556 | -0.002740100893 ✔\n0.1 | 0 | -0.097413777944 | -0.097413777967 ✔\n0.1 | 1 | -0.087380924068 | -0.087380924205 ✔\n0.1 | 2 | -0.077893174789 | -0.077893175145 ✔\n0.1 | 3 | -0.068950530107 | -0.068950530660 ✔\n0.1 | 4 | -0.060552990023 | -0.060552989095 ✔\n0.1 | 5 | -0.052700554537 | -0.052700557255 ✔\n0.1 | 6 | -0.045393223648 | -0.045393222818 ✔\n0.1 | 7 | -0.038630997356 | -0.038631017157 ✔\n0.1 | 8 | -0.032413875662 | -0.032413886246 ✔\n0.1 | 9 | -0.026741858566 | -0.026742018376 ✔\nTest Summary: | Pass Total\n<ψₙ|H|ψₙ> = ∫ψₙ*Hψₙdx = Eₙ | 40 40\n","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"CurrentModule = Antique","category":"page"},{"location":"HydrogenAtom/#Hydrogen-Atom","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The hydrogen atom is the simplest 2-body Coulomb system.","category":"page"},{"location":"HydrogenAtom/#Definitions","page":"Hydrogen Atom","title":"Definitions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Z is the atomic number. The domains of the potential and the wave functions are 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.","category":"page"},{"location":"HydrogenAtom/#Schrödinger-Equation","page":"Hydrogen Atom","title":"Schrödinger Equation","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"HydrogenAtom/#Hamiltonian","page":"Hydrogen Atom","title":"Hamiltonian","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, mu=left(frac1m_mathrme+frac1m_mathrmpright)^-1 is the reduced mass of electron mathrme and proton mathrmp. mu = m_mathrme holds in the limit m_mathrmprightarrowinfty. ","category":"page"},{"location":"HydrogenAtom/#Potential","page":"Hydrogen Atom","title":"Potential","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"V(r; Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\n endaligned","category":"page"},{"location":"HydrogenAtom/#Eigen-Values","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"E(; n=1, Z=Z, Eₕ=Eₕ)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" E_n = -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2 = -fracZ^22n^2 E_mathrmh","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, E_mathrmh is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, E_mathrmh = 27211386245988(53)mathrmeV from here.","category":"page"},{"location":"HydrogenAtom/#Eigen-Functions","page":"Hydrogen Atom","title":"Eigen Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ψ(r, θ, φ; n=1, l=0, m=0, Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" psi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)","category":"page"},{"location":"HydrogenAtom/#Radial-Functions","page":"Hydrogen Atom","title":"Radial Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"R(r; n=1, l=0, Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" R_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_0right)^3 left(frac2Zrn a_0right)^l exp left(-fracZrn a_0right) L_n+l^2l+1 left(frac2Zrn a_0right)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Laguerre polynomials are defined as L_n(x) = frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right), and associated Laguerre polynomials are defined as L_n^k(x) = fracmathrmd^kmathrmdx^k L_n(x). Note that, replace 2n(n+l) with 2n(n+l)^3 if Laguerre polynomials are defined as L_n(x) = mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).","category":"page"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"L(x; n=0, k=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n L_n^k(x)\n = fracmathrmd^kmathrmdx^k L_n(x) \n = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n = (-1)^k L_n-k^(k)(x)\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Examples:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n L_0^0(x) = 1 \n L_1^0(x) = 1 - x \n L_1^1(x) = 1 \n L_2^0(x) = 1 - 2 x + 12 x^2 \n L_2^1(x) = 2 - x \n L_2^2(x) = 1 \n L_3^0(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^1(x) = 3 - 3 x + 12 x^2 \n L_3^2(x) = 3 - x \n L_3^3(x) = 1 \n L_4^0(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 512 x^4 \n L_4^1(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_4^2(x) = 6 - 4 x + 12 x^2 \n L_4^3(x) = 4 - x \n L_4^4(x) = 1 \n vdots\n endaligned","category":"page"},{"location":"HydrogenAtom/#Spherical-Harmonics","page":"Hydrogen Atom","title":"Spherical Harmonics","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Y(θ, φ; l=0, m=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" Y_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Note that some variants are connected by ","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"i^m+m sqrtfrac(l-m)(l+m) P_l^m = (-1)^fracm+m2 sqrtfrac(l-m)(l+m) P_l^m = (-1)^m sqrtfrac(l-m)(l+m) P_l^m","category":"page"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials","page":"Hydrogen Atom","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"P(x; n=0, m=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n P_n^m(x)\n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Legendre polynomials are defined as P_n(x) = frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right. Note that P_l^-m = (-1)^m frac(l-m)(l+m) P_l^m for m0. (It is not compatible with P_k^m(t) = (-1)^mleft( 1-t^2 right)^m2 fracmathrmd^m P_k(t)mathrmdt^m caused by (-1)^m.) The specific formulae are given below.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Examples:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n P_0^0(x) = 1 \n P_1^0(x) = x \n P_1^1(x) = left(+1right)sqrt1-x^2 \n P_2^0(x) = -12 + 32 x^2 \n P_2^1(x) = left(-3 xright)sqrt1-x^2 \n P_2^2(x) = 3 - 6 x \n P_3^0(x) = -32 x + 52 x^3 \n P_3^1(x) = left(32 - 152 x^2right)sqrt1-x^2 \n P_3^2(x) = 15 x - 30 x^2 \n P_3^3(x) = left(15 - 30 xright)sqrt1-x^2 \n P_4^0(x) = 38 - 154 x^2 + 358 x^4 \n P_4^1(x) = left(- 152 x + 352 x^3right)sqrt1-x^2 \n P_4^2(x) = -152 + 15 x + 1052 x^2 - 105 x^3 \n P_4^3(x) = left(105 x - 210 x^2right)sqrt1-x^2 \n P_4^4(x) = 105 - 420 x + 420 x^2 \n vdots\n endaligned","category":"page"},{"location":"HydrogenAtom/#References","page":"Hydrogen Atom","title":"References","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"cpprefjp, legendre, assoc_legendre, laguerre, assoc_laguerre\nThe Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.3 Table1, 18.5 Table1, 18.5.17, 18.3 Table1, 18.5 Table1, 18.5.12\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965), p.598 (c.1), p.598 (c.4), p.603 (d.13), p.603 (d.13)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968), p.79 (14.12), p.93 (16.19)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999), p.493 (B.72), p.494 Table, p.493 (B.72), p.483 (B.12), p.483 (B.12)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994), p.83 (4), p.83 (5), p.149 (21)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995), p.126 (4.28), p.96 Table3.1, p.126 (4.27), p.139 (4.88), p.140 Table4.4, p.139 (4.87), p.140 Table4.5\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997), p.195 Table6.1, p.196 (6.26), p.196 Table6.2, p.207 Table6.4\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008), p.234\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021), p.245 Problem 3.30.b, ","category":"page"},{"location":"HydrogenAtom/#Usage-and-Examples","page":"Hydrogen Atom","title":"Usage & Examples","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Antique\nH = antique(:HydrogenAtom, Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Parameters:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.Z\n1\n\njulia> H.Eₕ\n1.0\n\njulia> H.mₑ\n1.0\n\njulia> H.a₀\n1.0\n\njulia> H.