From 43c9715b80db41b4758d57c4b3250898c23f783e Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Mon, 20 May 2024 04:42:08 +0000 Subject: [PATCH] build based on aedf62a --- dev/.documenter-siteinfo.json | 2 +- dev/API/index.html | 46 +++++++++++++------------- dev/CoulombTwoBody/index.html | 12 +++---- dev/DeltaPotential/index.html | 4 +-- dev/HarmonicOscillator/index.html | 8 ++--- dev/HydrogenAtom/index.html | 12 +++---- dev/InfinitePotentialWell/index.html | 6 ++-- dev/InfinitePotentialWell3D/index.html | 6 ++-- dev/MorsePotential/index.html | 14 ++++---- dev/PoschlTeller/index.html | 8 ++--- dev/RigidRotor/index.html | 13 ++++---- dev/SphericalOscillator/index.html | 12 +++---- dev/index.html | 2 +- dev/search_index.js | 2 +- 14 files changed, 74 insertions(+), 73 deletions(-) diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index fe82efb..fba7beb 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-12T18:03:14","documenter_version":"1.4.1"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-20T04:42:03","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/dev/API/index.html b/dev/API/index.html index 5193021..5793dea 100644 --- a/dev/API/index.html +++ b/dev/API/index.html @@ -1,10 +1,10 @@ -API reference · Antique.jl
GitHub

API reference

Antique.CoulombTwoBody — Type

CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$z₁$ is the charge number of particle 1, $z₁$ is the charge number of particle 2, $m₁$ is the mass of particle 1, $m₂$ is the mass of particle 2, $m_\mathrm{e}$ is the electron mass (the unit of $m₁$ and $m₂$), $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.DeltaPotential — Type

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).

source
Antique.HarmonicOscillator — Type

HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)

$k$ is the force constant, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.HydrogenAtom — Type

HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$Z$ is the atomic number, $m_\mathrm{e}$ is the electron mass, $a_0$is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.InfinitePotentialWell — Type

InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

$L$ is the length of the box, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.InfinitePotentialWell3D — Type

InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)

$L_x,L_y,L_z$ are the lengths of the box in $x$,$y$,$z$-direction, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.MorsePotential — Type

MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)

$r_\mathrm{e}$ is the equilibrium bond distance, $D_\mathrm{e}$ is the the well depth , $k$ is the force constant, $\mu$ is the reduced mass and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.RigidRotor — Type

RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)

$m₁$ and $m₂$ are mass of two particles, $R$ is the distance, and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.SphericalOscillator — Type

HarmonicOscillator(k=1.0, μ=1.0, ℏ=1.0)

$k$ is the force constant, $μ$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.E — Method

E(model::CoulombTwoBody; n=1)

\[E_n -= -\frac{(z_1 z_2)^2}{2n^2} \frac{\mu}{m_\mathrm{e}} E_\mathrm{h},\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2, $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source
Antique.E — Method

E(model::DeltaPotential)

\[E = - \frac{m\alpha^2}{2\hbar^2}\]

source
Antique.E — Method

E(model::HarmonicOscillator; n=0)

\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

source
Antique.E — Method

E(model::HydrogenAtom; n=1)

\[E_n +API reference · Antique.jl

GitHub

API reference

Antique.CoulombTwoBody — Type

CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$z₁$ is the charge number of particle 1, $z₁$ is the charge number of particle 2, $m₁$ is the mass of particle 1, $m₂$ is the mass of particle 2, $m_\mathrm{e}$ is the electron mass (the unit of $m₁$ and $m₂$), $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.DeltaPotential — Type

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).

source
Antique.HarmonicOscillator — Type

HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)

$k$ is the force constant, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.HydrogenAtom — Type

HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$Z$ is the atomic number, $m_\mathrm{e}$ is the electron mass, $a_0$is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.InfinitePotentialWell — Type

InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

$L$ is the length of the box, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.InfinitePotentialWell3D — Type

InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)

$L_x,L_y,L_z$ are the lengths of the box in $x$,$y$,$z$-direction, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.MorsePotential — Type

MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)

$r_\mathrm{e}$ is the equilibrium bond distance, $D_\mathrm{e}$ is the the well depth , $k$ is the force constant, $\mu$ is the reduced mass and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.RigidRotor — Type

RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)

$m₁$ and $m₂$ are mass of two particles, $R$ is the distance, and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.SphericalOscillator — Type

HarmonicOscillator(k=1.0, μ=1.0, ℏ=1.0)

$k$ is the force constant, $μ$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.E — Method

E(model::CoulombTwoBody; n=1)

\[E_n += -\frac{(z_1 z_2)^2}{2n^2} \frac{\mu}{m_\mathrm{e}} E_\mathrm{h},\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2, $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source
Antique.E — Method

E(model::DeltaPotential)

\[E = - \frac{m\alpha^2}{2\hbar^2}\]

source
Antique.E — Method

E(model::HarmonicOscillator; n=0)

\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

source
Antique.E — Method

E(model::HydrogenAtom; n=1)

\[E_n = -\frac{m_\mathrm{e} e^4 Z^2}{2n^2(4\pi\varepsilon_0)^2\hbar^2} -= -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source
Antique.E — Method

E(model::InfinitePotentialWell3D; nx=1, ny=1, nz=1)

\[E_{n_x,n_y,n_z} = \frac{\hbar^2 \pi^2}{2 m} \left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right)\]

source
Antique.E — Method

E(model::InfinitePotentialWell; n=1)

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

source
Antique.E — Method

E(model::MorsePotential; n=0)

\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]

where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.

source
Antique.E — Method

E(model::PoschlTeller; n=0)

\[E_n = -\frac{\hbar^2}{m x_0^2}\frac{\mu^2}{2},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.

source
Antique.E — Method

E(model::RigidRotor; l=0)

\[E_l -= \frac{\hbar^2}{2I}l(l+1),\]

where $I=\mu R^2$ is the moment of inertia, $R$ is the distance, and $\mu$ is the reduced mass of the two particles.

source
Antique.E — Method

E(model::SphericalOscillator; n=0, l=0)

\[E_{nl} -= \left(2n + l + \frac{3}{2}\right)\hbar \omega,\]

where $\omega = \sqrt{k/\mu}$.

source
Antique.H — Method

H(model::HarmonicOscillator, x; n=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} += -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source
Antique.E — Method

E(model::InfinitePotentialWell3D; nx=1, ny=1, nz=1)

\[E_{n_x,n_y,n_z} = \frac{\hbar^2 \pi^2}{2 m} \left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right)\]

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Antique.E — Method

E(model::InfinitePotentialWell; n=1)

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

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Antique.E — Method

E(model::MorsePotential; n=0)

\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]

where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.

source
Antique.E — Method

E(model::PoschlTeller; n=0)

\[E_n = -\frac{\hbar^2}{m x_0^2}\frac{\mu^2}{2},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.

source
Antique.E — Method

E(model::RigidRotor; l=0)

\[E_l += \frac{\hbar^2}{2I}l(l+1),\]

where $I=\mu R^2$ is the moment of inertia, $R$ is the distance, and $\mu$ is the reduced mass of the two particles.

source
Antique.E — Method

E(model::SphericalOscillator; n=0, l=0)

\[E_{nl} += \left(2n + l + \frac{3}{2}\right)\hbar \omega,\]

where $\omega = \sqrt{k/\mu}$.

source
Antique.H — Method

H(model::HarmonicOscillator, x; n=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} H_{n}(x) &:= (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2} \\ &= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}. @@ -20,7 +20,7 @@ H_{8}(x) &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}, \\ H_{9}(x) &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}, \\ &\vdots -\end{aligned}\]

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Antique.L — Method

L(model::CoulombTwoBody, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source
Antique.L — Method

L(model::CoulombTwoBody, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -43,7 +43,7 @@ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots -\end{aligned}\]

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Antique.L — Method

L(model::HydrogenAtom, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

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Antique.L — Method

L(model::HydrogenAtom, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -66,7 +66,7 @@ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots -\end{aligned}\]

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Antique.L — Method

L(model::MorsePotential, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source
Antique.L — Method

L(model::MorsePotential, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ @@ -88,7 +88,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots -\end{aligned}\]

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Antique.L — Method

L(model::SphericalOscillator, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

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Antique.L — Method

L(model::SphericalOscillator, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ @@ -110,7 +110,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots -\end{aligned}\]

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Antique.P — Method

P(model::CoulombTwoBody, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

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Antique.P — Method

P(model::CoulombTwoBody, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -132,7 +132,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

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Antique.P — Method

P(model::HydrogenAtom, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

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Antique.P — Method

P(model::HydrogenAtom, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -154,9 +154,9 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

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Antique.P — Method

P(model::PoschlTeller, x; n=0, m=0)

Associated Legendre polynomials are the associated Legendre functions for integer indices. Here we use the same notation of the associated Legendre functions as in the model HydrogenAtom.

```math \begin{aligned} Pn^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} Pn(x) \ +\end{aligned}\]

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Antique.P — Method

P(model::PoschlTeller, x; n=0, m=0)

Associated Legendre polynomials are the associated Legendre functions for integer indices. Here we use the same notation of the associated Legendre functions as in the model HydrogenAtom.

```math \begin{aligned} Pn^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} Pn(x) \ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \ - &= \frac{1}{2^n} (1-x^2)^{m/2} \sum_{j=0}^{\left\lfloor\frac{n-m}{2}\right\rfloor} (-1)^j \frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \end{aligned} ```

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Antique.P — Method

P(model::RigidRotor, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} + &= \frac{1}{2^n} (1-x^2)^{m/2} \sum_{j=0}^{\left\lfloor\frac{n-m}{2}\right\rfloor} (-1)^j \frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \end{aligned} ```

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Antique.P — Method

P(model::RigidRotor, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -178,7 +178,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

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Antique.P — Method

P(model::SphericalOscillator, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source
Antique.P — Method

P(model::SphericalOscillator, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -200,39 +200,39 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

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Antique.R — Method

R(model::CoulombTwoBody, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_\mu}\right)^3} \left(\frac{2Zr}{n a_\mu}\right)^l \exp \left(-\frac{Zr}{n a_\mu}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_\mu}\right),\]

where $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$, $a_\mu = a_0 \frac{m_\mathrm{e}}{\mu}$, Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

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Antique.R — Method

R(model::HydrogenAtom, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_0}\right)^3} \left(\frac{2Zr}{n a_0}\right)^l \exp \left(-\frac{Zr}{n a_0}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_0}\right),\]

where Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

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Antique.R — Method

R(model::SphericalOscillator, r; n=0, l=0)

\[R_{nl}(r) = \sqrt{ \frac{\gamma^{3/2}}{2\sqrt{\pi}}} \sqrt{\frac{2^{n+l+3} n!}{(2n+2l+1)!!}} \xi^l \exp\left(-\xi^2/2\right)L_{n}^{(l+\frac{1}{2})} \left(\xi^2\right),\]

where $\gamma = \mu\omega/\hbar$ and $\xi = \sqrt{\gamma}r = \sqrt{\mu\omega/\hbar}r$ are defined. The generalized Laguerre polynomials are defined as $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$. The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::CoulombTwoBody, r)

\[\begin{aligned} +\end{aligned}\]

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Antique.R — Method

R(model::CoulombTwoBody, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_\mu}\right)^3} \left(\frac{2Zr}{n a_\mu}\right)^l \exp \left(-\frac{Zr}{n a_\mu}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_\mu}\right),\]

where $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$, $a_\mu = a_0 \frac{m_\mathrm{e}}{\mu}$, Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

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Antique.R — Method

R(model::HydrogenAtom, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_0}\right)^3} \left(\frac{2Zr}{n a_0}\right)^l \exp \left(-\frac{Zr}{n a_0}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_0}\right),\]

where Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

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Antique.R — Method

R(model::SphericalOscillator, r; n=0, l=0)

\[R_{nl}(r) = \sqrt{ \frac{\gamma^{3/2}}{2\sqrt{\pi}}} \sqrt{\frac{2^{n+l+3} n!}{(2n+2l+1)!!}} \xi^l \exp\left(-\xi^2/2\right)L_{n}^{(l+\frac{1}{2})} \left(\xi^2\right),\]

where $\gamma = \mu\omega/\hbar$ and $\xi = \sqrt{\gamma}r = \sqrt{\mu\omega/\hbar}r$ are defined. The generalized Laguerre polynomials are defined as $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$. The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::CoulombTwoBody, r)

\[\begin{aligned} V(r) &= - \frac{Ze^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{Z}{r/a_0} &= - \frac{Z}{r/a_0} E_\mathrm{h}, -\end{aligned}\]

The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::DeltaPotential, x)

\[V(x) = -\alpha \delta(x).\]

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Antique.V — Method

V(model::HarmonicOscillator, x)

\[V(x) +\end{aligned}\]

The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::DeltaPotential, x)

\[V(x) = -\alpha \delta(x).\]

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Antique.V — Method

V(model::HarmonicOscillator, x)

\[V(x) = \frac{1}{2} k x^2 = \frac{1}{2} m \omega^2 x^2 -= \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/m}$ is the angular frequency and $\xi = \sqrt{\frac{m\omega}{\hbar}}x$.

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Antique.V — Method

V(model::HydrogenAtom, r)

\[\begin{aligned} += \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/m}$ is the angular frequency and $\xi = \sqrt{\frac{m\omega}{\hbar}}x$.

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Antique.V — Method

V(model::HydrogenAtom, r)

\[\begin{aligned} V(r) &= - \frac{Ze^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{Z}{r/a_0} &= - \frac{Z}{r/a_0} E_\mathrm{h}, -\end{aligned}\]

The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::InfinitePotentialWell; x)

\[V(x) = +\end{aligned}\]

The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::InfinitePotentialWell; x)

\[V(x) = \left\{ \begin{array}{ll} \infty & x \lt 0, L \lt x \\ 0 & 0 \leq x \leq L \end{array} -\right.\]

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Antique.V — Method

V(model::InfinitePotentialWell3D, x,y,z)

\[V(x,y,z) = +\right.\]

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Antique.V — Method

V(model::InfinitePotentialWell3D, x,y,z)

\[V(x,y,z) = \left\{ \begin{array}{ll} 0 & 0 \leq x \leq L_x \ \mathrm{and}\ 0 \leq y \leq L_y \ \mathrm{and}\ 0 \leq z \leq L_z \\ \infty & \mathrm{elsewhere} \end{array} -\right.\]

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Antique.V — Method

V(model::MorsePotential, r)

\[V(r) = D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::PoschlTeller, x)

\[\begin{aligned} +\right.\]

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Antique.V — Method

V(model::MorsePotential, r)

\[V(r) = D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::PoschlTeller, x)

\[\begin{aligned} V(x) &= -\frac{\hbar^2}{m x_0^2} \frac{\lambda(\lambda+1)}{2} \mathrm{sech}(x)^2 &= -\frac{\hbar^2}{m x_0^2} \frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}(x)^2}. -\end{aligned}\]

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Antique.V — Method

V(model::RigidRotor, r)

\[\begin{aligned} +\end{aligned}\]

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Antique.V — Method

V(model::RigidRotor, r)

\[\begin{aligned} V(r) &= 0, -\end{aligned}\]

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Antique.V — Method

V(model::SphericalOscillator, r)

\[V(r) +\end{aligned}\]

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Antique.V — Method

V(model::SphericalOscillator, r)

\[V(r) = \frac{1}{2} k r^2 = \frac{1}{2} \mu \omega^2 r^2 -= \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/\mu}$ is the angular frequency and $\xi = \sqrt{\frac{\mu\omega}{\hbar}}r$.

