Merge pull request #16 from ajarifi/main

add Delta potential

README.md

@@ -65,9 +65,10 @@ This is the guideline for adding new models.
 6. Write the code in that file. First we need to create a structure `struct ModelName` with the same name as the model name (The best way is Find & Replace). Create V, E, ψ and other functions. Because the function names conflict, you must always give the structure as an argument. Multi-dispatch avoids conflict. We recommend using Revice.jl while coding. Run `include("./developer/revice.jl")` on the REPL or use dev.ipynb.
 7. Add test code test/ModelName.jl. At a minimum, it is recommended to check the normalization and the orthogonality of wavefunction using QuadGK.jl. All tests will be executed by executing `include("./developer/test.jl")`. It will take about 2 minutes to complete.
 8. Add documentation. Add either docs/ModelName.md or docs/jmd/ModelName.jmd (if you have a jmd file, the md file will be automatically generated). Include at least the definition of the Hamiltonian and the analytical solutions (eigenvalues and eigenfunctions).
-9. Execute `include("./developer/docs.jl")` to compile. Please check docs/build/*.html in your browser.
-10. Push the code.
-11. Submit a pull request on GitHub.
+9. Add the new model into `pages=[...]` in docs/make.jl.
+10. Execute `include("./developer/docs.jl")` to compile. Please check docs/build/*.html in your browser.
+11. Push the code.
+12. Submit a pull request on GitHub.
 
 ## Acknowledgment
 

docs/jmd2md.jl

@@ -1,7 +1,7 @@
 using Antique
 using Weave
 
-for file in Antique.models # [:InfinitePotentialWell :HarmonicOscillator :MorsePotential :HydrogenAtom]
+for file in Antique.models # [:DeltaPotential]
   weave("./src/jmd/$file.jmd", doctype="github", out_path="./src/", fig_path="./assets/fig/")
   text = Antique.load("./src/$file.md")
   # remove ```  after include(...s)

docs/make.jl

@@ -24,6 +24,7 @@ makedocs(;
     "Harmonic Oscillator"     => "HarmonicOscillator.md"   ,
     "Morse Potential"         => "MorsePotential.md"       ,
     "Hydrogen Atom"           => "HydrogenAtom.md"         ,
+    "Delta Potential"         => "DeltaPotential.md"       ,
   ],
 )
 