ℏ\n1.0","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eigen values:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.E(n=1)\n-0.5\n\njulia> H.E(n=2)\n-0.125","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Wave length (n=2rightarrow1, the first line of the Lyman series):","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eₕ2nm⁻¹ = 2.1947463136320e-2 # https://physics.nist.gov/cgi-bin/cuu/CCValue?hrminv\nprintln(\"ΔE = \", H.E(n=2) - H.E(n=1), \" Eₕ\")\nprintln(\"λ = \", ((H.E(n=2)-H.E(n=1))*Eₕ2nm⁻¹)^-1, \" nm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ΔE = 0.375 Eₕ\nλ = 121.50227341098497 nm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Hyperfine Splitting:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"# constants: https://doi.org/10.1103/RevModPhys.93.025010\ne = 1.602176634e-19 # C https://physics.nist.gov/cgi-bin/cuu/Value?e\nh = 6.62607015e-34 # J Hz-1 https://physics.nist.gov/cgi-bin/cuu/Value?h\nc = 299792458 # m s-1 https://physics.nist.gov/cgi-bin/cuu/Value?c\na0 = 5.29177210903e-11 # m https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0\nµ0 = 1.25663706212e-6 # N A-2 https://physics.nist.gov/cgi-bin/cuu/Value?mu0\nµB = 9.2740100783e-24 # J T-1 https://physics.nist.gov/cgi-bin/cuu/Value?mub\nµN = 5.0507837461e-27 # J T-1 https://physics.nist.gov/cgi-bin/cuu/Value?mun\nge = 2.00231930436256 # https://physics.nist.gov/cgi-bin/cuu/Value?gem\ngp = 5.5856946893 # https://physics.nist.gov/cgi-bin/cuu/Value?gp\n\n# calculation: https://doi.org/10.1119/1.12733\nδ = abs(H.ψ(0,0,0))^2\nΔE = 2 / 3 * µ0 * µN * µB * gp * ge * δ * a0^(-3)\nprintln(\"1/π = \", 1/π)\nprintln(\"<δ(r)> = \", δ, \" a₀⁻³\")\nprintln(\"<δ(r)> = \", δ * a0^(-3), \" m⁻³\")\nprintln(\"ΔE = \", ΔE, \" J\")\nprintln(\"ν = ΔE/h = \", ΔE / h * 1e-6, \" MHz\")\nprintln(\"λ = hc/ΔE = \", h*c/ΔE*100, \" cm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"1/π = 0.3183098861837907\n<δ(r)> = 0.3183098861837908 a₀⁻³\n<δ(r)> = 2.1480615849063944e30 m⁻³\nΔE = 9.427622831641132e-25 J\nν = ΔE/h = 1422.8075794882932 MHz\nλ = hc/ΔE = 21.070485027063118 cm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h},~E_n/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nplot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve, Energy levels:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nfor n in 0:10\n plot!(0.0:0.01:15, r -> H.E(n=n) > H.V(r) ? H.E(n=n) : NaN, lc=n, lw=1, label=\"\") # energy level\nend\nplot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Radial functions:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\\$\", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=300)\nfor n in 1:3\n for l in 0:n-1\n plot!(0:0.01:20, r->r^2*H.R(r,n=n,l=l)^2, lc=n, lw=2, ls=[:solid,:dash,:dot,:dashdot,:dashdotdot][l+1], label=\"\\$n = $n, l=$l\\$\")\n end\nend\nplot!()","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/#Testing","page":"Hydrogen Atom","title":"Testing","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials-P_nm(x)","page":"Hydrogen Atom","title":"Associated Legendre Polynomials P_n^m(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n P_n^m(x)\n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=0 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_0^0(x)\n = 1\n = 1 \n = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_1^0(x)\n = frac12 fracmathrmdmathrmdx left( -1 + x^2 right)\n = x \n = x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_1^1(x)\n = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac12 fracmathrmdmathrmdx left( -1 + x^2 right)\n = left( 1 - x^2 right)^frac12 \n = left( 1 - x^2 right)^frac12\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_2^0(x)\n = frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n = frac-12 + frac32 x^2 \n = frac-12 + frac32 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_2^1(x)\n = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n = 3 left( 1 - x^2 right)^frac12 x \n = 3 left( 1 - x^2 right)^frac12 x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_2^2(x)\n = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n = 3 - 3 x^2 \n = 3 - 3 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_3^0(x)\n = frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n = - frac32 x + frac52 x^3 \n = - frac32 x + frac52 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_3^1(x)\n = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n = - frac32 left( 1 - x^2 right)^frac12 + frac152 left( 1 - x^2 right)^frac12 x^2 \n = - frac32 left( 1 - x^2 right)^frac12 + frac152 left( 1 - x^2 right)^frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_3^2(x)\n = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n = 15 x - 15 x^3 \n = 15 x - 15 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_3^3(x)\n = left( 1 - x^2 right)^frac32 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n = 15 left( 1 - x^2 right)^frac32 \n = 15 left( 1 - x^2 right)^frac32\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^0(x)\n = frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = frac38 - frac154 x^2 + frac358 x^4 \n = frac38 - frac154 x^2 + frac358 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^1(x)\n = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = - frac152 left( 1 - x^2 right)^frac12 x + frac352 left( 1 - x^2 right)^frac12 x^3 \n = - frac152 left( 1 - x^2 right)^frac12 x + frac352 left( 1 - x^2 right)^frac12 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^2(x)\n = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = frac-152 + 60 x^2 - frac1052 x^4 \n = frac-152 + 60 x^2 - frac1052 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^3(x)\n = left( 1 - x^2 right)^frac32 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = 105 left( 1 - x^2 right)^frac32 x \n = 105 left( 1 - x^2 right)^frac32 x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=4 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n P_4^4(x)\n = left( 1 - x^2 right)^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n = 105 left( 1 - x^2 right)^2 \n = 105 left( 1 - x^2 right)^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Test Summary: | Pass Total\nPₙᵐ(x) = √(1-x²)ᵐ dᵐ/dxᵐ Pₙ(x); Pₙ(x) = 1/(2ⁿn!) dⁿ/dxⁿ (x²-1)ⁿ | 15 15","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-P_nm(x)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of P_n^m(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_-1^1 P_i^m(x) P_j^m(x) mathrmdx = frac2(j+m)(2j+1)(j-m) delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" m | i | j | analytical | numerical \n-- | -- | -- | ----------------- | ----------------- \n 0 | 0 | 0 | 2.000000000000 | 2.000000000000 ✔\n 0 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 1 | 0.666666666667 | 0.666666666667 ✔\n 0 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 2 | 0.400000000000 | 0.400000000000 ✔\n 0 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 3 | 0.285714285714 | 0.285714285714 ✔\n 0 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 4 | 0.222222222222 | 0.222222222222 ✔\n 0 | 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 0 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n 0 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 5 | 0.181818181818 | 0.181818181818 ✔\n 0 | 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 6 | 0.153846153846 | 0.153846153846 ✔\n 0 | 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 5 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 7 | 0.133333333333 | 0.133333333333 ✔\n 0 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 0 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 8 | 0.117647058824 | 0.117647058824 ✔\n 0 | 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 0 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 0 | 9 | 9 | 0.105263157895 | 0.105263157895 ✔\n 1 | 1 | 1 | 1.333333333333 | 1.333333333333 ✔\n 1 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 2 | 2.400000000000 | 2.400000000000 ✔\n 1 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 3 | 3.428571428571 | 3.428571428571 ✔\n 1 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 4 | 4.444444444444 | 4.444444444444 ✔\n 1 | 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 5 | 5.454545454545 | 5.454545454545 ✔\n 1 | 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 6 | 6.461538461538 | 6.461538461538 ✔\n 1 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 7 | 7.466666666667 | 7.466666666667 ✔\n 1 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 8 | 8.470588235294 | 8.470588235294 ✔\n 1 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 9 | 9.473684210526 | 9.473684210526 ✔\n 2 | 2 | 2 | 9.600000000000 | 9.600000000000 ✔\n 2 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 3 | 34.285714285714 | 34.285714285714 ✔\n 2 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 4 | 80.000000000000 | 80.000000000000 ✔\n 2 | 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 5 | 152.727272727273 | 152.727272727273 ✔\n 2 | 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 6 | 258.461538461538 | 258.461538461538 ✔\n 2 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 7 | 403.200000000000 | 403.200000000000 ✔\n 2 | 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 8 | 8 | 592.941176470588 | 592.941176470588 ✔\n 2 | 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 9 | 833.684210526316 | 833.684210526316 ✔\n 3 | 3 | 3 | 205.714285714286 | 205.714285714286 ✔\n 3 | 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 9 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 3 | 4 | 4 | 1120.000000000000 | 1120.000000000000 ✔\n 3 | 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 3 | 5 | 5 | 3665.454545454545 | 3665.454545454545 ✔\n 3 | 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 6 | 9304.615384615385 | 9304.615384615387 ✔\n 3 | 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 9 | 0.000000000000 | -0.000000000002 ✔\n 3 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 7 | 20160.000000000000 | 20160.000000000004 ✔\n 3 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 9 | 0.000000000000 | -0.000000000003 ✔\n 3 | 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 8 | 39134.117647058825 | 39134.117647058825 ✔\n 3 | 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 6 | 0.000000000000 | -0.000000000002 ✔\n 3 | 9 | 7 | 0.000000000000 | -0.000000000003 ✔\n 3 | 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 9 | 70029.473684210534 | 70029.473684210505 ✔\n 4 | 4 | 4 | 8960.000000000000 | 8960.000000000002 ✔\n 4 | 4 | 5 | 0.000000000000 | -0.000000000002 ✔\n 4 | 4 | 6 | 0.000000000000 | -0.000000000001 ✔\n 4 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 8 | 0.000000000000 | 0.000000000007 ✔\n 4 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 4 | 0.000000000000 | -0.000000000002 ✔\n 4 | 5 | 5 | 65978.181818181823 | 65978.181818181838 ✔\n 4 | 5 | 6 | 0.000000000000 | -0.000000000001 ✔\n 4 | 5 | 7 | 0.000000000000 | -0.000000000058 ✔\n 4 | 5 | 8 | 0.000000000000 | -0.000000000002 ✔\n 4 | 5 | 9 | 0.000000000000 | -0.000000000007 ✔\n 4 | 6 | 4 | 0.000000000000 | -0.000000000001 ✔\n 4 | 6 | 5 | 0.000000000000 | -0.000000000001 ✔\n 4 | 6 | 6 | 279138.461538461561 | 279138.461538461503 ✔\n 4 | 6 | 7 | 0.000000000000 | -0.000000000018 ✔\n 4 | 6 | 8 | 0.000000000000 | 0.000000000055 ✔\n 4 | 6 | 9 | 0.000000000000 | 0.000000000029 ✔\n 4 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 5 | 0.000000000000 | -0.000000000058 ✔\n 4 | 7 | 6 | 0.000000000000 | -0.000000000018 ✔\n 4 | 7 | 7 | 887040.000000000000 | 887040.000000000000 ✔\n 4 | 7 | 8 | 0.000000000000 | 0.000000000031 ✔\n 4 | 7 | 9 | 0.000000000000 | 0.000000000104 ✔\n 4 | 8 | 4 | 0.000000000000 | 0.000000000007 ✔\n 4 | 8 | 5 | 0.000000000000 | -0.000000000002 ✔\n 4 | 8 | 6 | 0.000000000000 | 0.000000000055 ✔\n 4 | 8 | 7 | 0.000000000000 | 0.000000000031 ✔\n 4 | 8 | 8 | 2348047.058823529165 | 2348047.058823529631 ✔\n 4 | 8 | 9 | 0.000000000000 | -0.000000000015 ✔\n 4 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 5 | 0.000000000000 | -0.000000000007 ✔\n 4 | 9 | 6 | 0.000000000000 | 0.000000000029 ✔\n 4 | 9 | 7 | 0.000000000000 | 0.000000000104 ✔\n 4 | 9 | 8 | 0.000000000000 | -0.000000000015 ✔\n 4 | 9 | 9 | 5462298.947368421592 | 5462298.947368418798 ✔\n 5 | 5 | 5 | 659781.818181818235 | 659781.818181818351 ✔\n 5 | 5 | 6 | 0.000000000000 | -0.000000000002 ✔\n 5 | 5 | 7 | 0.000000000000 | 0.000000000233 ✔\n 5 | 5 | 8 | 0.000000000000 | 0.000000000567 ✔\n 5 | 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 5 | 0.000000000000 | -0.000000000002 ✔\n 5 | 6 | 6 | 6141046.153846153989 | 6141046.153846156783 ✔\n 5 | 6 | 7 | 0.000000000000 | 0.000000000250 ✔\n 5 | 6 | 8 | 0.000000000000 | 0.000000001630 ✔\n 5 | 6 | 9 | 0.000000000000 | 0.000000000931 ✔\n 5 | 7 | 5 | 0.000000000000 | 0.000000000233 ✔\n 5 | 7 | 6 | 0.000000000000 | 0.000000000250 ✔\n 5 | 7 | 7 | 31933440.000000000000 | 31933440.000000000000 ✔\n 5 | 7 | 8 | 0.000000000000 | 0.000000002503 ✔\n 5 | 7 | 9 | 0.000000000000 | 0.000000003725 ✔\n 5 | 8 | 5 | 0.000000000000 | 0.000000000567 ✔\n 5 | 8 | 6 | 0.000000000000 | 0.000000001630 ✔\n 5 | 8 | 7 | 0.000000000000 | 0.000000002503 ✔\n 5 | 8 | 8 | 122098447.058823525906 | 122098447.058823525906 ✔\n 5 | 8 | 9 | 0.000000000000 | -0.000000001397 ✔\n 5 | 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 6 | 0.000000000000 | 0.000000000931 ✔\n 5 | 9 | 7 | 0.000000000000 | 0.000000003725 ✔\n 5 | 9 | 8 | 0.000000000000 | -0.000000001397 ✔\n 5 | 9 | 9 | 382360926.315789461136 | 382360926.315789461136 ✔\nTest Summary: | Pass Total\n∫Pᵢᵐ(x)Pⱼᵐ(x)dx = 2(j+m)!/(2j+1)(j-m)! δᵢⱼ | 355 355","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-Y_{lm}(\\theta,\\varphi)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of Y_lm(thetavarphi)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_0^2pi\nint_0^pi\nY_lm(thetavarphi)^* Y_lm(thetavarphi) sin(theta)\nmathrmdtheta mathrmdvarphi\n= delta_ll delta_mm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"l₁ | l₂ | m₁ | m₂ | analytical | numerical \n-- | -- | -- | -- | ----------------- | ----------------- \n 0 | 0 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 0 | 2 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 0 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | -1 | -1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 1 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 1 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 1 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 1 | 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | -1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | -1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 0 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 2 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | -2 | -2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | -2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | -2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | -2 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | -1 | -1 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | -1 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 0 | -2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | -2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 2 | -2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 2 | 2 | 1.000000000000 | 1.000000000000 ✔\nTest Summary: | Pass Total\n∫Yₗ₁ₘ₁(θ,φ)Yₗ₂ₘ₂(θ,φ)sinθdθdφ = δₗ₁ₗ₂δₘ₁ₘ₂ | 81 81","category":"page"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials-L_n{k}(x)","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials L_n^k(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n L_n^k(x)\n = fracmathrmd^kmathrmdx^k L_n(x) \n = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n = (-1)^k L_n-k^(k)(x)\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=0 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_0^0(x)\n = e^x e^ - x\n = 1 \n = 1 \n = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_1^0(x)\n = e^x fracmathrmdmathrmdx x e^ - x\n = 1 - x \n = 1 - x \n = 1 - x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_1^1(x)\n = fracmathrmdmathrmdx e^x fracmathrmdmathrmdx x e^ - x\n = -1 \n = -1 \n = -1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_2^0(x)\n = frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n = 1 - 2 x + frac12 x^2 \n = 1 - 2 x + frac12 x^2 \n = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_2^1(x)\n = fracmathrmdmathrmdx frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n = -2 + x \n = -2 + x \n = -2 + x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_2^2(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n = 1 \n = 1 \n = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_3^0(x)\n = frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = 1 - frac16 x^3 - 3 x + frac32 x^2 \n = 1 - frac16 x^3 - 3 x + frac32 x^2 \n = 1 - frac16 x^3 - 3 x + frac32 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_3^1(x)\n = fracmathrmdmathrmdx frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = -3 + 3 x - frac12 x^2 \n = -3 + 3 x - frac12 x^2 \n = -3 + 3 x - frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_3^2(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = 3 - x \n = 3 - x \n = 3 - x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_3^3(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n = -1 \n = -1 \n = -1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^0(x)\n = frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4 \n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4 \n = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^1(x)\n = fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = -4 + frac16 x^3 + 6 x - 2 x^2 \n = -4 + frac16 x^3 + 6 x - 2 x^2 \n = -4 + frac16 x^3 + 6 x - 2 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^2(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = 6 - 4 x + frac12 x^2 \n = 6 - 4 x + frac12 x^2 \n = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^3(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = -4 + x \n = -4 + x \n = -4 + x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=4 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n L_4^4(x)\n = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n = 1 \n = 1 \n = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Test Summary: | Pass Total\nLₙᵏ(x) = dᵏ/dxᵏ Lₙ(x); Lₙ(x) = 1/(n!) eˣ dⁿ/dxⁿ e⁻ˣ xⁿ | 15 15","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-L_n{k}(x)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of L_n^k(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_0^infty mathrme^-x x^k L_i^k(x) L_j^k(x) mathrmdx = fraci(i-k) delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Replace n+k with n for the definition of Wolfram MathWorld.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" i | j | k | analytical | numerical \n-- | -- | -- | ----------------- | ----------------- \n 0 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 0 | 1.000000000000 | 1.000000000000 ✔\n 1 | 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 1.000000000000 | 1.000000000000 ✔\n 2 | 2 | 1 | 2.000000000000 | 2.000000000000 ✔\n 2 | 2 | 2 | 2.000000000000 | 2.000000000000 ✔\n 2 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 0 | 1.000000000000 | 1.000000000000 ✔\n 3 | 3 | 1 | 3.000000000000 | 3.000000000000 ✔\n 3 | 3 | 2 | 6.000000000000 | 6.000000000000 ✔\n 3 | 3 | 3 | 6.000000000000 | 6.000000000000 ✔\n 3 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 0 | 1.000000000000 | 1.000000000000 ✔\n 4 | 4 | 1 | 4.000000000000 | 4.000000000000 ✔\n 4 | 4 | 2 | 12.000000000000 | 12.000000000000 ✔\n 4 | 4 | 3 | 24.000000000000 | 24.000000000000 ✔\n 4 | 4 | 4 | 24.000000000000 | 24.000000000000 ✔\n 4 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 4 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 5 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 0 | 1.000000000000 | 1.000000000000 ✔\n 5 | 5 | 1 | 5.000000000000 | 4.999999999999 ✔\n 5 | 5 | 2 | 20.000000000000 | 20.000000000000 ✔\n 5 | 5 | 3 | 60.000000000000 | 60.000000000000 ✔\n 5 | 5 | 4 | 120.000000000000 | 120.000000000000 ✔\n 5 | 5 | 5 | 120.000000000000 | 120.000000000000 ✔\n 5 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 6 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 2 | 0.000000000000 | -0.000000000000 ✔\n 6 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 0 | 1.000000000000 | 1.000000000000 ✔\n 6 | 6 | 1 | 6.000000000000 | 6.000000000000 ✔\n 6 | 6 | 2 | 30.000000000000 | 30.000000000000 ✔\n 6 | 6 | 3 | 120.000000000000 | 119.999999999978 ✔\n 6 | 6 | 4 | 360.000000000000 | 359.999999999996 ✔\n 6 | 6 | 5 | 720.000000000000 | 720.000000000000 ✔\n 6 | 6 | 6 | 720.000000000000 | 720.000000000000 ✔\n 6 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 2 | 0.000000000000 | -0.000000000000 ✔\n 6 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n 7 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 7 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔\n 7 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 7 | 3 | 3 | 0.000000000000 | -0.000000000000 ✔\n 7 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 7 | 5 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 6 | 0.000000000000 | 0.000000000000 ✔\n 7 | 7 | 0 | 1.000000000000 | 1.000000000000 ✔\n 7 | 7 | 1 | 7.000000000000 | 7.000000000000 ✔\n 7 | 7 | 2 | 42.000000000000 | 42.000000000000 ✔\n 7 | 7 | 3 | 210.000000000000 | 210.000000000000 ✔\n 7 | 7 | 4 | 840.000000000000 | 840.000000000000 ✔\n 7 | 7 | 5 | 2520.000000000000 | 2519.999999999775 ✔\n 7 | 7 | 6 | 5040.000000000000 | 5039.999999999985 ✔\n 7 | 7 | 7 | 5040.000000000000 | 5040.000000000000 ✔\nTest Summary: | Pass Total\n∫exp(-x)xᵏLᵢᵏ(x)Lⱼᵏ(x)dx = (2i+k)!/(i+k)! δᵢⱼ | 204 204","category":"page"},{"location":"HydrogenAtom/#Normalization-of-R_{nl}(r)","page":"Hydrogen Atom","title":"Normalization of R_nl(r)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int R_nl(r)^2 r^2 mathrmdr = 1","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" n | l | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 0 | 1.000000000000 | 1.000000000000 ✔\n 2 | 0 | 1.000000000000 | 1.000000000000 ✔\n 2 | 1 | 1.000000000000 | 1.000000000000 ✔\n 3 | 0 | 1.000000000000 | 1.000000000000 ✔\n 3 | 1 | 1.000000000000 | 0.999999999999 ✔\n 3 | 2 | 1.000000000000 | 1.000000000000 ✔\n 4 | 0 | 1.000000000000 | 1.000000000000 ✔\n 4 | 1 | 1.000000000000 | 1.000000000000 ✔\n 4 | 2 | 1.000000000000 | 1.000000000000 ✔\n 4 | 3 | 1.000000000000 | 1.000000000000 ✔\n 5 | 0 | 1.000000000000 | 1.000000000000 ✔\n 5 | 1 | 1.000000000000 | 1.000000000000 ✔\n 5 | 2 | 1.000000000000 | 1.000000000000 ✔\n 5 | 3 | 1.000000000000 | 1.000000000000 ✔\n 5 | 4 | 1.000000000000 | 1.000000000000 ✔\n 6 | 0 | 1.000000000000 | 1.000000000000 ✔\n 6 | 1 | 1.000000000000 | 1.000000000000 ✔\n 6 | 2 | 1.000000000000 | 1.000000000000 ✔\n 6 | 3 | 1.000000000000 | 1.000000000000 ✔\n 6 | 4 | 1.000000000000 | 1.000000000000 ✔\n 6 | 5 | 1.000000000000 | 1.000000000000 ✔\n 7 | 0 | 1.000000000000 | 1.000000000000 ✔\n 7 | 1 | 1.000000000000 | 1.000000000000 ✔\n 7 | 2 | 1.000000000000 | 1.000000000000 ✔\n 7 | 3 | 1.000000000000 | 1.000000000000 ✔\n 7 | 4 | 1.000000000000 | 1.000000000000 ✔\n 7 | 5 | 1.000000000000 | 1.000000000000 ✔\n 7 | 6 | 1.000000000000 | 1.000000000000 ✔\n 8 | 0 | 1.000000000000 | 1.000000000000 ✔\n 8 | 1 | 1.000000000000 | 1.000000000000 ✔\n 8 | 2 | 1.000000000000 | 1.000000000000 ✔\n 8 | 3 | 1.000000000000 | 1.000000000000 ✔\n 8 | 4 | 1.000000000000 | 1.000000000000 ✔\n 8 | 5 | 1.000000000000 | 1.000000000000 ✔\n 8 | 6 | 1.000000000000 | 1.000000000000 ✔\n 8 | 7 | 1.000000000000 | 1.000000000000 ✔\n 9 | 0 | 1.000000000000 | 1.000000000000 ✔\n 9 | 1 | 1.000000000000 | 1.000000000000 ✔\n 9 | 2 | 1.000000000000 | 1.000000000000 ✔\n 9 | 3 | 1.000000000000 | 1.000000000000 ✔\n 9 | 4 | 1.000000000000 | 1.000000000000 ✔\n 9 | 5 | 1.000000000000 | 1.000000000000 ✔\n 9 | 6 | 1.000000000000 | 