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Antique.Y — Method

Y(model::CoulombTwoBody, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.Y — Method

Y(model::HydrogenAtom, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.Y — Method

Y(model::RigidRotor, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.Y — Method

Y(model::SphericalOscillator, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.nₘₐₓ — Method

nₘₐₓ(model::MorsePotential)

\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]

where $\omega = \sqrt{k/µ}$ is defined.

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Antique.nₘₐₓ — Method

nₘₐₓ(model::PoschlTeller)

\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]

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Antique.ψ — Method

ψ(model::CoulombTwoBody, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

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Antique.ψ — Method

ψ(model::DeltaPotential, x)

\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]

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Antique.ψ — Method

ψ(model::HarmonicOscillator, x; n=0)

\[\psi_n(x) = A_n H_n(\xi) \exp{\left( -\frac{\xi^2}{2} \right)},\]

where $\omega = \sqrt{k/m}$, $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, $A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$, $H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$ are defined.

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Antique.ψ — Method

ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

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Antique.ψ — Method

ψ(model::InfinitePotentialWell, x; n=1)

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

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Antique.ψ — Method

ψ(model::InfinitePotentialWell3D, x,y,z; nx=1, ny=1, nz=1)

The wave functions can be expressed as products of wave functions in a one-dimensional box.

\[\psi_{n_x,n_y,n_z}(x,y,z) = \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) = \sqrt{\frac{8}{L_xL_yL_z}} \sin\left(\frac{n_x\pi x}{L_x}\right) \sin\left(\frac{n_y\pi y}{L_y}\right) \sin\left(\frac{n_z\pi z}{L_z}\right)\]

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Antique.ψ — Method

ψ(model::MorsePotential, r; n=0)

\[\psi_n(r) = N_n z^{\lambda-n-1/2} \mathrm{e}^{-z/2} L_n^{(2\lambda-2n-1)}(\xi),\]

$N_n = \sqrt{\frac{n!(2\lambda-2n-1)a}{\Gamma(2\lambda-n)}}$, $\lambda = \frac{\sqrt{2\mu D_\mathrm{e}}}{a\hbar}$, $a = \sqrt{\frac{k}{2Dₑ}}$, $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$, $\xi := 2\lambda\mathrm{e}^{-a(r-r_e)}$ are defined. The domain is $0\leq r \lt \infty$.

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Antique.ψ — Method

ψ(model::PoschlTeller, x; n=0)

\[\psi_n(x) = P_\lambda^{\mu}(\mathrm{tanh}(x/x_0)) \sqrt{\mu\frac{\Gamma(\lambda-\mu+1)}{\Gamma(\lambda+\mu+1)}},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$ and $P_\lambda^{\mu}$ are the associated Legendre functions.

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Antique.ψ — Method

ψ(model::RigidRotor, θ, φ; l=0, m=0)

\[\psi_{lm}(\theta,\varphi) = Y_{lm}(\theta,\varphi)\]

The wave function is the spherical harmonics. The domain is $0\leq \theta \lt \pi$ and $0\leq \varphi \lt 2\pi$.

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Antique.ψ — Method

ψ(model::SphericalOscillator, r, θ, φ; n=0, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

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+= \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/\mu}$ is the angular frequency and $\xi = \sqrt{\frac{\mu\omega}{\hbar}}r$.

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Antique.Y — Method

Y(model::CoulombTwoBody, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.Y — Method

Y(model::HydrogenAtom, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.Y — Method

Y(model::RigidRotor, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.Y — Method

Y(model::SphericalOscillator, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.nₘₐₓ — Method

nₘₐₓ(model::MorsePotential)

\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]

where $\omega = \sqrt{k/µ}$ is defined.

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Antique.nₘₐₓ — Method

nₘₐₓ(model::PoschlTeller)

\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]

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Antique.ψ — Method

ψ(model::CoulombTwoBody, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

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Antique.ψ — Method

ψ(model::DeltaPotential, x)

\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]

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Antique.ψ — Method

ψ(model::HarmonicOscillator, x; n=0)

\[\psi_n(x) = A_n H_n(\xi) \exp{\left( -\frac{\xi^2}{2} \right)},\]

where $\omega = \sqrt{k/m}$, $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, $A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$, $H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$ are defined.

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Antique.ψ — Method

ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

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Antique.ψ — Method

ψ(model::InfinitePotentialWell, x; n=1)

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

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Antique.ψ — Method

ψ(model::InfinitePotentialWell3D, x,y,z; nx=1, ny=1, nz=1)

The wave functions can be expressed as products of wave functions in a one-dimensional box.

\[\psi_{n_x,n_y,n_z}(x,y,z) = \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) = \sqrt{\frac{8}{L_xL_yL_z}} \sin\left(\frac{n_x\pi x}{L_x}\right) \sin\left(\frac{n_y\pi y}{L_y}\right) \sin\left(\frac{n_z\pi z}{L_z}\right)\]

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Antique.ψ — Method

ψ(model::MorsePotential, r; n=0)

\[\psi_n(r) = N_n z^{\lambda-n-1/2} \mathrm{e}^{-z/2} L_n^{(2\lambda-2n-1)}(\xi),\]

$N_n = \sqrt{\frac{n!(2\lambda-2n-1)a}{\Gamma(2\lambda-n)}}$, $\lambda = \frac{\sqrt{2\mu D_\mathrm{e}}}{a\hbar}$, $a = \sqrt{\frac{k}{2Dₑ}}$, $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$, $\xi := 2\lambda\mathrm{e}^{-a(r-r_e)}$ are defined. The domain is $0\leq r \lt \infty$.

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Antique.ψ — Method

ψ(model::PoschlTeller, x; n=0)

\[\psi_n(x) = P_\lambda^{\mu}(\mathrm{tanh}(x/x_0)) \sqrt{\mu\frac{\Gamma(\lambda-\mu+1)}{\Gamma(\lambda+\mu+1)}},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$ and $P_\lambda^{\mu}$ are the associated Legendre functions.

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Antique.ψ — Method

ψ(model::RigidRotor, θ, φ; l=0, m=0)

\[\psi_{lm}(\theta,\varphi) = Y_{lm}(\theta,\varphi)\]

The wave function is the spherical harmonics. The domain is $0\leq \theta \lt \pi$ and $0\leq \varphi \lt 2\pi$.

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Antique.ψ — Method

ψ(model::SphericalOscillator, r, θ, φ; n=0, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

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diff --git a/dev/CoulombTwoBody/index.html b/dev/CoulombTwoBody/index.html index a76c23b..99361a3 100644 --- a/dev/CoulombTwoBody/index.html +++ b/dev/CoulombTwoBody/index.html @@ -1,11 +1,11 @@ -Coulomb 2-Body System · Antique.jl
GitHub

Coulomb 2-Body System

This is the model of two particles interacting through Coulomb forces such as positronium, muonium, hydrogen atoms, deuterium atoms, etc.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r),\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.

Parameters

Antique.CoulombTwoBody — Type

CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$z₁$ is the charge number of particle 1, $z₁$ is the charge number of particle 2, $m₁$ is the mass of particle 1, $m₂$ is the mass of particle 2, $m_\mathrm{e}$ is the electron mass (the unit of $m₁$ and $m₂$), $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::CoulombTwoBody, r)

\[\begin{aligned} +Coulomb 2-Body System · Antique.jl

GitHub

Coulomb 2-Body System

This is the model of two particles interacting through Coulomb forces such as positronium, muonium, hydrogen atoms, deuterium atoms, etc.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r),\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.

Parameters

Antique.CoulombTwoBody — Type

CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$z₁$ is the charge number of particle 1, $z₁$ is the charge number of particle 2, $m₁$ is the mass of particle 1, $m₂$ is the mass of particle 2, $m_\mathrm{e}$ is the electron mass (the unit of $m₁$ and $m₂$), $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::CoulombTwoBody, r)

\[\begin{aligned} V(r) &= - \frac{Ze^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{Z}{r/a_0} &= - \frac{Z}{r/a_0} E_\mathrm{h}, -\end{aligned}\]

The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::CoulombTwoBody; n=1)

\[E_n -= -\frac{(z_1 z_2)^2}{2n^2} \frac{\mu}{m_\mathrm{e}} E_\mathrm{h},\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2, $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source

Eigen Functions

Antique.ψ — Method

ψ(model::CoulombTwoBody, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

source

Radial Functions

Antique.R — Method

R(model::CoulombTwoBody, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_\mu}\right)^3} \left(\frac{2Zr}{n a_\mu}\right)^l \exp \left(-\frac{Zr}{n a_\mu}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_\mu}\right),\]

where $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$, $a_\mu = a_0 \frac{m_\mathrm{e}}{\mu}$, Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

source

Associated Laguerre Polynomials

Antique.L — Method

L(model::CoulombTwoBody, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::CoulombTwoBody; n=1)

\[E_n += -\frac{(z_1 z_2)^2}{2n^2} \frac{\mu}{m_\mathrm{e}} E_\mathrm{h},\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2, $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source

Eigen Functions

Antique.ψ — Method

ψ(model::CoulombTwoBody, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

source

Radial Functions

Antique.R — Method

R(model::CoulombTwoBody, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_\mu}\right)^3} \left(\frac{2Zr}{n a_\mu}\right)^l \exp \left(-\frac{Zr}{n a_\mu}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_\mu}\right),\]

where $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$, $a_\mu = a_0 \frac{m_\mathrm{e}}{\mu}$, Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

source

Associated Laguerre Polynomials

Antique.L — Method

L(model::CoulombTwoBody, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -28,7 +28,7 @@ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots -\end{aligned}\]

source

Spherical Harmonics

Antique.Y — Method

Y(model::CoulombTwoBody, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::CoulombTwoBody, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source

Spherical Harmonics

Antique.Y — Method

Y(model::CoulombTwoBody, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::CoulombTwoBody, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -50,7 +50,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

source

Usage & Examples

Install Antique.jl 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 CoulombTwoBody and several parameters z₁, z₂, m₁, m₂, mₑ, a₀, Eₕ and ℏ are set as optional arguments.

using Antique
+\end{aligned}\]
source

Usage & Examples

Install Antique.jl 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 CoulombTwoBody and several parameters z₁, z₂, m₁, m₂, mₑ, a₀, Eₕ and ℏ are set as optional arguments.

using Antique
 Ps = CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

Parameters

julia> Ps.z₁-1
julia> Ps.z₂1
julia> Ps.m₁1.0
julia> Ps.m₂1.0
julia> Ps.mₑ1.0
julia> Ps.a₀1.0
julia> Ps.Eₕ1.0
julia> Ps.ℏ1.0

Eigen Values

Examples of positronium:

julia> E(Ps, n=1)-0.25
julia> E(Ps, n=2)-0.0625

Mass and Charge Dependence

The values of masses are cited from the 2018 CODATA recommended values, E. Tiesinga, et al., Rev. Mod. Phys. 93, 025010 (2021).

me = 1.0           # me #
 mµ = 206.7682830   # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mmusme
 mp = 1836.15267343 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mpsme
@@ -1869,4 +1869,4 @@
  3 |  3 |  2 |  2 |  2 | -1 |    0.000000000000 |    0.000000000000 ✔
  3 |  3 |  2 |  2 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
  3 |  3 |  2 |  2 |  2 |  1 |    0.000000000000 |   -0.000000000000 ✔
- 3 |  3 |  2 |  2 |  2 |  2 |    1.000000000000 |    1.000222366721 ✔
+ 3 | 3 | 2 | 2 | 2 | 2 | 1.000000000000 | 1.000222366721 ✔ diff --git a/dev/DeltaPotential/index.html b/dev/DeltaPotential/index.html index e33f42b..7bf20c7 100644 --- a/dev/DeltaPotential/index.html +++ b/dev/DeltaPotential/index.html @@ -1,5 +1,5 @@ -Delta Potential · Antique.jl
GitHub

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.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.DeltaPotential — Type

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).

source

Potential

Antique.V — Method

V(model::DeltaPotential, x)

\[V(x) = -\alpha \delta(x).\]

source

Eigen Values

Antique.E — Method

E(model::DeltaPotential)

\[E = - \frac{m\alpha^2}{2\hbar^2}\]

source

Eigen Functions

Antique.ψ — Method

ψ(model::DeltaPotential, x)

\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]

source

Usage & Examples

Install Antique.jl 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.

using Antique
+Delta Potential · Antique.jl
GitHub
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.
Definitions
This model is described with the time-independent Schrödinger equation
\[  \hat{H} \psi(x) = E \psi(x),\]
and the Hamiltonian
\[  \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]
Parameters are specified with the following struct.
Parameters
Antique.DeltaPotential — Type
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).
source
Potential
Antique.V — Method
V(model::DeltaPotential, x)
\[V(x) = -\alpha \delta(x).\]
source
Eigen Values
Antique.E — Method
E(model::DeltaPotential)
\[E = - \frac{m\alpha^2}{2\hbar^2}\]
source
Eigen Functions
Antique.ψ — Method
ψ(model::DeltaPotential, x)
\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]
source
Usage & Examples
Install Antique.jl 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.
using Antique
 DP = DeltaPotential(α=1.0, m=1.0, ℏ=1.0)
Parameters:
julia> DP.α1.0
julia> DP.m1.0
julia> DP.ℏ1.0
Eigen values:
julia> E(DP)-0.5
Wave functions:
using CairoMakie
 
 # setting
@@ -54,4 +54,4 @@
 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 ✔

+7.0 | 7.0 | 7.0 |    1.000000000000 |    1.000000000000 ✔
diff --git a/dev/HarmonicOscillator/index.html b/dev/HarmonicOscillator/index.html index 7877d4d..ae3c9f9 100644 --- a/dev/HarmonicOscillator/index.html +++ b/dev/HarmonicOscillator/index.html @@ -1,8 +1,8 @@ -Harmonic Oscillator · Antique.jl
GitHub

Harmonic Oscillator

The harmonic oscillator is the most frequently used model in quantum physics.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.HarmonicOscillator — Type

HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)

$k$ is the force constant, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::HarmonicOscillator, x)

\[V(x) +Harmonic Oscillator · Antique.jl

GitHub

Harmonic Oscillator

The harmonic oscillator is the most frequently used model in quantum physics.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.HarmonicOscillator — Type

HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)

$k$ is the force constant, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::HarmonicOscillator, x)

\[V(x) = \frac{1}{2} k x^2 = \frac{1}{2} m \omega^2 x^2 -= \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/m}$ is the angular frequency and $\xi = \sqrt{\frac{m\omega}{\hbar}}x$.

source

Eigen Values

Antique.E — Method

E(model::HarmonicOscillator; n=0)

\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

source

Eigen Functions

Antique.ψ — Method

ψ(model::HarmonicOscillator, x; n=0)

\[\psi_n(x) = A_n H_n(\xi) \exp{\left( -\frac{\xi^2}{2} \right)},\]

where $\omega = \sqrt{k/m}$, $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, $A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$, $H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$ are defined.

source

Hermite Polynomials

Antique.H — Method

H(model::HarmonicOscillator, x; n=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} += \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/m}$ is the angular frequency and $\xi = \sqrt{\frac{m\omega}{\hbar}}x$.

source

Eigen Values

Antique.E — Method

E(model::HarmonicOscillator; n=0)

\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

source

Eigen Functions

Antique.ψ — Method

ψ(model::HarmonicOscillator, x; n=0)

\[\psi_n(x) = A_n H_n(\xi) \exp{\left( -\frac{\xi^2}{2} \right)},\]

where $\omega = \sqrt{k/m}$, $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, $A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$, $H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$ are defined.

source

Hermite Polynomials

Antique.H — Method

H(model::HarmonicOscillator, x; n=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} H_{n}(x) &:= (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2} \\ &= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}. @@ -18,7 +18,7 @@ H_{8}(x) &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}, \\ H_{9}(x) &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}, \\ &\vdots -\end{aligned}\]

source

Reference

Usage & Examples

Install Antique.jl 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 HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.

using Antique
+\end{aligned}\]
source

Reference

Usage & Examples

Install Antique.jl 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 HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.

using Antique
 HO = HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)

Parameters:

julia> HO.k1.0
julia> HO.m1.0
julia> HO.ℏ1.0

Eigen values:

julia> E(HO, n=0)0.5
julia> E(HO, n=1)1.5

Potential energy curve:

using CairoMakie
 
 f = Figure()
@@ -443,4 +443,4 @@
 5.0 |  6 |   14.534441853749 |   14.534428572309 ✔
 5.0 |  7 |   16.770509831248 |   16.770492175222 ✔
 5.0 |  8 |   19.006577808748 |   19.006555152416 ✔
-5.0 |  9 |   21.242645786248 |   21.242617504750 ✔
+5.0 | 9 | 21.242645786248 | 21.242617504750 ✔ diff --git a/dev/HydrogenAtom/index.html b/dev/HydrogenAtom/index.html index 6570969..afcfbbb 100644 --- a/dev/HydrogenAtom/index.html +++ b/dev/HydrogenAtom/index.html @@ -1,12 +1,12 @@ -Hydrogen Atom · Antique.jl
GitHub

Hydrogen Atom

The hydrogen atom is the simplest Coulomb 2-body system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r),\]

where $\mu=\left(\frac{1}{m_\mathrm{e}}+\frac{1}{m_\mathrm{p}}\right)^{-1}$ is the reduced mass of electron $\mathrm{e}$ and proton $\mathrm{p}$. $\mu = m_\mathrm{e}$ holds in the limit $m_\mathrm{p}\rightarrow\infty$. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.

Parameters

Antique.HydrogenAtom — Type

HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$Z$ is the atomic number, $m_\mathrm{e}$ is the electron mass, $a_0$is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::HydrogenAtom, r)

\[\begin{aligned} +Hydrogen Atom · Antique.jl

GitHub

Hydrogen Atom

The hydrogen atom is the simplest Coulomb 2-body system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r),\]

where $\mu=\left(\frac{1}{m_\mathrm{e}}+\frac{1}{m_\mathrm{p}}\right)^{-1}$ is the reduced mass of electron $\mathrm{e}$ and proton $\mathrm{p}$. $\mu = m_\mathrm{e}$ holds in the limit $m_\mathrm{p}\rightarrow\infty$. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.

Parameters

Antique.HydrogenAtom — Type

HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$Z$ is the atomic number, $m_\mathrm{e}$ is the electron mass, $a_0$is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::HydrogenAtom, r)

\[\begin{aligned} V(r) &= - \frac{Ze^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{Z}{r/a_0} &= - \frac{Z}{r/a_0} E_\mathrm{h}, -\end{aligned}\]

The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::HydrogenAtom; n=1)

\[E_n +\end{aligned}\]

The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::HydrogenAtom; n=1)

\[E_n = -\frac{m_\mathrm{e} e^4 Z^2}{2n^2(4\pi\varepsilon_0)^2\hbar^2} -= -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source

Eigen Functions

Antique.ψ — Method

ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

source

Radial Functions

Antique.R — Method

R(model::HydrogenAtom, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_0}\right)^3} \left(\frac{2Zr}{n a_0}\right)^l \exp \left(-\frac{Zr}{n a_0}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_0}\right),\]

where Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

source

Associated Laguerre Polynomials

Antique.L — Method

L(model::HydrogenAtom, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} += -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source

Eigen Functions

Antique.ψ — Method

ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

source

Radial Functions

Antique.R — Method

R(model::HydrogenAtom, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_0}\right)^3} \left(\frac{2Zr}{n a_0}\right)^l \exp \left(-\frac{Zr}{n a_0}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_0}\right),\]

where Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

source

Associated Laguerre Polynomials

Antique.L — Method

L(model::HydrogenAtom, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -29,7 +29,7 @@ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots -\end{aligned}\]

source

Spherical Harmonics

Antique.Y — Method

Y(model::HydrogenAtom, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::HydrogenAtom, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source

Spherical Harmonics

Antique.Y — Method

Y(model::HydrogenAtom, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::HydrogenAtom, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -51,7 +51,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

source

References

Usage & Examples

Install Antique.jl 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 HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.

using Antique
+\end{aligned}\]
source

References

Usage & Examples

Install Antique.jl 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 HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.

using Antique
 H = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)

Parameters:

julia> H.Z1
julia> H.Eₕ1.0
julia> H.mₑ1.0
julia> H.a₀1.0
julia> H.ℏ1.0

Eigen values:

julia> E(H, n=1)-0.5
julia> E(H, n=2)-0.125

Wave length ($n=2\rightarrow1$, the first line of the Lyman series):

Eₕ2nm⁻¹ = 2.1947463136320e-2 # https://physics.nist.gov/cgi-bin/cuu/CCValue?hrminv
 println("ΔE = ", E(H,n=2) - E(H,n=1), " Eₕ")
 println("λ  = ", ((E(H,n=2)-E(H,n=1))*Eₕ2nm⁻¹)^-1, " nm")
ΔE = 0.375 Eₕ
@@ -1306,4 +1306,4 @@
  3 |  3 |  2 |  2 |  2 | -1 |    0.000000000000 |   -0.000000000000 ✔
  3 |  3 |  2 |  2 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
  3 |  3 |  2 |  2 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
- 3 |  3 |  2 |  2 |  2 |  2 |    1.000000000000 |    1.000300628566 ✔
+ 3 | 3 | 2 | 2 | 2 | 2 | 1.000000000000 | 1.000300628566 ✔ diff --git a/dev/InfinitePotentialWell/index.html b/dev/InfinitePotentialWell/index.html index 4eb5517..c07b0a1 100644 --- a/dev/InfinitePotentialWell/index.html +++ b/dev/InfinitePotentialWell/index.html @@ -1,11 +1,11 @@ -Infinite Potential Well · Antique.jl
GitHub

Infinite Potential Well (Particle in a Box)

The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.InfinitePotentialWell — Type

InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

$L$ is the length of the box, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::InfinitePotentialWell; x)

\[V(x) = +Infinite Potential Well · Antique.jl

GitHub

Infinite Potential Well (Particle in a Box)

The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.InfinitePotentialWell — Type

InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

$L$ is the length of the box, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::InfinitePotentialWell; x)

\[V(x) = \left\{ \begin{array}{ll} \infty & x \lt 0, L \lt x \\ 0 & 0 \leq x \leq L \end{array} -\right.\]

source

Eigen Values

Antique.E — Method

E(model::InfinitePotentialWell; n=1)

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

source

Eigen Functions

Antique.ψ — Method

ψ(model::InfinitePotentialWell, x; n=1)

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

source

Proofs

Usage & Examples

Install Antique.jl 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 InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.

using Antique
+\right.\]
source

Eigen Values

Antique.E — Method

E(model::InfinitePotentialWell; n=1)

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

source

Eigen Functions

Antique.ψ — Method

ψ(model::InfinitePotentialWell, x; n=1)

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

source

Proofs

Usage & Examples

Install Antique.jl 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 InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.

using Antique
 IPW = InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

Parameters:

julia> IPW.L1.0
julia> IPW.m1.0
julia> IPW.ℏ1.0

Eigen values:

julia> E(IPW, n=1)4.934802200544679
julia> E(IPW, n=2)19.739208802178716

Wave functions:

using CairoMakie
 
 # settings
@@ -379,4 +379,4 @@
 0.1 |  1 |  986.960431582781 |  986.960440108936 ✔
 0.5 |  1 |   39.478417274195 |   39.478417604357 ✔
 1.0 |  1 |    9.869604318963 |    9.869604401089 ✔
-7.0 |  1 |    0.201420496383 |    0.201420497981 ✔
+7.0 | 1 | 0.201420496383 | 0.201420497981 ✔ diff --git a/dev/InfinitePotentialWell3D/index.html b/dev/InfinitePotentialWell3D/index.html index 0d56d3d..3f403eb 100644 --- a/dev/InfinitePotentialWell3D/index.html +++ b/dev/InfinitePotentialWell3D/index.html @@ -1,11 +1,11 @@ -Infinite Potential Well 3D · Antique.jl
GitHub

Infinite Potential Well 3D (Particle in a 3D-Box)

The three-dimensional infinite potential well (particle in a 3D-box) is one of the simplest models for a three-dimensional quantum mechanical system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x,y,z) = E \psi(x,y,z),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial x ^2} + \frac{\partial^2}{\partial y ^2} + \frac{\partial^2}{\partial z ^2}\right) + V(x,y,z).\]

Parameters are specified with the following struct.

Parameters

Antique.InfinitePotentialWell3D — Type

InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)

$L_x,L_y,L_z$ are the lengths of the box in $x$,$y$,$z$-direction, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::InfinitePotentialWell3D, x,y,z)

\[V(x,y,z) = +Infinite Potential Well 3D · Antique.jl

GitHub

Infinite Potential Well 3D (Particle in a 3D-Box)

The three-dimensional infinite potential well (particle in a 3D-box) is one of the simplest models for a three-dimensional quantum mechanical system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x,y,z) = E \psi(x,y,z),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial x ^2} + \frac{\partial^2}{\partial y ^2} + \frac{\partial^2}{\partial z ^2}\right) + V(x,y,z).\]

Parameters are specified with the following struct.

Parameters

Antique.InfinitePotentialWell3D — Type

InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)

$L_x,L_y,L_z$ are the lengths of the box in $x$,$y$,$z$-direction, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::InfinitePotentialWell3D, x,y,z)

\[V(x,y,z) = \left\{ \begin{array}{ll} 0 & 0 \leq x \leq L_x \ \mathrm{and}\ 0 \leq y \leq L_y \ \mathrm{and}\ 0 \leq z \leq L_z \\ \infty & \mathrm{elsewhere} \end{array} -\right.\]

source

Eigen Values

Antique.E — Method

E(model::InfinitePotentialWell3D; nx=1, ny=1, nz=1)

\[E_{n_x,n_y,n_z} = \frac{\hbar^2 \pi^2}{2 m} \left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right)\]

source

Eigen Functions

Antique.ψ — Method

ψ(model::InfinitePotentialWell3D, x,y,z; nx=1, ny=1, nz=1)

The wave functions can be expressed as products of wave functions in a one-dimensional box.

\[\psi_{n_x,n_y,n_z}(x,y,z) = \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) = \sqrt{\frac{8}{L_xL_yL_z}} \sin\left(\frac{n_x\pi x}{L_x}\right) \sin\left(\frac{n_y\pi y}{L_y}\right) \sin\left(\frac{n_z\pi z}{L_z}\right)\]

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are supported. In this system, the model is generated by InfinitePotentialWell3D and several parameters Lx, Ly, Lz, m and ℏ are set as optional arguments.

using Antique
+\right.\]
source

Eigen Values

Antique.E — Method

E(model::InfinitePotentialWell3D; nx=1, ny=1, nz=1)

\[E_{n_x,n_y,n_z} = \frac{\hbar^2 \pi^2}{2 m} \left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right)\]

source

Eigen Functions

Antique.ψ — Method

ψ(model::InfinitePotentialWell3D, x,y,z; nx=1, ny=1, nz=1)

The wave functions can be expressed as products of wave functions in a one-dimensional box.

\[\psi_{n_x,n_y,n_z}(x,y,z) = \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) = \sqrt{\frac{8}{L_xL_yL_z}} \sin\left(\frac{n_x\pi x}{L_x}\right) \sin\left(\frac{n_y\pi y}{L_y}\right) \sin\left(\frac{n_z\pi z}{L_z}\right)\]

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are supported. In this system, the model is generated by InfinitePotentialWell3D and several parameters Lx, Ly, Lz, m and ℏ are set as optional arguments.

using Antique
 IPW3D = InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)

Parameters:

julia> IPW3D.Lx1.0
julia> IPW3D.Ly1.0
julia> IPW3D.Lz1.0
julia> IPW3D.m1.0
julia> IPW3D.ℏ1.0

Eigen values:

julia> E(IPW3D, nx=1, ny=1, nz=1)14.804406601634037
julia> E(IPW3D, nx=2, ny=1, nz=1)29.608813203268074
julia> E(IPW3D, nx=1, ny=2, nz=1)29.608813203268074
julia> E(IPW3D, nx=1, ny=1, nz=2)29.608813203268074
julia> E(IPW3D, nx=2, ny=2, nz=1)44.41321980490211
julia> E(IPW3D, nx=2, ny=1, nz=2)44.41321980490211
julia> E(IPW3D, nx=1, ny=2, nz=2)44.41321980490211
julia> E(IPW3D, nx=2, ny=2, nz=2)59.21762640653615

Wave functions:

The wave functions of the 3D infinite potential well can be expressed as a product of wave functions in 1D infinite potential wells for each of the three directions $x$,$y$,$z$. As the representation of 3D (non-spherical) wave functions easily becomes unclear, we refer to the documentation of the one-dimensional counterpart: InfinitePotentialWell (...)

Testing

Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.