docs/src/DeltaPotential.md

@@ -0,0 +1,139 @@
+```@meta
+CurrentModule = Antique
+```
+
+# Delta Potential
+
+The Delta potential is one of the simplest models for quantum mechanical system in 1D.
+It always has one bound state and its wave function has a cusp at the origin.
+
+## Defitions
+
+This model is described with the time-independent Schrödinger equation
+
+```math
+  \hat{H} \psi(x) = E \psi(x),
+```
+
+and the Hamiltonian
+```math
+  \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).
+```
+
+Parameters are specified with the following struct.
+
+#### Parameters
+
+```@docs
+DeltaPotential
+```
+
+#### Potential
+```@docs
+V(::DeltaPotential)
+```
+
+#### Eigen Values
+```@docs
+E(::DeltaPotential)
+```
+
+#### Eigen Functions
+```@docs
+ψ(::DeltaPotential)
+```
+
+## Usage & Examples
+
+[Install Antique.jl](@ref Install) for the first use and run `using Antique` before each use. The energy `E()`, wavefunction `ψ()`, potential `V()` and some other functions are suppoted. In this system, the model is generated by `DeltaPotential` and several parameters `α`, `m` and `ℏ` are set as optional arguments.
+
+```julia
+using Antique
+DP = DeltaPotential(α=1.0, m=1.0, ℏ=1.0)
+```
+
+
+
+
+Parameters:
+
+```julia
+julia> DP.α
+1.0
+
+julia> DP.m
+1.0
+
+julia> DP.ℏ
+1.0
+```
+
+
+
+Eigen values:
+
+```julia
+julia> E(DP)
+-0.5
+```
+
+
+
+Wave functions:
+
+```julia
+DP = DeltaPotential(α=0.1, m=0.5, ℏ=0.1)
+x = LinRange(-2,2,500);
+
+using Plots
+plot(x, x->ψ(DP,x), linewidth=3)
+plot!(xlim=[-2,2], ylim=[0,2.5], legend=false)
+plot!(xlabel="x", ylabel="ψ(x)", title="Delta Potential")
+```
+
+![](./assets/fig//DeltaPotential_4_1.png)
+
+
+
+## Testing
+
+Unit testing and Integration testing were done using numerical integration ([QuadGK.jl](https://juliamath.github.io/QuadGK.jl/stable/)). The test script is [here](https://github.com/ohno/Antique.jl/blob/main/test/DeltaPotential.jl).
+
+#### Normalization of $\psi(x)$
+
+```math
+\int_{-\infty}^{\infty} \psi^\ast(x) \psi(x) ~\mathrm{d}x = 1
+```
+
+```
+  α |   m |   ℏ |        analytical |         numerical 
+--- | --- | --- | ----------------- | ----------------- 
+0.1 | 0.1 | 0.1 |    1.000000000000 |    1.000000000000 ✔
+0.1 | 0.1 | 1.0 |    1.000000000000 |    1.000000000000 ✔
+0.1 | 0.1 | 7.0 |    1.000000000000 |    1.000004676239 ✔
+0.1 | 1.0 | 0.1 |    1.000000000000 |    1.000000000000 ✔
+0.1 | 1.0 | 1.0 |    1.000000000000 |    1.000000000000 ✔
+0.1 | 1.0 | 7.0 |    1.000000000000 |    0.999999999999 ✔
+0.1 | 7.0 | 0.1 |    1.000000000000 |    1.000000000000 ✔
+0.1 | 7.0 | 1.0 |    1.000000000000 |    1.000000000000 ✔
+0.1 | 7.0 | 7.0 |    1.000000000000 |    1.000000000000 ✔
+1.0 | 0.1 | 0.1 |    1.000000000000 |    1.000000000000 ✔
+1.0 | 0.1 | 1.0 |    1.000000000000 |    1.000000000000 ✔
+1.0 | 0.1 | 7.0 |    1.000000000000 |    0.999999999999 ✔
+1.0 | 1.0 | 0.1 |    1.000000000000 |    1.000000000000 ✔
+1.0 | 1.0 | 1.0 |    1.000000000000 |    1.000000000000 ✔
+1.0 | 1.0 | 7.0 |    1.000000000000 |    1.000000000000 ✔
+1.0 | 7.0 | 0.1 |    1.000000000000 |    1.000000000000 ✔
+1.0 | 7.0 | 1.0 |    1.000000000000 |    1.000000000000 ✔
+1.0 | 7.0 | 7.0 |    1.000000000000 |    1.000000000000 ✔
+7.0 | 0.1 | 0.1 |    1.000000000000 |    1.000000000000 ✔
+7.0 | 0.1 | 1.0 |    1.000000000000 |    1.000000000000 ✔
+7.0 | 0.1 | 7.0 |    1.000000000000 |    1.000000000000 ✔
+7.0 | 1.0 | 0.1 |    1.000000000000 |    1.000000000000 ✔
+7.0 | 1.0 | 1.0 |    1.000000000000 |    1.000000000000 ✔
+7.0 | 1.0 | 7.0 |    1.000000000000 |    1.000000000000 ✔
+7.0 | 7.0 | 0.1 |    1.000000000000 |    1.000000000000 ✔
+7.0 | 7.0 | 1.0 |    1.000000000000 |    1.000000000000 ✔
+7.0 | 7.0 | 7.0 |    1.000000000000 |    1.000000000000 ✔
+
+```

docs/src/InfinitePotentialWell.md

@@ -17,7 +17,7 @@ The infinite potential well (particle in a box) is the simplest model for quantu
 
 #### Hamiltonian
 ```math
-  \hat{H} = \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x)
+  \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x)
 ```
 