1.000000000000 ✔\n 9 | 7 | 1.000000000000 | 1.000000000000 ✔\n 9 | 8 | 1.000000000000 | 1.000000000000 ✔\nTest Summary: | Pass Total\n∫|Rₙₗ(r)|²r²dr = δₙ₁ₙ₂δₗ₁ₗ₂ | 45 45","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-r","page":"Hydrogen Atom","title":"Expected Value of r","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle r rangle\n= int r R_n_1 l_1(r)^2 r^2 mathrmdr\n= fraca_mu2Z left 3n^2 - l(l+1) right \na_mu = a_0 fracm_mathrmemu \nfrac1mu = frac1m_mathrme + frac1m_mathrmp","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22\n Quan­tum Me­chan­ics for En­gi­neers by Leon van Dom­me­len","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" n | l | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 0 | 1.500000000000 | 1.500000000000 ✔\n 2 | 0 | 6.000000000000 | 6.000000000000 ✔\n 2 | 1 | 5.000000000000 | 5.000000000000 ✔\n 3 | 0 | 13.500000000000 | 13.500000000000 ✔\n 3 | 1 | 12.500000000000 | 12.500000000000 ✔\n 3 | 2 | 10.500000000000 | 10.500000000000 ✔\n 4 | 0 | 24.000000000000 | 23.999999999999 ✔\n 4 | 1 | 23.000000000000 | 22.999999999999 ✔\n 4 | 2 | 21.000000000000 | 21.000000000000 ✔\n 4 | 3 | 18.000000000000 | 18.000000000000 ✔\n 5 | 0 | 37.500000000000 | 37.500000000000 ✔\n 5 | 1 | 36.500000000000 | 36.500000000000 ✔\n 5 | 2 | 34.500000000000 | 34.500000000000 ✔\n 5 | 3 | 31.500000000000 | 31.500000000000 ✔\n 5 | 4 | 27.500000000000 | 27.499999999943 ✔\n 6 | 0 | 54.000000000000 | 54.000000000001 ✔\n 6 | 1 | 53.000000000000 | 53.000000000001 ✔\n 6 | 2 | 51.000000000000 | 51.000000000000 ✔\n 6 | 3 | 48.000000000000 | 48.000000000000 ✔\n 6 | 4 | 44.000000000000 | 44.000000000000 ✔\n 6 | 5 | 39.000000000000 | 39.000000000000 ✔\n 7 | 0 | 73.500000000000 | 73.500000000000 ✔\n 7 | 1 | 72.500000000000 | 72.500000000000 ✔\n 7 | 2 | 70.500000000000 | 70.500000000000 ✔\n 7 | 3 | 67.500000000000 | 67.500000000000 ✔\n 7 | 4 | 63.500000000000 | 63.500000000000 ✔\n 7 | 5 | 58.500000000000 | 58.500000000000 ✔\n 7 | 6 | 52.500000000000 | 52.499999999992 ✔\n 8 | 0 | 96.000000000000 | 96.000000000001 ✔\n 8 | 1 | 95.000000000000 | 94.999999999999 ✔\n 8 | 2 | 93.000000000000 | 93.000000000000 ✔\n 8 | 3 | 90.000000000000 | 90.000000000000 ✔\n 8 | 4 | 86.000000000000 | 86.000000000000 ✔\n 8 | 5 | 81.000000000000 | 81.000000000000 ✔\n 8 | 6 | 75.000000000000 | 75.000000000000 ✔\n 8 | 7 | 68.000000000000 | 68.000000000000 ✔\n 9 | 0 | 121.500000000000 | 121.500000000001 ✔\n 9 | 1 | 120.500000000000 | 120.500000000000 ✔\n 9 | 2 | 118.500000000000 | 118.500000000001 ✔\n 9 | 3 | 115.500000000000 | 115.500000000000 ✔\n 9 | 4 | 111.500000000000 | 111.499999999998 ✔\n 9 | 5 | 106.500000000000 | 106.499999999999 ✔\n 9 | 6 | 100.500000000000 | 100.500000000000 ✔\n 9 | 7 | 93.500000000000 | 93.500000000000 ✔\n 9 | 8 | 85.500000000000 | 85.500000000000 ✔\nTest Summary: | Pass Total\n∫r|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)/2Z × [3n²-l(l+1)]; 1/μ = 1/mₑ + 1/mₚ | 45 45","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-r2","page":"Hydrogen Atom","title":"Expected Value of r^2","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle r^2 rangle\n= int r^2 R_n_1 l_1(r)^2 r^2 mathrmdr\n= fraca_mu^22Z^2 n^2 left 5n^2 + 1 - 3l(l+1) right \na_mu = a_0 fracm_mathrmemu \nfrac1mu = frac1m_mathrme + frac1m_mathrmp","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22\n Quan­tum Me­chan­ics for En­gi­neers by Leon van Dom­me­len","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" n | l | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 0 | 3.000000000000 | 3.000000000000 ✔\n 2 | 0 | 42.000000000000 | 42.000000000000 ✔\n 2 | 1 | 30.000000000000 | 30.000000000000 ✔\n 3 | 0 | 207.000000000000 | 207.000000000000 ✔\n 3 | 1 | 180.000000000000 | 180.000000000000 ✔\n 3 | 2 | 126.000000000000 | 126.000000000000 ✔\n 4 | 0 | 648.000000000000 | 647.999999999903 ✔\n 4 | 1 | 600.000000000000 | 599.999999999936 ✔\n 4 | 2 | 504.000000000000 | 503.999999999975 ✔\n 4 | 3 | 360.000000000000 | 359.999999999996 ✔\n 5 | 0 | 1575.000000000000 | 1574.999999999999 ✔\n 5 | 1 | 1500.000000000000 | 1499.999999999998 ✔\n 5 | 2 | 1350.000000000000 | 1350.000000000000 ✔\n 5 | 3 | 1125.000000000000 | 1125.000000000003 ✔\n 5 | 4 | 825.000000000000 | 825.000000000000 ✔\n 6 | 0 | 3258.000000000000 | 3257.999999999997 ✔\n 6 | 1 | 3150.000000000000 | 3149.999999999992 ✔\n 6 | 2 | 2934.000000000000 | 2933.999999999998 ✔\n 6 | 3 | 2610.000000000000 | 2610.000000000033 ✔\n 6 | 4 | 2178.000000000000 | 2178.000000000008 ✔\n 6 | 5 | 1638.000000000000 | 1638.000000000000 ✔\n 7 | 0 | 6027.000000000000 | 6026.999999999992 ✔\n 7 | 1 | 5880.000000000000 | 5880.000000000003 ✔\n 7 | 2 | 5586.000000000000 | 5585.999999999990 ✔\n 7 | 3 | 5145.000000000000 | 5144.999999999992 ✔\n 7 | 4 | 4557.000000000000 | 4556.999999999997 ✔\n 7 | 5 | 3822.000000000000 | 3821.999999999999 ✔\n 7 | 6 | 2940.000000000000 | 2940.000000000001 ✔\n 8 | 0 | 10272.000000000000 | 10272.000000000029 ✔\n 8 | 1 | 10080.000000000000 | 10079.999999999995 ✔\n 8 | 2 | 9696.000000000000 | 9695.999999999993 ✔\n 8 | 3 | 9120.000000000000 | 9120.000000000011 ✔\n 8 | 4 | 8352.000000000000 | 8352.000000000002 ✔\n 8 | 5 | 7392.000000000000 | 7392.000000000010 ✔\n 8 | 6 | 6240.000000000000 | 6240.000000000000 ✔\n 8 | 7 | 4896.000000000000 | 4896.000000000008 ✔\n 9 | 0 | 16443.000000000000 | 16443.000000000102 ✔\n 9 | 1 | 16200.000000000000 | 16200.000000000040 ✔\n 9 | 2 | 15714.000000000000 | 15714.000000000149 ✔\n 9 | 3 | 14985.000000000000 | 14984.999999999918 ✔\n 9 | 4 | 14013.000000000000 | 14012.999999999545 ✔\n 9 | 5 | 12798.000000000000 | 12797.999999999807 ✔\n 9 | 6 | 11340.000000000000 | 11339.999999999945 ✔\n 9 | 7 | 9639.000000000000 | 9638.999999999991 ✔\n 9 | 8 | 7695.000000000000 | 7694.999999999998 ✔\nTest Summary: | Pass Total\n∫r²|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)²/2Z² × n²[5n²+1-3l(l+1)]; 1/μ = 1/mₑ + 1/mₚ | 45 45","category":"page"},{"location":"HydrogenAtom/#Virial-Theorem","page":"Hydrogen Atom","title":"Virial Theorem","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The virial theorem 2langle T rangle + langle V rangle = 0 and the definition of Hamiltonian langle H rangle = langle T rangle + langle V rangle derive langle H rangle = frac12 langle V rangle and langle H rangle = -langle T rangle.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"frac12 int psi_n^ast(x) V(x) psi_n(x) mathrmdx = E_n","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" n | analytical | numerical \n-- | ----------------- | ----------------- \n 1 | -1.000000000000 | -1.000000000000 ✔\n 2 | -0.250000000000 | -0.250000000000 ✔\n 3 | -0.111111111111 | -0.111111111111 ✔\n 4 | -0.062500000000 | -0.062500000000 ✔\n 5 | -0.040000000000 | -0.040000000000 ✔\n 6 | -0.027777777778 | -0.027777777778 ✔\n 7 | -0.020408163265 | -0.020408163265 ✔\n 8 | -0.015625000000 | -0.015625000000 ✔\n 9 | -0.012345679012 | -0.012345679012 ✔\n10 | -0.010000000000 | -0.010000000000 ✔\nTest Summary: | Pass Total\n<ψₙ|V|ψₙ> / 2 = Eₙ | 10 10","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-\\psi_n(r,\\theta,\\varphi)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of psi_n(rthetavarphi)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int psi_i^ast(rthetavarphi) psi_j(rthetavarphi) r^2 sin(theta) mathrmdr mathrmdtheta mathrmdvarphi = delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n₁ | n₂ | l₁ | l₂ | m₁ | m₂ | analytical | numerical \n-- | -- | -- | -- | -- | -- | ----------------- | ----------------- \n 1 | 1 | 0 | 0 | 0 | 0 | 1.000000000000 | 1.000000000252 ✔\n 1 | 2 | 0 | 0 | 0 | 0 | 0.000000000000 | -0.000000011223 ✔\n 1 | 2 | 0 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 0 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 2 | 0 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 0 | 0 | 0 | 0.000000000000 | -0.000000045661 ✔\n 1 | 3 | 