Normalization & Orthogonality of $\psi_{n_x,n_y,n_z}(x,y,z)$

\[\int_{0}^{L_x}\int_{0}^{L_y}\int_{0}^{L_z} \psi_{i_x,i_y,i_z}^\ast(x,y,z) \psi_{j_x,j_y,j_z}(x,y,z) ~\mathrm{d}x \mathrm{d}y\mathrm{d}z = \delta_{i_x,j_x}\delta_{i_y,j_y}\delta_{i_z,j_z}\]

ix | iy | iz | jx | jy | jz |        analytical |         numerical 
 -- | -- | -- | -- | -- | -- | ----------------- | ----------------- 
  1 |  1 |  1 |  1 |  1 |  1 |    1.000000000000 |    0.999949984990 ✔
@@ -124,4 +124,4 @@
   2 |   1 |   1 |   20.954151942424 |   21.521220707931 ✔
   2 |   1 |   2 |   22.555743156203 |   23.166154774779 ✔
   2 |   2 |   1 |   24.557732170327 |   25.222322358339 ✔
-  2 |   2 |   2 |   26.159323384853 |   26.867256425188 ✔
+ 2 | 2 | 2 | 26.159323384853 | 26.867256425188 ✔ diff --git a/dev/MorsePotential/index.html b/dev/MorsePotential/index.html index 046368d..67292db 100644 --- a/dev/MorsePotential/index.html +++ b/dev/MorsePotential/index.html @@ -1,5 +1,5 @@ -Morse Potential · Antique.jl
GitHub

Morse Potential

The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(r) = E \psi(r),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + V(r)\]

Parameters are specified with the following struct.

Parameters

Antique.MorsePotential — Type

MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)

$r_\mathrm{e}$ is the equilibrium bond distance, $D_\mathrm{e}$ is the the well depth , $k$ is the force constant, $\mu$ is the reduced mass and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::MorsePotential, r)

\[V(r) = D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::MorsePotential; n=0)

\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]

where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.

source

Number of Bound States

Antique.nₘₐₓ — Method

nₘₐₓ(model::MorsePotential)

\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]

where $\omega = \sqrt{k/µ}$ is defined.

source

Eigen Functions

Antique.ψ — Method

ψ(model::MorsePotential, r; n=0)

\[\psi_n(r) = N_n z^{\lambda-n-1/2} \mathrm{e}^{-z/2} L_n^{(2\lambda-2n-1)}(\xi),\]

$N_n = \sqrt{\frac{n!(2\lambda-2n-1)a}{\Gamma(2\lambda-n)}}$, $\lambda = \frac{\sqrt{2\mu D_\mathrm{e}}}{a\hbar}$, $a = \sqrt{\frac{k}{2Dₑ}}$, $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$, $\xi := 2\lambda\mathrm{e}^{-a(r-r_e)}$ are defined. The domain is $0\leq r \lt \infty$.

source

Generalized Laguerre Polynomials

Note

The generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, not the associated Laguerre polynomials $L_n^{k}(x)$, are used in this model.

Antique.L — Method

L(model::MorsePotential, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +Morse Potential · Antique.jl

GitHub

Morse Potential

The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(r) = E \psi(r),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + V(r)\]

Parameters are specified with the following struct.

Parameters

Antique.MorsePotential — Type

MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)

$r_\mathrm{e}$ is the equilibrium bond distance, $D_\mathrm{e}$ is the the well depth , $k$ is the force constant, $\mu$ is the reduced mass and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::MorsePotential, r)

\[V(r) = D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::MorsePotential; n=0)

\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]

where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.

source

Number of Bound States

Antique.nₘₐₓ — Method

nₘₐₓ(model::MorsePotential)

\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]

where $\omega = \sqrt{k/µ}$ is defined.

source

Eigen Functions

Antique.ψ — Method

ψ(model::MorsePotential, r; n=0)

\[\psi_n(r) = N_n z^{\lambda-n-1/2} \mathrm{e}^{-z/2} L_n^{(2\lambda-2n-1)}(\xi),\]

$N_n = \sqrt{\frac{n!(2\lambda-2n-1)a}{\Gamma(2\lambda-n)}}$, $\lambda = \frac{\sqrt{2\mu D_\mathrm{e}}}{a\hbar}$, $a = \sqrt{\frac{k}{2Dₑ}}$, $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$, $\xi := 2\lambda\mathrm{e}^{-a(r-r_e)}$ are defined. The domain is $0\leq r \lt \infty$.

source

Generalized Laguerre Polynomials

Note

The generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, not the associated Laguerre polynomials $L_n^{k}(x)$, are used in this model.

Antique.L — Method

L(model::MorsePotential, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ @@ -21,7 +21,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots -\end{aligned}\]

source

References

Usage & Examples

Install Antique.jl 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 MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.

# Parameters for H₂⁺
+\end{aligned}\]
source

References

Usage & Examples

Install Antique.jl 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 MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.

# 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
@@ -766,10 +766,10 @@
  9 |  8 |    0.000000000000 |    0.000000000000 ✔
  9 |  9 |    1.000000000000 |    1.000000000015 ✔

Eigen Values

\[ \begin{aligned} E_n - &= \int \psi^\ast_n(r) \hat{H} \psi_n(r) \mathrm{d}x \\ - &= \int \psi^\ast_n(r) \left[ \hat{V} + \hat{T} \right] \psi(r) \mathrm{d}x \\ - &= \int \psi^\ast_n(r) \left[ V(r) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} r^{2}} \right] \psi(r) \mathrm{d}x \\ - &\simeq \int \psi^\ast_n(r) \left[ V(r)\psi(r) -\frac{\hbar^2}{2m} \frac{\psi(r+\Delta r) - 2\psi(r) + \psi(r-\Delta r)}{\Delta r^{2}} \right] \mathrm{d}x. + &= \int \psi^\ast_n(r) \hat{H} \psi_n(r) \mathrm{d}r \\ + &= \int \psi^\ast_n(r) \left[ \hat{V} + \hat{T} \right] \psi(r) \mathrm{d}r \\ + &= \int \psi^\ast_n(r) \left[ V(r) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} r^{2}} \right] \psi(r) \mathrm{d}r \\ + &\simeq \int \psi^\ast_n(r) \left[ V(r)\psi(r) -\frac{\hbar^2}{2m} \frac{\psi(r+\Delta r) - 2\psi(r) + \psi(r-\Delta r)}{\Delta r^{2}} \right] \mathrm{d}r. \end{aligned}\]

Where, the difference formula for the 2nd-order derivative:

\[\begin{aligned} % 2\psi(r) % + \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} \Delta r^{2} @@ -849,4 +849,4 @@ 0.1 | 6 | -0.045393223648 | -0.045393222818 ✔ 0.1 | 7 | -0.038630997356 | -0.038631017157 ✔ 0.1 | 8 | -0.032413875662 | -0.032413886246 ✔ -0.1 | 9 | -0.026741858566 | -0.026742018376 ✔

+0.1 | 9 | -0.026741858566 | -0.026742018376 ✔ diff --git a/dev/PoschlTeller/index.html b/dev/PoschlTeller/index.html index 783789f..ee19c7e 100644 --- a/dev/PoschlTeller/index.html +++ b/dev/PoschlTeller/index.html @@ -1,11 +1,11 @@ -Pöschl-Teller Potential · Antique.jl
GitHub

Pöschl-Teller Potential

The Pöschl-Teller potential is one of the few potentials for which the quantum mechanical Schrödinger equation has an analytical solution. It has a finite number of bound states, which can be inferred easily from its potential strength parameter λ. It is defined for one-dimensional systems.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2 m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} - \frac{\hbar^2}{m x_0^2} \frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}(x/x_0)^2}.\]

After introducing the dimensionless variables

\[ x^\ast \equiv x/x_0,\qquad E^\ast \equiv \frac{\hbar^2}{m x_0^2} E\]

the Schrödinger equation reduces to

\[ \hat{H}^\ast \psi(x^\ast) = E^\ast \psi(x^\ast),\]

with

\[ \hat{H}^\ast = - \frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{d}{x^\ast}^2} - \frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}(x^\ast)^2}.\]

Parameters are specified within the following struct.

Parameters

Potential

Antique.V — Method

V(model::PoschlTeller, x)

\[\begin{aligned} +Pöschl-Teller Potential · Antique.jl

GitHub

Pöschl-Teller Potential

The Pöschl-Teller potential is one of the few potentials for which the quantum mechanical Schrödinger equation has an analytical solution. It has a finite number of bound states, which can be inferred easily from its potential strength parameter λ. It is defined for one-dimensional systems.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2 m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} - \frac{\hbar^2}{m x_0^2} \frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}(x/x_0)^2}.\]

After introducing the dimensionless variables

\[ x^\ast \equiv x/x_0,\qquad E^\ast \equiv \frac{\hbar^2}{m x_0^2} E\]

the Schrödinger equation reduces to

\[ \hat{H}^\ast \psi(x^\ast) = E^\ast \psi(x^\ast),\]

with

\[ \hat{H}^\ast = - \frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{d}{x^\ast}^2} - \frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}(x^\ast)^2}.\]

Parameters are specified within the following struct.

Parameters

Potential

Antique.V — Method

V(model::PoschlTeller, x)

\[\begin{aligned} V(x) &= -\frac{\hbar^2}{m x_0^2} \frac{\lambda(\lambda+1)}{2} \mathrm{sech}(x)^2 &= -\frac{\hbar^2}{m x_0^2} \frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}(x)^2}. -\end{aligned}\]

source

Number of Bound States

Antique.nₘₐₓ — Method

nₘₐₓ(model::PoschlTeller)

\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]

source

Eigen Values

Antique.E — Method

E(model::PoschlTeller; n=0)

\[E_n = -\frac{\hbar^2}{m x_0^2}\frac{\mu^2}{2},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.

source

Eigen Functions

Antique.ψ — Method

ψ(model::PoschlTeller, x; n=0)

\[\psi_n(x) = P_\lambda^{\mu}(\mathrm{tanh}(x/x_0)) \sqrt{\mu\frac{\Gamma(\lambda-\mu+1)}{\Gamma(\lambda+\mu+1)}},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$ and $P_\lambda^{\mu}$ are the associated Legendre functions.

source

Associated Legendre Polynomials

Antique.P — Method

P(model::PoschlTeller, x; n=0, m=0)

Associated Legendre polynomials are the associated Legendre functions for integer indices. Here we use the same notation of the associated Legendre functions as in the model HydrogenAtom.

```math \begin{aligned} Pn^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} Pn(x) \ +\end{aligned}\]

source

Number of Bound States

Antique.nₘₐₓ — Method

nₘₐₓ(model::PoschlTeller)

\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]

source

Eigen Values

Antique.E — Method

E(model::PoschlTeller; n=0)

\[E_n = -\frac{\hbar^2}{m x_0^2}\frac{\mu^2}{2},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.

source

Eigen Functions

Antique.ψ — Method

ψ(model::PoschlTeller, x; n=0)

\[\psi_n(x) = P_\lambda^{\mu}(\mathrm{tanh}(x/x_0)) \sqrt{\mu\frac{\Gamma(\lambda-\mu+1)}{\Gamma(\lambda+\mu+1)}},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$ and $P_\lambda^{\mu}$ are the associated Legendre functions.

source

Associated Legendre Polynomials

Antique.P — Method

P(model::PoschlTeller, x; n=0, m=0)

Associated Legendre polynomials are the associated Legendre functions for integer indices. Here we use the same notation of the associated Legendre functions as in the model HydrogenAtom.

```math \begin{aligned} Pn^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} Pn(x) \ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \ - &= \frac{1}{2^n} (1-x^2)^{m/2} \sum_{j=0}^{\left\lfloor\frac{n-m}{2}\right\rfloor} (-1)^j \frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \end{aligned} ```

source

Usage & Examples

Install Antique.jl 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 PoschlTeller and the parameters λ, m, ℏ, x₀.

using Antique
+  &= \frac{1}{2^n} (1-x^2)^{m/2} \sum_{j=0}^{\left\lfloor\frac{n-m}{2}\right\rfloor} (-1)^j \frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}.   \end{aligned}   ```
source

Usage & Examples

Install Antique.jl 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 PoschlTeller and the parameters λ, m, ℏ, x₀.

using Antique
 PT = PoschlTeller(λ=4.0)

Parameters:

julia> PT.λ4.0
julia> PT.m1.0
julia> PT.ℏ1.0
julia> PT.x₀1.0

Number of bound states:

julia> nₘₐₓ(PT)3

Eigen values:

julia> E(PT, n=0)-8.0
julia> E(PT, n=1)-4.5
julia> E(PT, n=2)-2.0
julia> E(PT, n=3)-0.5

Potential energy curve:

using CairoMakie
 
 f = Figure()
@@ -237,4 +237,4 @@
 3.0 |  2 | 2.7 |1.0 |1.0 |  -0.500000000000 |   -0.500000668186 ✔
 3.0 |  2 | 2.7 |1.0 |2.7 |  -0.500000000000 |   -0.500000668186 ✔
 3.0 |  2 | 2.7 |2.7 |1.0 |  -0.500000000000 |   -0.500000668186 ✔
-3.0 |  2 | 2.7 |2.7 |2.7 |  -0.500000000000 |   -0.500000668186 ✔
+3.0 | 2 | 2.7 |2.7 |2.7 | -0.500000000000 | -0.500000668186 ✔ diff --git a/dev/RigidRotor/index.html b/dev/RigidRotor/index.html index e5229fa..e3382d5 100644 --- a/dev/RigidRotor/index.html +++ b/dev/RigidRotor/index.html @@ -1,12 +1,13 @@ Rigid Rotor · Antique.jl
GitHub

Rigid rotor

The linear rigid rotor model can be used in quantum mechanics to predict the rotational energy of a diatomic molecule. The rotational energy depends on the moment of inertia for the system, $I$.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]

and the Hamiltonian

\[\begin{aligned} \hat{H} &= - \frac{\hbar^2}{2\mu} \nabla^2 + V(r), \\ - &= - \frac{\hbar^2}{2I} \left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \left(\sin\theta \frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2} \right] -\end{aligned}\]

where $I=\mu R^2$ is the moment of intertia, $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of two particles and $R$ is the distance between the two particles. Parameters are specified with the following struct.

Parameters

Antique.RigidRotor — Type

RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)

$m₁$ and $m₂$ are mass of two particles, $R$ is the distance, and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::RigidRotor, r)

\[\begin{aligned} + &= - \frac{\hbar^2}{2I} \left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \left(\sin\theta \frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2} \right], \\ + &= \frac{L^2}{2I} +\end{aligned}\]

where $I=\mu R^2$ is the moment of intertia, $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of two particles, $R$ is the distance between the two particles, and $L^2$ is the angular momentum operator. Parameters are specified with the following struct.