 #### Potential

docs/src/index.md

@@ -70,6 +70,12 @@ There are more examples on each model page.
     </a>
     <code>HydrogenAtom</code>
   </div>
+    <div class="item">
+    <a target="_blank" href="./DeltaPotential">
+      <img src="assets/fig/DeltaPotential_4_1.png" alt="DeltaPotential"/>
+    </a>
+    <code>DeltaPotential</code>
+  </div>
 </div>
 ```
 
@@ -89,9 +95,10 @@ This is the guideline for adding new models.
 6. Write the code in that file. First we need to create a structure `struct ModelName` with the same name as the model name (The best way is Find & Replace). Create V, E, ψ and other functions. Because the function names conflict, you must always give the structure as an argument. Multi-dispatch avoids conflict. We recommend using Revice.jl while coding. Run `include("./developer/revice.jl")` on the REPL or use dev.ipynb.
 7. Add test code test/ModelName.jl. At a minimum, it is recommended to check the normalization and the orthogonality of wavefunction using QuadGK.jl. All tests will be executed by executing `include("./developer/test.jl")`. It will take about 2 minutes to complete.
 8. Add documentation. Add either docs/ModelName.md or docs/jmd/ModelName.jmd (if you have a jmd file, the md file will be automatically generated). Include at least the definition of the Hamiltonian and the analytical solutions (eigenvalues and eigenfunctions).
-9. Execute `include("./developer/docs.jl")` to compile. Please check docs/build/*.html in your browser.
-10. Push the code.
-11. Submit a pull request on GitHub.
+9. Add the new model into `pages=[...]` in docs/make.jl.
+10. Execute `include("./developer/docs.jl")` to compile. Please check docs/build/*.html in your browser.
+11. Push the code.
+12. Submit a pull request on GitHub.
 
 ## Acknowledgment
 

docs/src/jmd/DeltaPotential.jmd

@@ -0,0 +1,87 @@
+```@meta
+CurrentModule = Antique
+```
+
+# Delta Potential
+
+The Delta potential is one of the simplest models for quantum mechanical system in 1D.
+It always has one bound state and its wave function has a cusp at the origin.
+
+## Defitions
+
+This model is described with the time-independent Schrödinger equation
+
+```math
+  \hat{H} \psi(x) = E \psi(x),
+```
+
+and the Hamiltonian
+```math
+  \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).
+```
+
+Parameters are specified with the following struct.
+
+#### Parameters
+
+```@docs
+DeltaPotential
+```
+
+#### Potential
+```@docs
+V(::DeltaPotential)
+```
+
+#### Eigen Values
+```@docs
+E(::DeltaPotential)
+```
+
+#### Eigen Functions
+```@docs
+ψ(::DeltaPotential)
+```
+
+## Usage & Examples
+
+[Install Antique.jl](@ref Install) for the first use and run `using Antique` before each use. The energy `E()`, wavefunction `ψ()`, potential `V()` and some other functions are suppoted. In this system, the model is generated by `DeltaPotential` and several parameters `α`, `m` and `ℏ` are set as optional arguments.
+
+```julia; cache = :all; results = "hidden"
+using Antique
+DP = DeltaPotential(α=1.0, m=1.0, ℏ=1.0)
+```
+
+Parameters:
+
+```julia; term = true
+DP.α
+DP.m
+DP.ℏ
+```
+
+Eigen values:
+
+```julia; term = true
+E(DP)
+```
+
+Wave functions:
+
+```julia
+DP = DeltaPotential(α=0.1, m=0.5, ℏ=0.1)
+x = LinRange(-2,2,500);
+
+using Plots
+plot(x, x->ψ(DP,x), linewidth=3)
+plot!(xlim=[-2,2], ylim=[0,2.5], legend=false)
+plot!(xlabel="x", ylabel="ψ(x)", title="Delta Potential")
+```
+
+## Testing
+
+Unit testing and Integration testing were done using numerical integration ([QuadGK.jl](https://juliamath.github.io/QuadGK.jl/stable/)). The test script is [here](https://github.com/ohno/Antique.jl/blob/main/test/DeltaPotential.jl).
+
+```julia; line_width = 500
+println(Antique.load("../../test/result/DeltaPotential.log"))
+```

docs/src/jmd/InfinitePotentialWell.jmd

@@ -17,7 +17,7 @@ The infinite potential well (particle in a box) is the simplest model for quantu
 
 #### Hamiltonian
 ```math
-  \hat{H} = \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x)
+  \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x)
 ```
 
 #### Potential

src/Antique.jl

@@ -6,6 +6,7 @@ module Antique
     :HarmonicOscillator,
     :MorsePotential,
     :HydrogenAtom,
+    :DeltaPotential,
   ]
 
   # for Julia 1.1

src/DeltaPotential.jl

@@ -0,0 +1,62 @@
+export DeltaPotential, V, E, ψ
+
+# Parameters
+@kwdef struct DeltaPotential
+  α = 1.0
+  m = 1.0
+  ℏ = 1.0
+end
+
+# Potential
+function V(model::DeltaPotential, x)
+  return x==0 ? -Inf : 0
+end
+
+# Energy
+function E(model::DeltaPotential)
+  α = model.α
+  m = model.m
+  ℏ = model.ℏ
+  return -(m*α^2)/(2*ℏ^2)
+end
+
+# Wave Function
+function ψ(model::DeltaPotential, x)
+  α = model.α
+  m = model.m
+  ℏ = model.ℏ
+  return sqrt(m*α)/ℏ * exp.(-m*α*abs.(x)/ℏ^2)
+end
+
+# Documents
+
+@doc raw"""
+`DeltaPotential(α=1.0, m=1.0, ℏ=1.0)`
+
+``\alpha`` is the potential strength, ``m`` is the mass of particle and ``\hbar`` is the reduced Planck constant (Dirac's constant).
+
+""" DeltaPotential
+
+@doc raw"""
+`V(model::DeltaPotential; x)`
+
+```math
+  V(x) = -\alpha \delta(x).
+```
+""" V(model::DeltaPotential)
+
+@doc raw"""
+`E(model::DeltaPotential)`
+
+```math
+  E = - \frac{m\alpha^2}{2\hbar^2}
+```
+""" E(model::DeltaPotential)
+
+@doc raw"""
+`ψ(model::DeltaPotential, x)`
+
+```math
+   \psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}
+```
+""" ψ(model::DeltaPotential)