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | -2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0 | 2 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0 | 0 | 0 | 0 | 0.000000000000 | -0.000000011223 ✔\n 2 | 1 | 1 | 0 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 0 | 0 | 0 | 1.000000000000 | 1.000006970517 ✔\n 2 | 2 | 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 0 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 0 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | -1 | -1 | 1.000000000000 | 1.000002301351 ✔\n 2 | 2 | 1 | 1 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | 0 | 0 | 1.000000000000 | 1.000002301351 ✔\n 2 | 2 | 1 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1 | 1 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1 | 1 | 1 | 1 | 1.000000000000 | 1.000002301351 ✔\n 2 | 3 | 0 | 0 | 0 | 0 | 0.000000000000 | 0.000088519421 ✔\n 2 | 3 | 0 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 0 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 1 | -1 | -1 | 0.000000000000 | 0.000038730338 ✔\n 2 | 3 | 1 | 1 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 1 | 0 | 0 | 0.000000000000 | 0.000038730338 ✔\n 2 | 3 | 1 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 1 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 1 | 1 | 1 | 0.000000000000 | 0.000038730338 ✔\n 2 | 3 | 1 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | -1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | -1 | 2 | 0.000000000000 | -0.000000000272 ✔\n 2 | 3 | 1 | 2 | 0 | -2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | 1 | -2 | 0.000000000000 | 0.000000000272 ✔\n 2 | 3 | 1 | 2 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 3 | 1 | 2 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 3 | 1 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0 | 0 | 0 | 0 | 0.000000000000 | -0.000000045661 ✔\n 3 | 1 | 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 1 | 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 2 | 0 | -2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 2 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 2 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 1 | 2 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 1 | 2 | 0 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0 | 0 | 0 | 0 | 0.000000000000 | 0.000088519421 ✔\n 3 | 2 | 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 0 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 1 | 1 | -1 | -1 | 0.000000000000 | 0.000038730338 ✔\n 3 | 2 | 1 | 1 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 1 | 1 | 0 | 0 | 0.000000000000 | 0.000038730338 ✔\n 3 | 2 | 1 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 1 | 1 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 1 | 1 | 1 | 1 | 0.000000000000 | 0.000038730338 ✔\n 3 | 2 | 2 | 0 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 0 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | -2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | -2 | 1 | 0.000000000000 | 0.000000000272 ✔\n 3 | 2 | 2 | 1 | -1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 2 | 1 | 2 | -1 | 0.000000000000 | -0.000000000272 ✔\n 3 | 2 | 2 | 1 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 2 | 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 0 | 0 | 0 | 1.000000000000 | 1.002052594504 ✔\n 3 | 3 | 0 | 1 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 1 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 1 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | -2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 0 | 2 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 0 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | -1 | -1 | 1.000000000000 | 1.001223346388 ✔\n 3 | 3 | 1 | 1 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | 0 | 0 | 1.000000000000 | 1.001223346388 ✔\n 3 | 3 | 1 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 1 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 1 | 1 | 1 | 1.000000000000 | 1.001223346388 ✔\n 3 | 3 | 1 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | -1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | -1 | 2 | 0.000000000000 | 0.000000000308 ✔\n 3 | 3 | 1 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1 | 2 | 1 | -2 | 0.000000000000 | -0.000000000308 ✔\n 3 | 3 | 1 | 2 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 0 | -2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 0 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 0 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | -2 | 1 | 0.000000000000 | -0.000000000308 ✔\n 3 | 3 | 2 | 1 | -1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | -1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | 1 | -1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | 1 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 1 | 2 | -1 | 0.000000000000 | 0.000000000308 ✔\n 3 | 3 | 2 | 1 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 1 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | -2 | -2 | 1.000000000000 | 1.000300628566 ✔\n 3 | 3 | 2 | 2 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | -2 | 2 | 0.000000000000 | 0.000000193779 ✔\n 3 | 3 | 2 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -1 | -1 | 1.000000000000 | 1.000300628559 ✔\n 3 | 3 | 2 | 2 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -1 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | -1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 0 | 0 | 1.000000000000 | 1.000300628572 ✔\n 3 | 3 | 2 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 1 | -2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 1 | 1 | 1.000000000000 | 1.000300628559 ✔\n 3 | 3 | 2 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 2 | -2 | 0.000000000000 | 0.000000193779 ✔\n 3 | 3 | 2 | 2 | 2 | -1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 2 | 2 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 2 | 2 | 2 | 2 | 1.000000000000 | 1.000300628566 ✔\nTest Summary: | Pass Total\n<ψₙ₁ₗ₁ₘ₁|ψₙ₂ₗ₂ₘ₂> = δₙ₁ₙ₂δₗ₁ₗ₂δₘ₁ₘ₂ | 196 196\n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"CurrentModule = Antique","category":"page"},{"location":"HarmonicOscillator/#Harmonic-Oscillator","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The harmonic oscillator is the most frequently used model in quantum physics.","category":"page"},{"location":"HarmonicOscillator/#Definitions","page":"Harmonic Oscillator","title":"Definitions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"k is the spring constant. And omega = sqrtkm, xi = sqrtfracmomegahbarx, A_n = sqrtfrac1n 2^n sqrtfracmomegapihbar, H_n(x) = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 are used.","category":"page"},{"location":"HarmonicOscillator/#Schrödinger-Equation","page":"Harmonic Oscillator","title":"Schrödinger Equation","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatHpsi(x) = E psi(x)","category":"page"},{"location":"HarmonicOscillator/#Hamiltonian","page":"Harmonic Oscillator","title":"Hamiltonian","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n hatH\n = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x) \n = - frac12 hbaromega fracmathrmd^2mathrmdxi^2 + V(x)\n endaligned","category":"page"},{"location":"HarmonicOscillator/#Potential","page":"Harmonic Oscillator","title":"Potential","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"V(x; k=k, m=m, ω=sqrt(k/m))","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" V(x)\n = frac12 k x^2\n = frac12 m omega^2 x^2\n = frac12 hbar omega xi^2","category":"page"},{"location":"HarmonicOscillator/#Eigen-Values","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"E(n; k, m, ω=sqrt(k/m), ℏ=ℏ)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" E_n = hbar omega left( n + frac12 