Parameters

Antique.RigidRotor — Type

RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)

$m₁$ and $m₂$ are mass of two particles, $R$ is the distance, and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::RigidRotor, r)

\[\begin{aligned} V(r) &= 0, -\end{aligned}\]

source

Eigen Values

Antique.E — Method

E(model::RigidRotor; l=0)

\[E_l -= \frac{\hbar^2}{2I}l(l+1),\]

where $I=\mu R^2$ is the moment of inertia, $R$ is the distance, and $\mu$ is the reduced mass of the two particles.

source

Eigen Functions

Antique.ψ — Method

ψ(model::RigidRotor, θ, φ; l=0, m=0)

\[\psi_{lm}(\theta,\varphi) = Y_{lm}(\theta,\varphi)\]

The wave function is the spherical harmonics. The domain is $0\leq \theta \lt \pi$ and $0\leq \varphi \lt 2\pi$.

source

Spherical Harmonics

Antique.Y — Method

Y(model::RigidRotor, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::RigidRotor, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source

Eigen Values

Antique.E — Method

E(model::RigidRotor; l=0)

\[E_l += \frac{\hbar^2}{2I}l(l+1),\]

where $I=\mu R^2$ is the moment of inertia, $R$ is the distance, and $\mu$ is the reduced mass of the two particles.

source

Eigen Functions

Antique.ψ — Method

ψ(model::RigidRotor, θ, φ; l=0, m=0)

\[\psi_{lm}(\theta,\varphi) = Y_{lm}(\theta,\varphi)\]

The wave function is the spherical harmonics. The domain is $0\leq \theta \lt \pi$ and $0\leq \varphi \lt 2\pi$.

source

Spherical Harmonics

Antique.Y — Method

Y(model::RigidRotor, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::RigidRotor, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -28,7 +29,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

source

Usage & Examples

Install Antique.jl 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 RigidRotor and several parameters m₁, m₂, R and ℏ are set as optional arguments.

using Antique
+\end{aligned}\]
source

References

  • Anderson, J.M. Introduction to Quantum Chemistry, 1969, W.A. Benjamin, Inc, p.91-100
  • Chemistry Libre Texts

Usage & Examples

Install Antique.jl 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 RigidRotor and several parameters m₁, m₂, R and ℏ are set as optional arguments.

using Antique
 RR = RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)

Parameters:

julia> RR.m₁1.0
julia> RR.m₂1.0
julia> RR.R1.0
julia> RR.ℏ1.0

Eigen values:

julia> E(RR, l=0)0.0
julia> E(RR, l=1)2.0
julia> E(RR, l=2)6.0

Wave functions:

julia> ψ(RR, 0, 0, l=2, m=1)-0.0 + 0.0im
julia> ψ(RR, π/4, 0, l=2, m=1)-0.38627420202318963 + 0.0im
julia> ψ(RR, π/4, π/2, l=2, m=1)-2.3652473255044865e-17 - 0.38627420202318963im
using CairoMakie
 
 f = Figure(size=(400,400))
@@ -567,4 +568,4 @@
  2 |  2 |  2 | -1 |    0.000000000000 |   -0.000000000000 ✔
  2 |  2 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
  2 |  2 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
- 2 |  2 |  2 |  2 |    1.000000000000 |    1.000000000000 ✔
+ 2 | 2 | 2 | 2 | 1.000000000000 | 1.000000000000 ✔ diff --git a/dev/SphericalOscillator/index.html b/dev/SphericalOscillator/index.html index 9ab8ff5..4a56992 100644 --- a/dev/SphericalOscillator/index.html +++ b/dev/SphericalOscillator/index.html @@ -1,9 +1,9 @@ -Spherical Oscillator · Antique.jl
GitHub

Spherical Oscillator

The spherical oscillator (3D isotropic harmonic oscillator) is the most frequently used model in quantum physics. This model uses a spherical coordinate system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(r) = E \psi(r),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r).\]

Parameters are specified with the following struct.

Parameters

Antique.SphericalOscillator — Type

HarmonicOscillator(k=1.0, μ=1.0, ℏ=1.0)

$k$ is the force constant, $μ$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::SphericalOscillator, r)

\[V(r) +Spherical Oscillator · Antique.jl

GitHub

Spherical Oscillator

The spherical oscillator (3D isotropic harmonic oscillator) is the most frequently used model in quantum physics. This model uses a spherical coordinate system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(r) = E \psi(r),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r).\]

Parameters are specified with the following struct.

Parameters

Antique.SphericalOscillator — Type

HarmonicOscillator(k=1.0, μ=1.0, ℏ=1.0)

$k$ is the force constant, $μ$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::SphericalOscillator, r)

\[V(r) = \frac{1}{2} k r^2 = \frac{1}{2} \mu \omega^2 r^2 -= \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/\mu}$ is the angular frequency and $\xi = \sqrt{\frac{\mu\omega}{\hbar}}r$.

source

Eigen Values

Antique.E — Method

E(model::SphericalOscillator; n=0, l=0)

\[E_{nl} -= \left(2n + l + \frac{3}{2}\right)\hbar \omega,\]

where $\omega = \sqrt{k/\mu}$.

source

Eigen Functions

Antique.ψ — Method

ψ(model::SphericalOscillator, r, θ, φ; n=0, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

source

Radial Functions

Antique.R — Method

R(model::SphericalOscillator, r; n=0, l=0)

\[R_{nl}(r) = \sqrt{ \frac{\gamma^{3/2}}{2\sqrt{\pi}}} \sqrt{\frac{2^{n+l+3} n!}{(2n+2l+1)!!}} \xi^l \exp\left(-\xi^2/2\right)L_{n}^{(l+\frac{1}{2})} \left(\xi^2\right),\]

where $\gamma = \mu\omega/\hbar$ and $\xi = \sqrt{\gamma}r = \sqrt{\mu\omega/\hbar}r$ are defined. The generalized Laguerre polynomials are defined as $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$. The domain is $0\leq r \lt \infty$.

source

Generalized Laguerre Polynomials

Note

The generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, not the associated Laguerre polynomials $L_n^{k}(x)$, are used in this model.

Antique.L — Method

L(model::SphericalOscillator, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} += \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/\mu}$ is the angular frequency and $\xi = \sqrt{\frac{\mu\omega}{\hbar}}r$.

source

Eigen Values

Antique.E — Method

E(model::SphericalOscillator; n=0, l=0)

\[E_{nl} += \left(2n + l + \frac{3}{2}\right)\hbar \omega,\]

where $\omega = \sqrt{k/\mu}$.

source

Eigen Functions

Antique.ψ — Method

ψ(model::SphericalOscillator, r, θ, φ; n=0, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

source

Radial Functions

Antique.R — Method

R(model::SphericalOscillator, r; n=0, l=0)

\[R_{nl}(r) = \sqrt{ \frac{\gamma^{3/2}}{2\sqrt{\pi}}} \sqrt{\frac{2^{n+l+3} n!}{(2n+2l+1)!!}} \xi^l \exp\left(-\xi^2/2\right)L_{n}^{(l+\frac{1}{2})} \left(\xi^2\right),\]

where $\gamma = \mu\omega/\hbar$ and $\xi = \sqrt{\gamma}r = \sqrt{\mu\omega/\hbar}r$ are defined. The generalized Laguerre polynomials are defined as $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$. The domain is $0\leq r \lt \infty$.

source

Generalized Laguerre Polynomials

Note

The generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, not the associated Laguerre polynomials $L_n^{k}(x)$, are used in this model.

Antique.L — Method

L(model::SphericalOscillator, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ @@ -25,7 +25,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots -\end{aligned}\]

source

Spherical Harmonics

Antique.Y — Method

Y(model::SphericalOscillator, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::SphericalOscillator, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source

Spherical Harmonics

Antique.Y — Method

Y(model::SphericalOscillator, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::SphericalOscillator, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -47,7 +47,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

source

Usage & Examples

Install Antique.jl 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 SphericalOscillator and several parameters k, μ and ℏ are set as optional arguments.

using Antique
+\end{aligned}\]
source

Usage & Examples

Install Antique.jl 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 SphericalOscillator and several parameters k, μ and ℏ are set as optional arguments.

using Antique
 SO = SphericalOscillator(k=1.0, μ=1.0, ℏ=1.0)

Parameters:

julia> SO.k1.0
julia> SO.μ1.0
julia> SO.ℏ1.0

Eigen values:

julia> E(SO, n=0)1.5
julia> E(SO, n=1)3.5
julia> E(SO, n=2)5.5
julia> E(SO, n=0, l=1)2.5
julia> E(SO, n=1, l=1)4.5
julia> E(SO, n=2, l=1)6.5

Testing

Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.

Associated Legendre Polynomials $P_n^m(x)$

\[ \begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ @@ -1252,4 +1252,4 @@ 1 | 1 | 1 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔ 1 | 1 | 1 | 1 | 1 | -1 | 0.000000000000 | -0.000000000000 ✔ 1 | 1 | 1 | 1 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔ - 1 | 1 | 1 | 1 | 1 | 1 | 1.000000000000 | 0.999999951768 ✔

+ 1 | 1 | 1 | 1 | 1 | 1 | 1.000000000000 | 0.999999951768 ✔ diff --git a/dev/index.html b/dev/index.html index efd76df..5cca89f 100644 --- a/dev/index.html +++ b/dev/index.html @@ -29,4 +29,4 @@ MorsePotential -

Future Works

List of quantum-mechanical systems with analytical solutions

Developer's Guide

Here is the guideline for adding new models.

Acknowledgment

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

+

Future Works

List of quantum-mechanical systems with analytical solutions

Developer's Guide

Here is the guideline for adding new models.