right)","category":"page"},{"location":"HarmonicOscillator/#Eigen-Functions","page":"Harmonic Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"ψ(n, x; k=k, m=m, ω=sqrt(k/m), ℏ=ℏ)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" psi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials","page":"Harmonic Oscillator","title":"Hermite Polynomials","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"H(x; n=0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Examples:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n H_0(x) = 1 \n H_1(x) = 2 x \n H_2(x) = -2 + 4 x^2 \n H_3(x) = -12 x + 8 x^3 \n H_4(x) = 12 - 48 x^2 + 16 x^4 \n H_5(x) = 120 x - 160 x^3 + 32 x^5 \n H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n H_7(x) = -1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n H_9(x) = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n vdots\n endaligned","category":"page"},{"location":"HarmonicOscillator/#Reference","page":"Harmonic Oscillator","title":"Reference","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"DLMF 18.5.18\ncpprefjp\nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965) p.595 (a.4), (a.6)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968) p.71 (13.12)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999) p.491 (B.59)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994) p.152 (7.22)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995) p.41 Table 2.1, p.43 (2.70)\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008) p.293 Table 9.1\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)","category":"page"},{"location":"HarmonicOscillator/#Usage-and-Examples","page":"Harmonic Oscillator","title":"Usage & Examples","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Antique\nHO = antique(:HarmonicOscillator, k=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.k\n1.0\n\njulia> HO.m\n1.0\n\njulia> HO.ℏ\n1.0","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Eigen values:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.E(n=0)\n0.5\n\njulia> HO.E(n=1)\n1.5","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(-5:0.1:5, x -> HO.V(x), lw=2, label=\"\", xlabel=\"x\", ylabel=\"V(x)\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> HO.ψ(x, n=0), label=\"n=0\", lw=2)\nplot!(x -> HO.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> HO.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> HO.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> HO.ψ(x, n=4), label=\"n=4\", lw=2)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times0.5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 0:4\n # energy\n hline!([HO.E(n=n)], lc=:black, ls=:dash, label=\"\")\n plot!([-sqrt(2*HO.k*HO.E(n=n)),sqrt(2*HO.k*HO.E(n=n))], fill(HO.E(n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(x -> HO.E(n=n) + 0.5*HO.ψ(x,n=n), lc=n+1, lw=2, label=\"\")\nend\n# potential\nplot!(x -> HO.V(x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/#Testing","page":"Harmonic Oscillator","title":"Testing","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials-H_n(x)","page":"Harmonic Oscillator","title":"Hermite Polynomials H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=0 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_0(x)\n = e^x^2 e^ - x^2\n = 1 \n = 1\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=1 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_1(x)\n = - e^x^2 fracmathrmd e^ - x^2mathrmdx\n = 2 x \n = 2 x\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=2 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_2(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = -2 + 4 x^2 \n = -2 + 4 x^2\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=3 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_3(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 8 x^3 - 12 x \n = - 12 x + 8 x^3\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=4 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_4(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 12 - 48 x^2 + 16 x^4 \n = 12 - 48 x^2 + 16 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=5 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_5(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 120 x - 160 x^3 + 32 x^5 \n = 120 x - 160 x^3 + 32 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=6 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_6(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = -120 + 720 x^2 + 64 x^6 - 480 x^4 \n = -120 + 720 x^2 + 64 x^6 - 480 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=7 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_7(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = - 1680 x + 3360 x^3 + 128 x^7 - 1344 x^5 \n = - 1680 x + 3360 x^3 + 128 x^7 - 1344 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=8 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_8(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 1680 + 256 x^8 - 13440 x^2 - 3584 x^6 + 13440 x^4 \n = 1680 + 256 x^8 - 13440 x^2 - 3584 x^6 + 13440 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=9 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_9(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 30240 x - 80640 x^3 + 512 x^9 - 9216 x^7 + 48384 x^5 \n = 30240 x - 80640 x^3 + 512 x^9 - 9216 x^7 + 48384 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Test Summary: | Pass Total\nHₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ... | 10 10","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-H_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int_-infty^infty H_j(x) H_i(x) mathrme^-x^2 mathrmdx = sqrtpi 2^j j delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.772453850906 | 1.772453850906 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000001 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 3.544907701811 | 3.544907701811 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000014 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 14.179630807244 | 14.179630807244 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000011 ✔\n 2 | 9 | 0.000000000000 | -0.000000000002 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 85.077784843465 | 85.077784843465 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000139 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 680.622278747718 | 680.622278747718 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000002 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000063 ✔\n 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 6806.222787477181 | 6806.222787477180 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | 0.000000000009 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000001339 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000002 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 81674.673449726179 | 81674.673449726135 ✔\n 6 | 7 | 0.000000000000 | 0.000000000004 ✔\n 6 | 8 | 0.000000000000 | 0.000000000397 ✔\n 6 | 9 | 0.000000000000 | -0.000000000087 ✔\n 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | 0.000000000009 ✔\n 7 | 6 | 0.000000000000 | 0.000000000004 ✔\n 7 | 7 | 1143445.428296166472 | 1143445.428296166705 ✔\n 7 | 8 | 0.000000000000 | -0.000000000007 ✔\n 7 | 9 | 0.000000000000 | 0.000000011649 ✔\n 8 | 0 | 0.000000000000 | -0.000000000001 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000011 ✔\n 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 8 | 4 | 0.000000000000 | -0.000000000063 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000397 ✔\n 8 | 7 | 0.000000000000 | -0.000000000007 ✔\n 8 | 8 | 18295126.852738663554 | 18295126.852738667279 ✔\n 8 | 9 | 0.000000000000 | 0.000000001630 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000014 ✔\n 9 | 2 | 0.000000000000 | -0.000000000002 ✔\n 9 | 3 | 0.000000000000 | 0.000000000139 ✔\n 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000001339 ✔\n 9 | 6 | 0.000000000000 | -0.000000000087 ✔\n 9 | 7 | 0.000000000000 | 0.000000011649 ✔\n 9 | 8 | 0.000000000000 | 0.000000001630 ✔\n 9 | 9 | 329312283.349295914173 | 329312283.349295675755 ✔\nTest Summary: | Pass Total\n∫Hⱼ(x)Hᵢ(x)exp(-x²)dx = √π2ʲj!δᵢⱼ | 100 100","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000000 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | 0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000000 ✔\n 8 | 0 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000000 ✔\n 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n 9 | 3 | 0.000000000000 | 0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | 0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\nTest Summary: | Pass Total\n<ψᵢ|ψⱼ> = δᵢⱼ | 100 100","category":"page"},{"location":"HarmonicOscillator/#Virial-Theorem","page":"Harmonic Oscillator","title":"Virial Theorem","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The virial theorem langle T rangle = langle V rangle and the definition of Hamiltonian langle H rangle = langle T rangle + langle V rangle derive langle H rangle = 2 langle V rangle = 2 langle T rangle.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"2 int psi_n^ast(x) V(x) psi_n(x) mathrmdx = E_n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | 0.500000000000 | 0.500000000000 ✔\n0.1 | 1 | 1.500000000000 | 1.500000000000 ✔\n0.1 | 2 | 2.500000000000 | 2.500000000000 ✔\n0.1 | 3 | 3.500000000000 | 3.500000000000 ✔\n0.1 | 4 | 4.500000000000 | 4.500000000000 ✔\n0.1 | 5 | 5.500000000000 | 5.500000000000 ✔\n0.1 | 6 | 6.500000000000 | 6.500000000000 ✔\n0.1 | 7 | 7.500000000000 | 7.500000000000 ✔\n0.1 | 8 | 8.500000000000 | 8.500000000000 ✔\n0.1 | 9 | 9.500000000000 | 9.500000000000 ✔\n0.5 | 0 | 0.500000000000 | 0.500000000000 ✔\n0.5 | 1 | 1.500000000000 | 1.500000000000 ✔\n0.5 | 2 | 2.500000000000 | 2.500000000000 ✔\n0.5 | 3 | 3.500000000000 | 3.500000000000 ✔\n0.5 | 4 | 4.500000000000 | 4.500000000000 ✔\n0.5 | 5 | 5.500000000000 | 5.500000000000 ✔\n0.5 | 6 | 6.500000000000 | 6.500000000000 ✔\n0.5 | 7 | 7.500000000000 | 7.500000000000 ✔\n0.5 | 8 | 8.500000000000 | 8.500000000000 ✔\n0.5 | 9 | 9.500000000000 | 9.500000000000 ✔\n1.0 | 0 | 0.500000000000 | 0.500000000000 ✔\n1.0 | 1 | 1.500000000000 | 1.500000000000 ✔\n1.0 | 2 | 2.500000000000 | 2.500000000000 ✔\n1.0 | 3 | 3.500000000000 | 3.500000000000 ✔\n1.0 | 4 | 4.500000000000 | 4.500000000000 ✔\n1.0 | 5 | 5.500000000000 | 5.500000000000 ✔\n1.0 | 6 | 6.500000000000 | 6.500000000000 ✔\n1.0 | 7 | 7.500000000000 | 7.500000000000 ✔\n1.0 | 8 | 8.500000000000 | 8.500000000000 ✔\n1.0 | 9 | 9.500000000000 | 9.500000000000 ✔\n5.0 | 0 | 0.500000000000 | 0.500000000000 ✔\n5.0 | 1 | 1.500000000000 | 1.500000000000 ✔\n5.0 | 2 | 2.500000000000 | 2.500000000000 ✔\n5.0 | 3 | 3.500000000000 | 3.500000000000 ✔\n5.0 | 4 | 4.500000000000 | 4.500000000000 ✔\n5.0 | 5 | 5.500000000000 | 5.500000000000 ✔\n5.0 | 6 | 6.500000000000 | 6.500000000000 ✔\n5.0 | 7 | 7.500000000000 | 7.500000000000 ✔\n5.0 | 8 | 8.500000000000 | 8.500000000000 ✔\n5.0 | 9 | 9.500000000000 | 9.500000000000 ✔\nTest Summary: | Pass Total\n2 × <ψₙ|V|ψₙ> = Eₙ | 40 40","category":"page"},{"location":"HarmonicOscillator/#Eigen-Values-2","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n E_n\n = int psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int psi^ast_n(x) left V(x) - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int psi^ast_n(x) left V(x)psi(x) -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | 0.158113883008 | 0.158113879883 ✔\n0.1 | 1 | 0.474341649025 | 0.474341633410 ✔\n0.1 | 2 | 0.790569415042 | 0.790569374409 ✔\n0.1 | 3 | 1.106797181059 | 1.106797102928 ✔\n0.1 | 4 | 1.423024947076 | 1.423024818987 ✔\n0.1 | 5 | 1.739252713093 | 1.739252522506 ✔\n0.1 | 6 | 2.055480479109 | 2.055480213500 ✔\n0.1 | 7 | 2.371708245126 | 2.371707891950 ✔\n0.1 | 8 | 2.687936011143 | 2.687935558100 ✔\n0.1 | 9 | 3.004163777160 | 3.004163211450 ✔\n0.5 | 0 | 0.353553390593 | 0.353553374944 ✔\n0.5 | 1 | 1.060660171780 | 1.060660093649 ✔\n0.5 | 2 | 1.767766952966 | 1.767766749878 ✔\n0.5 | 3 | 2.474873734153 | 2.474873343556 ✔\n0.5 | 4 | 3.181980515339 | 3.181979874817 ✔\n0.5 | 5 | 3.889087296526 | 3.889086343463 ✔\n0.5 | 6 | 4.596194077713 | 4.596192749665 ✔\n0.5 | 7 | 5.303300858899 | 5.303299093519 ✔\n0.5 | 8 | 6.010407640086 | 6.010405374197 ✔\n0.5 | 9 | 6.717514421272 | 6.717511593266 ✔\n1.0 | 0 | 0.500000000000 | 0.499999968773 ✔\n1.0 | 1 | 1.500000000000 | 1.499999843774 ✔\n1.0 | 2 | 2.500000000000 | 2.499999593764 ✔\n1.0 | 3 | 3.500000000000 | 3.499999218732 ✔\n1.0 | 4 | 4.500000000000 | 4.499998718747 ✔\n1.0 | 5 | 5.500000000000 | 5.499998093755 ✔\n1.0 | 6 | 6.500000000000 | 6.499997343602 ✔\n1.0 | 7 | 7.500000000000 | 7.499996468887 ✔\n1.0 | 8 | 8.500000000000 | 8.499995468843 ✔\n1.0 | 9 | 9.500000000000 | 9.499994343445 ✔\n5.0 | 0 | 1.118033988750 | 1.118033832523 ✔\n5.0 | 1 | 3.354101966250 | 3.354101184969 ✔\n5.0 | 2 | 5.590169943749 | 5.590167912524 ✔\n5.0 | 3 | 7.826237921249 | 7.826234014984 ✔\n5.0 | 4 | 10.062305898749 | 10.062299492494 ✔\n5.0 | 5 | 12.298373876249 | 12.298364344997 ✔\n5.0 | 6 | 14.534441853749 | 14.534428572309 ✔\n5.0 | 7 | 16.770509831248 | 16.770492175222 ✔\n5.0 | 8 | 19.006577808748 | 19.006555152416 ✔\n5.0 | 9 | 21.242645786248 | 21.242617504750 ✔\nTest Summary: | Pass Total\n∫ψₙ*Hψₙdx = <ψₙ|H|ψₙ> = Eₙ | 40 40\n","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = Antique","category":"page"},{"location":"#Antique.jl","page":"Home","title":"Antique.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Self-contained, Well-Tested, Well-Documented Analytical Solutions of Quantum Mechanical Equations.","category":"page"},{"location":"#Install","page":"Home","title":"Install","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install this package, run the following code in your Jupyter Notebook:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg; Pkg.add(\"Antique\")","category":"page"},{"location":"#Usage-and-Examples","page":"Home","title":"Usage & Examples","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Install Antique.jl for the first use and run using Antique before each use. The function antique(model, parameters...) returns a module that has E, ψ, V and some other functions. Here are examples in hydrogen-like atom. The analytical notation of energy (eigen value of the Hamiltonian) is written as","category":"page"},{"location":"","page":"Home","title":"Home","text":"E_n = -fracZ^22n^2 E_mathrmh","category":"page"},{"location":"","page":"Home","title":"Home","text":"Hydrogen atom has symbol mathrmH and atomic number 1 (Z=1). Therefore the ground state (n=1) energy is -frac12 E_mathrmh.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Antique\nH = antique(:HydrogenAtom, Z=1)\nH.E(n=1)\n# output> -0.5","category":"page"},{"location":"","page":"Home","title":"Home","text":"Helium cation has symbol mathrmHe^+ and atomic number 2 (Z=2). Therefore the ground state (n=1) energy is -2 E_mathrmh.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Antique\nHe⁺ = antique(:HydrogenAtom, Z=2)\nHe⁺.E(n=1)\n# output> -2.0","category":"page"},{"location":"","page":"Home","title":"Home","text":"There are more examples on each model page.","category":"page"},{"location":"#Supported-Models","page":"Home","title":"Supported Models","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
\n
\n \n \"InfinitePotentialWell\"/\n \n :InfinitePotentialWell\n
\n
\n \n \"HarmonicOscillator\"/\n \n :HarmonicOscillator\n
\n
\n \n \"MorsePotential\"/\n \n :MorsePotential\n
\n
\n \n \"HydrogenAtom\"/\n \n :HydrogenAtom\n
\n
","category":"page"},{"location":"#Future-Works","page":"Home","title":"Future Works","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"List of quantum-mechanical systems with analytical solutions","category":"page"},{"location":"#Acknowledgment","page":"Home","title":"Acknowledgment","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package was named by @KB-satou and @ultimatile.","category":"page"}] }