Acknowledgment

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

diff --git a/dev/search_index.js b/dev/search_index.js index fee15d5..a21cc2d 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"CurrentModule = Antique","category":"page"},{"location":"PoschlTeller/#Pöschl-Teller-Potential","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"The Pöschl-Teller potential is one of the few potentials for which the quantum mechanical Schrödinger equation has an analytical solution. It has a finite number of bound states, which can be inferred easily from its potential strength parameter λ. It is defined for one-dimensional systems.","category":"page"},{"location":"PoschlTeller/#Definitions","page":"Pöschl-Teller Potential","title":"Definitions","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"and the Hamiltonian","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH = - frachbar^22 m fracmathrmd^2mathrmdx ^2 - frachbar^2m x_0^2 fraclambda(lambda+1)2 frac1mathrmcosh(xx_0)^2","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"After introducing the dimensionless variables","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" x^ast equiv xx_0qquad E^ast equiv frachbar^2m x_0^2 E","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"the Schrödinger equation reduces to","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH^ast psi(x^ast) = E^ast psi(x^ast)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"with","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH^ast = - frac12 fracmathrmd^2mathrmdx^ast^2 - fraclambda(lambda+1)2 frac1mathrmcosh(x^ast)^2","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Parameters are specified within the following struct.","category":"page"},{"location":"PoschlTeller/#Parameters","page":"Pöschl-Teller Potential","title":"Parameters","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.PoschlTeller","category":"page"},{"location":"PoschlTeller/#Antique.PoschlTeller-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.PoschlTeller","text":"PoschlTeller(λ=1.0, m=1.0, ℏ=1.0, x₀=1.0)\n\nlambda determines the potential strength.\n\n\n\n\n\n","category":"type"},{"location":"PoschlTeller/#Potential","page":"Pöschl-Teller Potential","title":"Potential","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.V(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.V-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.V","text":"V(model::PoschlTeller, x)\n\nbeginaligned\nV(x)\n= -frachbar^2m x_0^2 fraclambda(lambda+1)2 mathrmsech(x)^2\n= -frachbar^2m x_0^2 fraclambda(lambda+1)2 frac1mathrmcosh(x)^2\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Number-of-Bound-States","page":"Pöschl-Teller Potential","title":"Number of Bound States","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.nₘₐₓ(::PoschlTeller)","category":"page"},{"location":"PoschlTeller/#Antique.nₘₐₓ-Tuple{PoschlTeller}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::PoschlTeller)\n\nn_mathrmmax = leftlfloor lambda rightrfloor - 1\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Eigen-Values","page":"Pöschl-Teller Potential","title":"Eigen Values","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.E(::PoschlTeller)","category":"page"},{"location":"PoschlTeller/#Antique.E-Tuple{PoschlTeller}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.E","text":"E(model::PoschlTeller; n=0)\n\nE_n = -frachbar^2m x_0^2fracmu^22\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1.\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Eigen-Functions","page":"Pöschl-Teller Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.ψ(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.ψ-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.ψ","text":"ψ(model::PoschlTeller, x; n=0)\n\npsi_n(x) = P_lambda^mu(mathrmtanh(xx_0)) sqrtmufracGamma(lambda-mu+1)Gamma(lambda+mu+1)\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1 and P_lambda^mu are the associated Legendre functions.\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Associated-Legendre-Polynomials","page":"Pöschl-Teller Potential","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.P(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.P-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.P","text":"P(model::PoschlTeller, x; n=0, m=0)\n\nAssociated Legendre polynomials are the associated Legendre functions for integer indices. Here we use the same notation of the associated Legendre functions as in the model HydrogenAtom.\n\n```math \\begin{aligned} Pn^m(x) &= \\left( 1-x^2 \\right)^{m/2} \\frac{\\mathrm{d}^m}{\\mathrm{d}x^m} Pn(x) \\\n &= \\left( 1-x^2 \\right)^{m/2} \\frac{\\mathrm{d}^m}{\\mathrm{d}x^m} \\frac{1}{2^n n!} \\frac{\\mathrm{d}^n}{\\mathrm{d}x ^n} \\left[ \\left( x^2-1 \\right)^n \\right] \\\n &= \\frac{1}{2^n} (1-x^2)^{m/2} \\sum_{j=0}^{\\left\\lfloor\\frac{n-m}{2}\\right\\rfloor} (-1)^j \\frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \\end{aligned} ```\n\n\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Usage-and-Examples","page":"Pöschl-Teller Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Install Antique.jl 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 PoschlTeller and the parameters λ, m, ℏ, x₀.","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using Antique\nPT = PoschlTeller(λ=4.0)\n; # hide","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Parameters:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"PT.λ\nPT.m\nPT.ℏ\nPT.x₀","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Number of bound states:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"nₘₐₓ(PT)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Eigen values:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"E(PT, n=0)\nE(PT, n=1)\nE(PT, n=2)\nE(PT, n=3)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Potential energy curve:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using CairoMakie\n\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x)$\")\nlines!(ax, -6..6, x -> V(PT, x))\nf","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Wave functions:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using CairoMakie\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$\\psi(x)$\")\n\n# plot\nw0 = lines!(ax, -3..3, x -> ψ(PT, x, n=0))\nw1 = lines!(ax, -3..3, x -> ψ(PT, x, n=1))\nw2 = lines!(ax, -3..3, x -> ψ(PT, x, n=2))\nw3 = lines!(ax, -3..3, x -> ψ(PT, x, n=3))\n\n# legend\naxislegend(ax, [w0, w1, w2, w3], [L\"n=0\", L\"n=1\", L\"n=2\", L\"n=3\"], position=:lb)\n\nf","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x),~E_n,~\\psi_n(x) \\times 5 + E_n$\", aspect=1, limits=(-4,4,-10.5,1))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\nfor n in 0:3\n # classical turning point\n xE = acosh(sqrt(PT.λ*(PT.λ+1)/abs(E(PT,n=n))/2))\n # energy\n hlines!(ax, E(PT, n=n), color=:black, linewidth=1, linestyle=:dash)\n lines!(ax, [-xE,xE], fill(E(PT,n=n),2), color=:black, linewidth=2)\n # wave function\n lines!(ax, -4..4, x -> E(PT,n=n) + ψ(PT,x,n=n), linewidth=2)\nend\n\n#potential\nlines!(ax, -4..4, x -> V(PT,x), color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/PoschlTeller.png\", f) # hide\n; # hide","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"(Image: )","category":"page"},{"location":"PoschlTeller/#Testing","page":"Pöschl-Teller Potential","title":"Testing","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/PoschlTeller.log\"))","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":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"and the Hamiltonian","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters are specified with the following struct.","category":"page"},{"location":"InfinitePotentialWell/#Parameters","page":"Infinite Potential Well","title":"Parameters","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.InfinitePotentialWell","category":"page"},{"location":"InfinitePotentialWell/#Antique.InfinitePotentialWell-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.InfinitePotentialWell","text":"InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n\nL is the length of the box, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"InfinitePotentialWell/#Potential","page":"Infinite Potential Well","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.V(::InfinitePotentialWell, ::Any)","category":"page"},{"location":"InfinitePotentialWell/#Antique.V-Tuple{InfinitePotentialWell, Any}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.V","text":"V(model::InfinitePotentialWell; x)\n\nV(x) =\nleft\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\nright\n\n\n\n\n\n","category":"method"},{"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":"Antique.E(::InfinitePotentialWell)","category":"page"},{"location":"InfinitePotentialWell/#Antique.E-Tuple{InfinitePotentialWell}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.E","text":"E(model::InfinitePotentialWell; n=1)\n\nE_n = frachbar^2 n^2 pi^22 m L^2\n\n\n\n\n\n","category":"method"},{"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":"Antique.ψ(::InfinitePotentialWell, ::Any)","category":"page"},{"location":"InfinitePotentialWell/#Antique.ψ-Tuple{InfinitePotentialWell, Any}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell, x; n=1)\n\npsi_n(x) = sqrtfrac2L sin fracnpi xL\n\n\n\n\n\n","category":"method"},{"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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated 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 = InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n; #hide","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":"IPW.L\nIPW.m\nIPW.ℏ","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":"E(IPW, n=1)\nE(IPW, n=2)","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 CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$\\psi(x)$\")\n\n# plot\nw1 = lines!(ax, 0..1, x -> ψ(IPW, x, n=1))\nw2 = lines!(ax, 0..1, x -> ψ(IPW, x, n=2))\nw3 = lines!(ax, 0..1, x -> ψ(IPW, x, n=3))\nw4 = lines!(ax, 0..1, x -> ψ(IPW, x, n=4))\nw5 = lines!(ax, 0..1, x -> ψ(IPW, x, n=5))\n\n# legend\naxislegend(ax, [w1, w2, w3, w4, w5], [L\"n=1\", L\"n=2\", L\"n=3\", L\"n=4\", L\"n=5\"], position=:lb)\n\nf","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":"using CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x),~E_n,~\\psi_n(x) \\times 5 + E_n$\", aspect=1, limits=(-0.5,1.5,0,140))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\n# plot\nfor n in 1:5\n # energy\n lines!(ax, [0,IPW.L], fill(E(IPW,n=n),2), color=:black, linewidth=2)\n # wave function\n lines!(ax, 0..IPW.L, x -> E(IPW,n=n) + 5*ψ(IPW,x,n=n), linewidth=2)\nend\n\n#potential\nlines!(ax, [0,0,IPW.L,IPW.L], [140,0,0,140], color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/InfinitePotentialWell.png\", f) # hide\n; # hide","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/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/InfinitePotentialWell.log\"))","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"CurrentModule = Antique","category":"page"},{"location":"SphericalOscillator/#Spherical-Oscillator","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"The spherical oscillator (3D isotropic harmonic oscillator) is the most frequently used model in quantum physics. This model uses a spherical coordinate system.","category":"page"},{"location":"SphericalOscillator/#Definitions","page":"Spherical Oscillator","title":"Definitions","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":" hatH psi(r) = E psi(r)","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"and the Hamiltonian","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":" hatH = - frachbar^22mu nabla^2 + V(r)","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Parameters are specified with the following struct.","category":"page"},{"location":"SphericalOscillator/#Parameters","page":"Spherical Oscillator","title":"Parameters","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.SphericalOscillator","category":"page"},{"location":"SphericalOscillator/#Antique.SphericalOscillator-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.SphericalOscillator","text":"HarmonicOscillator(k=1.0, μ=1.0, ℏ=1.0)\n\nk is the force constant, μ is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"SphericalOscillator/#Potential","page":"Spherical Oscillator","title":"Potential","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.V(::SphericalOscillator, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.V-Tuple{SphericalOscillator, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.V","text":"V(model::SphericalOscillator, r)\n\nV(r)\n= frac12 k r^2\n= frac12 mu omega^2 r^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkmu is the angular frequency and xi = sqrtfracmuomegahbarr.\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Eigen-Values","page":"Spherical Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.E(::SphericalOscillator)","category":"page"},{"location":"SphericalOscillator/#Antique.E-Tuple{SphericalOscillator}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.E","text":"E(model::SphericalOscillator; n=0, l=0)\n\nE_nl\n= left(2n + l + frac32right)hbar omega\n\nwhere omega = sqrtkmu.\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Eigen-Functions","page":"Spherical Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.ψ(::SphericalOscillator, ::Any, ::Any, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.ψ-Tuple{SphericalOscillator, Any, Any, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.ψ","text":"ψ(model::SphericalOscillator, r, θ, φ; n=0, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Radial-Functions","page":"Spherical Oscillator","title":"Radial Functions","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.R(::SphericalOscillator, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.R-Tuple{SphericalOscillator, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.R","text":"R(model::SphericalOscillator, r; n=0, l=0)\n\nR_nl(r) = sqrt fracgamma^322sqrtpi sqrtfrac2^n+l+3 n(2n+2l+1) xi^l expleft(-xi^22right)L_n^(l+frac12) left(xi^2right)\n\nwhere gamma = muomegahbar and xi = sqrtgammar = sqrtmuomegahbarr are defined. The generalized Laguerre polynomials are defined as L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Generalized-Laguerre-Polynomials","page":"Spherical Oscillator","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"note: Note\nThe generalized Laguerre polynomials L_n^(alpha)(x), not the associated Laguerre polynomials L_n^k(x), are used in this model. ","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.L(::SphericalOscillator, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.L-Tuple{SphericalOscillator, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.L","text":"L(model::SphericalOscillator, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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 \nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Spherical-Harmonics","page":"Spherical Oscillator","title":"Spherical Harmonics","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.Y(::SphericalOscillator, ::Any, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.Y-Tuple{SphericalOscillator, Any, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.Y","text":"Y(model::SphericalOscillator, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Associated-Legendre-Polynomials","page":"Spherical Oscillator","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.P(::SphericalOscillator, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.P-Tuple{SphericalOscillator, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.P","text":"P(model::SphericalOscillator, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Usage-and-Examples","page":"Spherical Oscillator","title":"Usage & Examples","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Install Antique.jl 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 SphericalOscillator and several parameters k, μ and ℏ are set as optional arguments.","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"using Antique\nSO = SphericalOscillator(k=1.0, μ=1.0, ℏ=1.0)\n; # hide","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Parameters:","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"SO.k\nSO.μ\nSO.ℏ","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Eigen values:","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"E(SO, n=0)\nE(SO, n=1)\nE(SO, n=2)\nE(SO, n=0, l=1)\nE(SO, n=1, l=1)\nE(SO, n=2, l=1)","category":"page"},{"location":"SphericalOscillator/#Testing","page":"Spherical Oscillator","title":"Testing","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical 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":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/SphericalOscillator.log\"))","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"CurrentModule = Antique","category":"page"},{"location":"DeltaPotential/#Delta-Potential","page":"Delta Potential","title":"Delta Potential","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"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.","category":"page"},{"location":"DeltaPotential/#Definitions","page":"Delta Potential","title":"Definitions","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"and the Hamiltonian","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Parameters are specified with the following struct.","category":"page"},{"location":"DeltaPotential/#Parameters","page":"Delta Potential","title":"Parameters","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.DeltaPotential","category":"page"},{"location":"DeltaPotential/#Antique.DeltaPotential-DeltaPotential","page":"Delta Potential","title":"Antique.DeltaPotential","text":"DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n\nalpha is the potential strength, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"DeltaPotential/#Potential","page":"Delta Potential","title":"Potential","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.V(::DeltaPotential, ::Any)","category":"page"},{"location":"DeltaPotential/#Antique.V-Tuple{DeltaPotential, Any}-DeltaPotential","page":"Delta Potential","title":"Antique.V","text":"V(model::DeltaPotential, x)\n\nV(x) = -alpha delta(x)\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Eigen-Values","page":"Delta Potential","title":"Eigen Values","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.E(::DeltaPotential)","category":"page"},{"location":"DeltaPotential/#Antique.E-Tuple{DeltaPotential}-DeltaPotential","page":"Delta Potential","title":"Antique.E","text":"E(model::DeltaPotential)\n\nE = - fracmalpha^22hbar^2\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Eigen-Functions","page":"Delta Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.ψ(::DeltaPotential, ::Any)","category":"page"},{"location":"DeltaPotential/#Antique.ψ-Tuple{DeltaPotential, Any}-DeltaPotential","page":"Delta Potential","title":"Antique.ψ","text":"ψ(model::DeltaPotential, x)\n\npsi(x) = fracsqrtmalphahbar mathrme^-malpha xhbar^2\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Usage-and-Examples","page":"Delta Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Install Antique.jl 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.","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using Antique\nDP = DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Parameters:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"DP.α\nDP.m\nDP.ℏ","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Eigen values:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"E(DP)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Wave functions:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using CairoMakie\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$\\psi(x)$\")\n\n# plot\nw = lines!(ax, -5..5, x -> ψ(DP, x))\n\nf","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x),~E_n,~\\psi_n(x) \\times 5 + E_n$\", aspect=1, limits=(-5,5,-0.6,0.6))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\n# energy\nhlines!(ax, E(DP), color=:black, linewidth=1, linestyle=:dash)\n\n# wave function\nlines!(ax, -5..5, x -> E(DP) + ψ(DP,x), linewidth=2)\n\n#potential\nlines!(ax, [-5,0,0,0,5], [0,0,-1,0,0], color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/DeltaPotential.png\", f) # hide\n; # hide","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"(Image: )","category":"page"},{"location":"DeltaPotential/#Testing","page":"Delta Potential","title":"Testing","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/DeltaPotential.log\"))","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"CurrentModule = Antique","category":"page"},{"location":"CoulombTwoBody/#Coulomb-2-Body-System","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"This is the model of two particles interacting through Coulomb forces such as positronium, muonium, hydrogen atoms, deuterium atoms, etc.","category":"page"},{"location":"CoulombTwoBody/#Definitions","page":"Coulomb 2-Body System","title":"Definitions","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"and the Hamiltonian","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":" hatH = - frachbar^22mu nabla^2 + V(r)","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"where mu=left(frac1m_1+frac1m_2right)^-1 is the reduced mass of particle 1 and particle 2. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.","category":"page"},{"location":"CoulombTwoBody/#Parameters","page":"Coulomb 2-Body System","title":"Parameters","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.CoulombTwoBody","category":"page"},{"location":"CoulombTwoBody/#Antique.CoulombTwoBody-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.CoulombTwoBody","text":"CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nz₁ is the charge number of particle 1, z₁ is the charge number of particle 2, m₁ is the mass of particle 1, m₂ is the mass of particle 2, m_mathrme is the electron mass (the unit of m₁ and m₂), a_0 is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"CoulombTwoBody/#Potential","page":"Coulomb 2-Body System","title":"Potential","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.V(::CoulombTwoBody, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.V-Tuple{CoulombTwoBody, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.V","text":"V(model::CoulombTwoBody, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Eigen-Values","page":"Coulomb 2-Body System","title":"Eigen Values","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.E(::CoulombTwoBody)","category":"page"},{"location":"CoulombTwoBody/#Antique.E-Tuple{CoulombTwoBody}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.E","text":"E(model::CoulombTwoBody; n=1)\n\nE_n\n= -frac(z_1 z_2)^22n^2 fracmum_mathrme E_mathrmh\n\nwhere mu=left(frac1m_1+frac1m_2right)^-1 is the reduced mass of particle 1 and particle 2, 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.\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Eigen-Functions","page":"Coulomb 2-Body System","title":"Eigen Functions","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.ψ(::CoulombTwoBody, ::Any, ::Any, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.ψ-Tuple{CoulombTwoBody, Any, Any, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.ψ","text":"ψ(model::CoulombTwoBody, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Radial-Functions","page":"Coulomb 2-Body System","title":"Radial Functions","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.R(::CoulombTwoBody, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.R-Tuple{CoulombTwoBody, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.R","text":"R(model::CoulombTwoBody, r; n=1, l=0)\n\nR_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_muright)^3 left(frac2Zrn a_muright)^l exp left(-fracZrn a_muright) L_n+l^2l+1 left(frac2Zrn a_muright)\n\nwhere frac1mu = frac1m_1+frac1m_2, a_mu = a_0 fracm_mathrmemu, 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). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Associated-Laguerre-Polynomials","page":"Coulomb 2-Body System","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.L(::CoulombTwoBody, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.L-Tuple{CoulombTwoBody, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.L","text":"L(model::CoulombTwoBody, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_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)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Spherical-Harmonics","page":"Coulomb 2-Body System","title":"Spherical Harmonics","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.Y(::CoulombTwoBody, ::Any, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.Y-Tuple{CoulombTwoBody, Any, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.Y","text":"Y(model::CoulombTwoBody, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Associated-Legendre-Polynomials","page":"Coulomb 2-Body System","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.P(::CoulombTwoBody, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.P-Tuple{CoulombTwoBody, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.P","text":"P(model::CoulombTwoBody, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Usage-and-Examples","page":"Coulomb 2-Body System","title":"Usage & Examples","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Install Antique.jl 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 CoulombTwoBody and several parameters z₁, z₂, m₁, m₂, mₑ, a₀, Eₕ and ℏ are set as optional arguments.","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"using Antique\nPs = CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"CoulombTwoBody/#Parameters-2","page":"Coulomb 2-Body System","title":"Parameters","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Ps.z₁\nPs.z₂\nPs.m₁\nPs.m₂\nPs.mₑ\nPs.a₀\nPs.Eₕ\nPs.ℏ","category":"page"},{"location":"CoulombTwoBody/#Eigen-Values-2","page":"Coulomb 2-Body System","title":"Eigen Values","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Examples of positronium:","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"E(Ps, n=1)\nE(Ps, n=2)","category":"page"},{"location":"CoulombTwoBody/#Mass-and-Charge-Dependence","page":"Coulomb 2-Body System","title":"Mass and Charge Dependence","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"The values of masses are cited from the 2018 CODATA recommended values, E. Tiesinga, et al., Rev. Mod. Phys. 93, 025010 (2021).","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"me = 1.0 # me # \nmµ = 206.7682830 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mmusme\nmp = 1836.15267343 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mpsme\nmd = 3670.48296788 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mdsme\nmt = 5496.92153573 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mtsme\nmh = 5495.88528007 # me # https://physics.nist.gov/cgi-bin/cuu/Value?mhsme\nma = 7294.29954142 # me # https://physics.nist.gov/cgi-bin/cuu/Value?malsme\n\nPs = CoulombTwoBody(m₁=me, m₂=me)\nMu = CoulombTwoBody(m₁=me, m₂=mµ)\nH = CoulombTwoBody(m₁=me, m₂=mp)\nD = CoulombTwoBody(m₁=me, m₂=md)\nT = CoulombTwoBody(m₁=me, m₂=mt)\nBO = CoulombTwoBody(m₁=me, m₂=Inf)\n\nHe3⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=mh)\nHe4⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=ma)\nHe∞⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=Inf)\n\npμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mp)\ndμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=md)\ntμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mt)\nbμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=Inf)\nhμ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=mµ, m₂=mh)\nαμ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=mµ, m₂=ma)\n\nprintln(\" \\tE / Eₕ\")\nprintln(\"Ps \\t\", E(Ps))\nprintln(\"Mu \\t\", E(Mu))\nprintln(\"H \\t\", E(H))\nprintln(\"D \\t\", E(D))\nprintln(\"T \\t\", E(T))\nprintln(\"∞H \\t\", E(BO))\nprintln(\"³He⁺\\t\", E(He3⁺))\nprintln(\"⁴He⁺\\t\", E(He4⁺))\nprintln(\"∞He⁺\\t\", E(He∞⁺))\nprintln(\"pμ \\t\", E(pμ))\nprintln(\"dμ \\t\", E(dμ))\nprintln(\"tμ \\t\", E(tμ))\nprintln(\"bμ \\t\", E(bμ))\nprintln(\"hμ \\t\", E(hμ))\nprintln(\"αμ \\t\", E(αμ))","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"println(\" \\t<δ³(r)> / a₀⁻³\")\nprintln(\"1/8π =\\t\", 1/8/π)\nprintln(\"Ps \\t\", abs(ψ(Ps,0,0,0))^2)\nprintln(\"Mu \\t\", abs(ψ(Mu,0,0,0))^2)\nprintln(\"H \\t\", abs(ψ(H ,0,0,0))^2)\nprintln(\"D \\t\", abs(ψ(D ,0,0,0))^2)\nprintln(\"T \\t\", abs(ψ(T ,0,0,0))^2)\nprintln(\"∞H \\t\", abs(ψ(BO,0,0,0))^2)\nprintln(\"1/π = \\t\", 1/π)","category":"page"},{"location":"CoulombTwoBody/#Lifetime-of-Positronium","page":"Coulomb 2-Body System","title":"Lifetime of Positronium","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"The lifetime tau of positronium (Ps, mathrme^+mathrme^-) is written as","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"tau = frac1Gamma","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Gamma = 4 pi alpha^4 c a_0^2 langledelta^3(pmbr)rangle","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"where langledelta^3(pmbr)rangle = langlepsidelta^3(pmbr)psirangle = psi(pmb0)^2 = frac18pi a_0^-3 simeq 2685times10^29mathrmm^-3 is the value of probability density at the origin (r=0). Reference:","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"(7.169) in D. J. Griffiths, Introduction to Elementary Particles (John Wiley & Sons, Inc. 1987) ISBN 0-471-60386-4\nS. Berko, H. N. Pendleton, Annual Review of Nuclear and Particle Science, 30, 543 (1980)\nA. M. Frolov, S. I. Kryuchkov, and V. H. Smith, Jr., Phys. Rev. A, 51, 4514 (1995)","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"α = 7.2973525693e-3 # # https://physics.nist.gov/cgi-bin/cuu/Value?alph\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\n\nPs = CoulombTwoBody(z₁=1, z₂=-1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\nδ = abs(ψ(Ps,0,0,0))^2 * a₀^(-3)\nΓ = 4 * π * α^4 * c * a₀^2 * δ\nτ = 1/Γ\nprintln(\"<δ> = \", abs(ψ(Ps,0,0,0))^2, \" a₀⁻³\")\nprintln(\" = \", δ, \" m⁻³\")\nprintln(\"Γ = \", Γ / 1e9, \" GHz\")\nprintln(\"τ = \", τ / 1e-12, \" ps\")","category":"page"},{"location":"CoulombTwoBody/#Hyperfine-Splitting","page":"Coulomb 2-Body System","title":"Hyperfine Splitting","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"The hyperfine splitting of hydrogen atoms is given as","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Delta E (mathrmH) = -frac23 mu_0 gamma_mathrmp gamma_mathrme hbar^2 langledelta^3(pmbr)rangle","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"in Griffiths(1982). This fomula is not available for deuterium (D) and positronium (Ps). Because of the different spin between the proton and the deuteron for D, the contribution of positron-electron pair annihilation for Ps. Note the definition of gyromagnetic ratio. The mass of protons is used for all nucleons and nuclei:","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"beginaligned\ngamma_mathrme = frac-e2 m_mathrme g_mathrme \ngamma_mathrme^+ = frac+e2 m_mathrme g_mathrme \ngamma_mathrmmu = frac-e2 m_mathrmmu g_mathrmmu \ngamma_mathrmp = frac+e2 m_mathrmp g_mathrmp \ngamma_mathrmd = frac+e2 m_mathrmp g_mathrmd \ngamma_mathrmt = frac+e2 m_mathrmp g_mathrmt \ngamma_mathrmh = frac+2e2 m_mathrmp g_mathrmh \nendaligned","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"The value of probability density at the origin is langledelta^3(pmbr)rangle = langlepsidelta^3(pmbr)psirangle = psi(pmb0)^2 simeq frac1pi a_0^-3 simeq 2148times10^30mathrmm^-3 in Mu, H, D and T. This values are very different in Ps, ^3mathrmHe^+ and muonic hydrogen (mathrmpmu) due to the difference of reduced masses and charges. The energy can be converted to frequency (Hz) by v = Delta E h.","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"a₀ = 5.29177210903e-11 # m # https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0\nEₕ = 4.3597447222071e-18 # J # https://physics.nist.gov/cgi-bin/cuu/Value?hr\nℏ = 1.054571817e-34 # J s # https://physics.nist.gov/cgi-bin/cuu/Value?hbar\n\nme = 9.1093837015e-31 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?me\nmµ = 1.883531627e-28 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mmu\nmp = 1.67262192369e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mp\nmd = 3.3435837724e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?md\nmt = 5.0073567446e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mt\nmh = 5.0064127796e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mh\n\ne = 1.602176634e-19 # C # https://physics.nist.gov/cgi-bin/cuu/Value?e\nµ₀ = 1.25663706212e-6 # N A-2 # https://physics.nist.gov/cgi-bin/cuu/Value?mu0\nh = 6.62607015e-34 # J Hz-1 # https://physics.nist.gov/cgi-bin/cuu/Value?h\neV = 1.602176634e-19 # J # https://physics.nist.gov/cgi-bin/cuu/Value?evj\n\nge = 2.00231930436256 # https://physics.nist.gov/cgi-bin/cuu/Value?gem\ngµ = 2.0023318418 # https://physics.nist.gov/cgi-bin/cuu/Value?gmum\ngp = 5.5856946893 # https://physics.nist.gov/cgi-bin/cuu/Value?gp\ngd = 0.8574382338 # https://physics.nist.gov/cgi-bin/cuu/Value?gdn\ngt = 5.957924931 # https://physics.nist.gov/cgi-bin/cuu/Value?gtn\ngh = -4.255250615 # https://physics.nist.gov/cgi-bin/cuu/Value?ghn\n\nPs = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=me, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nMu = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mµ, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nH = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mp, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nD = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=md, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nT = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mt, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nhe = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=mh, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\npμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mp, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\n\nΔE_H = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gp / mp * abs(ψ(H,0,0,0))^2\nΔE_D = 1 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gd / mp * abs(ψ(D,0,0,0))^2\nΔE_T = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gt / mp * abs(ψ(H,0,0,0))^2\nΔE_Ps = 7/6 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * ge / me * abs(ψ(Ps,0,0,0))^2\nΔE_Mu = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gµ / mµ * abs(ψ(Mu,0,0,0))^2\nΔE_he = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gh / mp * abs(ψ(he,0,0,0))^2\nΔE_pµ = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * gµ / mµ * gp / mp * abs(ψ(pµ,0,0,0))^2\n\nprintln(\"H \\t\", ΔE_H / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"1420.405751768(1) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"D \\t\", ΔE_D / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"327.384352522(2) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"T \\t\", ΔE_T / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"1516.701470773(8) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"Ps\\t\", ΔE_Ps / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"203391.7(6) MHz (https://doi.org/10.48550/arXiv.hep-ph/0310099)\")\nprintln(\"Mu\\t\", ΔE_Mu / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"4463.30278(5) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"³He⁺\\t\", ΔE_he / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"-8665.649867(10) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"µp\\t\", ΔE_pµ / h / 1e12, \" THz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", 0.182725*eV / h / 1e12 ,\" THz (https://doi.org/10.1119/1.12733, https://doi.org/10.1016/j.nimb.2012.04.001)\")","category":"page"},{"location":"CoulombTwoBody/#Testing","page":"Coulomb 2-Body System","title":"Testing","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","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":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/CoulombTwoBody.log\"))","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":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"and the Hamiltonian","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters are specified with the following struct.","category":"page"},{"location":"HarmonicOscillator/#Parameters","page":"Harmonic Oscillator","title":"Parameters","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.HarmonicOscillator","category":"page"},{"location":"HarmonicOscillator/#Antique.HarmonicOscillator-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.HarmonicOscillator","text":"HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)\n\nk is the force constant, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"HarmonicOscillator/#Potential","page":"Harmonic Oscillator","title":"Potential","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.V(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.V-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.V","text":"V(model::HarmonicOscillator, x)\n\nV(x)\n= frac12 k x^2\n= frac12 m omega^2 x^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkm is the angular frequency and xi = sqrtfracmomegahbarx.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Eigen-Values","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.E(::HarmonicOscillator)","category":"page"},{"location":"HarmonicOscillator/#Antique.E-Tuple{HarmonicOscillator}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.E","text":"E(model::HarmonicOscillator; n=0)\n\nE_n = hbar omega left( n + frac12 right)\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Eigen-Functions","page":"Harmonic Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.ψ(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.ψ-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.ψ","text":"ψ(model::HarmonicOscillator, x; n=0)\n\npsi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)\n\nwhere 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 defined.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Hermite-Polynomials","page":"Harmonic Oscillator","title":"Hermite Polynomials","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.H(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.H-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.H","text":"H(model::HarmonicOscillator, x; n=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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\nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated 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 = HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"HO.k\nHO.m\nHO.ℏ","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Eigen values:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"E(HO, n=0)\nE(HO, n=1)","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 CairoMakie\n\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x)$\")\nlines!(ax, -5..5, x -> V(HO, x))\nf","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 CairoMakie\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$\\psi(x)$\")\n\n# plot\nw0 = lines!(ax, -5..5, x -> ψ(HO, x, n=0))\nw1 = lines!(ax, -5..5, x -> ψ(HO, x, n=1))\nw2 = lines!(ax, -5..5, x -> ψ(HO, x, n=2))\nw3 = lines!(ax, -5..5, x -> ψ(HO, x, n=3))\nw4 = lines!(ax, -5..5, x -> ψ(HO, x, n=4))\n\n# legend\naxislegend(ax, [w0, w1, w2, w3, w4], [L\"n=0\", L\"n=1\", L\"n=2\", L\"n=3\", L\"n=4\"], position=:lb)\n\nf","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 CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x),~E_n,~\\psi_n(x) \\times 5 + E_n$\", aspect=1, limits=(-5,5,0,5.2))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\nfor n in 0:4\n # classical turning point\n xE = sqrt(2*HO.k*E(HO, n=n))\n # energy\n lines!(ax, [-xE,xE], fill(E(HO,n=n),2), color=:black, linewidth=2)\n hlines!(ax, E(HO, n=n), color=:black, linewidth=1, linestyle=:dash)\n # wave function\n lines!(ax, -5..5, x -> E(HO,n=n) + 0.5*ψ(HO,x,n=n), linewidth=2)\nend\n\n#potential\nlines!(ax, -5..5, x -> V(HO, x), color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/HarmonicOscillator.png\", f) # hide\n; # hide","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/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/HarmonicOscillator.log\"))","category":"page"},{"location":"API/","page":"API reference","title":"API reference","text":"CurrentModule = Antique","category":"page"},{"location":"API/#API-reference","page":"API reference","title":"API reference","text":"","category":"section"},{"location":"API/","page":"API reference","title":"API reference","text":"Order = [:type, :function]\nModules = [Antique]","category":"page"},{"location":"API/#Antique.CoulombTwoBody","page":"API reference","title":"Antique.CoulombTwoBody","text":"CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nz₁ is the charge number of particle 1, z₁ is the charge number of particle 2, m₁ is the mass of particle 1, m₂ is the mass of particle 2, m_mathrme is the electron mass (the unit of m₁ and m₂), a_0 is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.DeltaPotential","page":"API reference","title":"Antique.DeltaPotential","text":"DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n\nalpha is the potential strength, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.HarmonicOscillator","page":"API reference","title":"Antique.HarmonicOscillator","text":"HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)\n\nk is the force constant, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.HydrogenAtom","page":"API reference","title":"Antique.HydrogenAtom","text":"HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nZ is the atomic number, m_mathrme is the electron mass, a_0is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.InfinitePotentialWell","page":"API reference","title":"Antique.InfinitePotentialWell","text":"InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n\nL is the length of the box, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.InfinitePotentialWell3D","page":"API reference","title":"Antique.InfinitePotentialWell3D","text":"InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)\n\nL_xL_yL_z are the lengths of the box in x,y,z-direction, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.MorsePotential","page":"API reference","title":"Antique.MorsePotential","text":"MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)\n\nr_mathrme is the equilibrium bond distance, D_mathrme is the the well depth , k is the force constant, mu is the reduced mass and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.PoschlTeller","page":"API reference","title":"Antique.PoschlTeller","text":"PoschlTeller(λ=1.0, m=1.0, ℏ=1.0, x₀=1.0)\n\nlambda determines the potential strength.\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.RigidRotor","page":"API reference","title":"Antique.RigidRotor","text":"RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)\n\nm₁ and m₂ are mass of two particles, R is the distance, and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.SphericalOscillator","page":"API reference","title":"Antique.SphericalOscillator","text":"HarmonicOscillator(k=1.0, μ=1.0, ℏ=1.0)\n\nk is the force constant, μ is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.E-Tuple{CoulombTwoBody}","page":"API reference","title":"Antique.E","text":"E(model::CoulombTwoBody; n=1)\n\nE_n\n= -frac(z_1 z_2)^22n^2 fracmum_mathrme E_mathrmh\n\nwhere mu=left(frac1m_1+frac1m_2right)^-1 is the reduced mass of particle 1 and particle 2, 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.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{DeltaPotential}","page":"API reference","title":"Antique.E","text":"E(model::DeltaPotential)\n\nE = - fracmalpha^22hbar^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{HarmonicOscillator}","page":"API reference","title":"Antique.E","text":"E(model::HarmonicOscillator; n=0)\n\nE_n = hbar omega left( n + frac12 right)\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{HydrogenAtom}","page":"API reference","title":"Antique.E","text":"E(model::HydrogenAtom; n=1)\n\nE_n\n= -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2\n= -fracZ^22n^2 E_mathrmh\n\nwhere 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.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{InfinitePotentialWell3D}","page":"API reference","title":"Antique.E","text":"E(model::InfinitePotentialWell3D; nx=1, ny=1, nz=1)\n\nE_n_xn_yn_z = frachbar^2 pi^22 m left(fracn_x^2L_x^2 + fracn_y^2L_y^2 + fracn_z^2L_z^2right)\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{InfinitePotentialWell}","page":"API reference","title":"Antique.E","text":"E(model::InfinitePotentialWell; n=1)\n\nE_n = frachbar^2 n^2 pi^22 m L^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{MorsePotential}","page":"API reference","title":"Antique.E","text":"E(model::MorsePotential; n=0)\n\nE_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2\n\nwhere omega = sqrtkµ and chi = frachbaromega4D_mathrme are defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{PoschlTeller}","page":"API reference","title":"Antique.E","text":"E(model::PoschlTeller; n=0)\n\nE_n = -frachbar^2m x_0^2fracmu^22\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{RigidRotor}","page":"API reference","title":"Antique.E","text":"E(model::RigidRotor; l=0)\n\nE_l\n= frachbar^22Il(l+1)\n\nwhere I=mu R^2 is the moment of inertia, R is the distance, and mu is the reduced mass of the two particles.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{SphericalOscillator}","page":"API reference","title":"Antique.E","text":"E(model::SphericalOscillator; n=0, l=0)\n\nE_nl\n= left(2n + l + frac32right)hbar omega\n\nwhere omega = sqrtkmu.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.H-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.H","text":"H(model::HarmonicOscillator, x; n=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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\nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{CoulombTwoBody, Any}","page":"API reference","title":"Antique.L","text":"L(model::CoulombTwoBody, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_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)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.L","text":"L(model::HydrogenAtom, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_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)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.L","text":"L(model::MorsePotential, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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 \nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{SphericalOscillator, Any}","page":"API reference","title":"Antique.L","text":"L(model::SphericalOscillator, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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 \nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{CoulombTwoBody, Any}","page":"API reference","title":"Antique.P","text":"P(model::CoulombTwoBody, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.P","text":"P(model::HydrogenAtom, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.P","text":"P(model::PoschlTeller, x; n=0, m=0)\n\nAssociated Legendre polynomials are the associated Legendre functions for integer indices. Here we use the same notation of the associated Legendre functions as in the model HydrogenAtom.\n\n```math \\begin{aligned} Pn^m(x) &= \\left( 1-x^2 \\right)^{m/2} \\frac{\\mathrm{d}^m}{\\mathrm{d}x^m} Pn(x) \\\n &= \\left( 1-x^2 \\right)^{m/2} \\frac{\\mathrm{d}^m}{\\mathrm{d}x^m} \\frac{1}{2^n n!} \\frac{\\mathrm{d}^n}{\\mathrm{d}x ^n} \\left[ \\left( x^2-1 \\right)^n \\right] \\\n &= \\frac{1}{2^n} (1-x^2)^{m/2} \\sum_{j=0}^{\\left\\lfloor\\frac{n-m}{2}\\right\\rfloor} (-1)^j \\frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \\end{aligned} ```\n\n\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{RigidRotor, Any}","page":"API reference","title":"Antique.P","text":"P(model::RigidRotor, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{SphericalOscillator, Any}","page":"API reference","title":"Antique.P","text":"P(model::SphericalOscillator, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.R-Tuple{CoulombTwoBody, Any}","page":"API reference","title":"Antique.R","text":"R(model::CoulombTwoBody, r; n=1, l=0)\n\nR_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_muright)^3 left(frac2Zrn a_muright)^l exp left(-fracZrn a_muright) L_n+l^2l+1 left(frac2Zrn a_muright)\n\nwhere frac1mu = frac1m_1+frac1m_2, a_mu = a_0 fracm_mathrmemu, 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). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.R-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.R","text":"R(model::HydrogenAtom, r; n=1, l=0)\n\nR_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)\n\nwhere 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). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.R-Tuple{SphericalOscillator, Any}","page":"API reference","title":"Antique.R","text":"R(model::SphericalOscillator, r; n=0, l=0)\n\nR_nl(r) = sqrt fracgamma^322sqrtpi sqrtfrac2^n+l+3 n(2n+2l+1) xi^l expleft(-xi^22right)L_n^(l+frac12) left(xi^2right)\n\nwhere gamma = muomegahbar and xi = sqrtgammar = sqrtmuomegahbarr are defined. The generalized Laguerre polynomials are defined as L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{CoulombTwoBody, Any}","page":"API reference","title":"Antique.V","text":"V(model::CoulombTwoBody, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{DeltaPotential, Any}","page":"API reference","title":"Antique.V","text":"V(model::DeltaPotential, x)\n\nV(x) = -alpha delta(x)\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.V","text":"V(model::HarmonicOscillator, x)\n\nV(x)\n= frac12 k x^2\n= frac12 m omega^2 x^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkm is the angular frequency and xi = sqrtfracmomegahbarx.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.V","text":"V(model::HydrogenAtom, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{InfinitePotentialWell, Any}","page":"API reference","title":"Antique.V","text":"V(model::InfinitePotentialWell; x)\n\nV(x) =\nleft\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{InfinitePotentialWell3D, Any, Any, Any}","page":"API reference","title":"Antique.V","text":"V(model::InfinitePotentialWell3D, x,y,z)\n\nV(xyz) =\nleft\n beginarrayll\n 0 0 leq x leq L_x mathrmand 0 leq y leq L_y mathrmand 0 leq z leq L_z \n infty mathrmelsewhere\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.V","text":"V(model::MorsePotential, r)\n\nV(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)\n\nwhere a = sqrtfrack2Dₑ is defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.V","text":"V(model::PoschlTeller, x)\n\nbeginaligned\nV(x)\n= -frachbar^2m x_0^2 fraclambda(lambda+1)2 mathrmsech(x)^2\n= -frachbar^2m x_0^2 fraclambda(lambda+1)2 frac1mathrmcosh(x)^2\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{RigidRotor, Any}","page":"API reference","title":"Antique.V","text":"V(model::RigidRotor, r)\n\nbeginaligned\n V(r)\n = 0\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{SphericalOscillator, Any}","page":"API reference","title":"Antique.V","text":"V(model::SphericalOscillator, r)\n\nV(r)\n= frac12 k r^2\n= frac12 mu omega^2 r^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkmu is the angular frequency and xi = sqrtfracmuomegahbarr.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{CoulombTwoBody, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::CoulombTwoBody, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{HydrogenAtom, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::HydrogenAtom, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{RigidRotor, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::RigidRotor, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{SphericalOscillator, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::SphericalOscillator, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.nₘₐₓ-Tuple{MorsePotential}","page":"API reference","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::MorsePotential)\n\nn_mathrmmax = leftlfloor frac2 D_e - omegaomega rightrfloor\n\nwhere omega = sqrtkµ is defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.nₘₐₓ-Tuple{PoschlTeller}","page":"API reference","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::PoschlTeller)\n\nn_mathrmmax = leftlfloor lambda rightrfloor - 1\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{CoulombTwoBody, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::CoulombTwoBody, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{DeltaPotential, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::DeltaPotential, x)\n\npsi(x) = fracsqrtmalphahbar mathrme^-malpha xhbar^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::HarmonicOscillator, x; n=0)\n\npsi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)\n\nwhere 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 defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{HydrogenAtom, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{InfinitePotentialWell, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell, x; n=1)\n\npsi_n(x) = sqrtfrac2L sin fracnpi xL\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{InfinitePotentialWell3D, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell3D, x,y,z; nx=1, ny=1, nz=1)\n\nThe wave functions can be expressed as products of wave functions in a one-dimensional box.\n\npsi_n_xn_yn_z(xyz) = psi_n_x(x)psi_n_y(y)psi_n_z(z) = sqrtfrac8L_xL_yL_z sinleft(fracn_xpi xL_xright) sinleft(fracn_ypi yL_yright) sinleft(fracn_zpi zL_zright)\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::MorsePotential, r; n=0)\n\npsi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)\n\nN_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), lambda = fracsqrt2mu D_mathrmeahbar, a = sqrtfrack2Dₑ, L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight), xi = 2lambdamathrme^-a(r-r_e) are defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::PoschlTeller, x; n=0)\n\npsi_n(x) = P_lambda^mu(mathrmtanh(xx_0)) sqrtmufracGamma(lambda-mu+1)Gamma(lambda+mu+1)\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1 and P_lambda^mu are the associated Legendre functions.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{RigidRotor, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::RigidRotor, θ, φ; l=0, m=0)\n\npsi_lm(thetavarphi) = Y_lm(thetavarphi)\n\nThe wave function is the spherical harmonics. The domain is 0leq theta lt pi and 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{SphericalOscillator, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::SphericalOscillator, r, θ, φ; n=0, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"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":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH psi(r) = E psi(r)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"and the Hamiltonian","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters are specified with the following struct.","category":"page"},{"location":"MorsePotential/#Parameters","page":"Morse Potential","title":"Parameters","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.MorsePotential","category":"page"},{"location":"MorsePotential/#Antique.MorsePotential-MorsePotential","page":"Morse Potential","title":"Antique.MorsePotential","text":"MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)\n\nr_mathrme is the equilibrium bond distance, D_mathrme is the the well depth , k is the force constant, mu is the reduced mass and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"MorsePotential/#Potential","page":"Morse Potential","title":"Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.V(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.V-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.V","text":"V(model::MorsePotential, r)\n\nV(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)\n\nwhere a = sqrtfrack2Dₑ is defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Eigen-Values","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.E(::MorsePotential)","category":"page"},{"location":"MorsePotential/#Antique.E-Tuple{MorsePotential}-MorsePotential","page":"Morse Potential","title":"Antique.E","text":"E(model::MorsePotential; n=0)\n\nE_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2\n\nwhere omega = sqrtkµ and chi = frachbaromega4D_mathrme are defined.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Number-of-Bound-States","page":"Morse Potential","title":"Number of Bound States","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.nₘₐₓ(::MorsePotential)","category":"page"},{"location":"MorsePotential/#Antique.nₘₐₓ-Tuple{MorsePotential}-MorsePotential","page":"Morse Potential","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::MorsePotential)\n\nn_mathrmmax = leftlfloor frac2 D_e - omegaomega rightrfloor\n\nwhere omega = sqrtkµ is defined.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Eigen-Functions","page":"Morse Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.ψ(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.ψ-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.ψ","text":"ψ(model::MorsePotential, r; n=0)\n\npsi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)\n\nN_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), lambda = fracsqrt2mu D_mathrmeahbar, a = sqrtfrack2Dₑ, L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight), xi = 2lambdamathrme^-a(r-r_e) are defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials","page":"Morse Potential","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"note: Note\nThe generalized Laguerre polynomials L_n^(alpha)(x), not the associated Laguerre polynomials L_n^k(x), are used in this model. ","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.L(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.L-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.L","text":"L(model::MorsePotential, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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 \nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated 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":"# 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\nDₑ = - 0.5 - (-0.602634619106539878727562156289)\nk = 2*((-1.1026342144949464615+1/2.00) - (-0.602634619106539878727562156289)) / (2.00 - rₑ)^2\nµ = 1/(1/1836.15267343 + 1/1836.15267343)\nℏ = 1.0\n\nusing Antique\nMP = MorsePotential(rₑ=rₑ, Dₑ=Dₑ, k=k, µ=µ, ℏ=ℏ)\n; #hide","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"MP.rₑ\nMP.Dₑ\nMP.k\nMP.µ\nMP.ℏ","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Number of bound states:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"nₘₐₓ(MP)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Eigen values:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"E(MP, n=0)\nE(MP, n=1)","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 CairoMakie\n\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r$\", ylabel=L\"$V(r)$\", limits=(0.0,20.0,-0.11,0.1))\nlines!(ax, 0.1:0.01:20, r -> V(MP, r))\nf","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 CairoMakie\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r$\", ylabel=L\"$\\psi(r)$\")\n\n# plot\nw0 = lines!(ax, 0..5, x -> ψ(MP, x, n=0))\nw1 = lines!(ax, 0..5, x -> ψ(MP, x, n=1))\nw2 = lines!(ax, 0..5, x -> ψ(MP, x, n=2))\nw3 = lines!(ax, 0..5, x -> ψ(MP, x, n=3))\nw4 = lines!(ax, 0..5, x -> ψ(MP, x, n=4))\n\n# legend\naxislegend(ax, [w0, w1, w2, w3, w4], [L\"n=0\", L\"n=1\", L\"n=2\", L\"n=3\", L\"n=4\"], position=:lb)\n\nf","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":"using Antique\n\n# https://physics.nist.gov/cgi-bin/cuu/Value?mmusme\nm = 206.7682830\nμ = 1 / (1/m + 1/m)\nMP = MorsePotential(μ=μ)\n# @show nₘₐₓ(MP)\n\nusing CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r$\", ylabel=L\"$V(r),~E_n,~\\psi_n(r) \\times 5 + E_n$\", aspect=1, limits=(0.5,10,-0.105,0.007))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\nprintln(nₘₐₓ(MP))\n\nfor n in 0:nₘₐₓ(MP)\n # energy\n EMP = E(MP, n=n)\n lines!(ax, 0.1:0.01:15, r -> EMP > V(MP, r) ? EMP : NaN, color=:black, linewidth=2)\n hlines!(ax, E(MP, n=n), color=:black, linewidth=1, linestyle=:dash)\n # wave function\n lines!(ax, 0..10, x -> E(MP,n=n) + 0.0065*ψ(MP,x,n=n), linewidth=2)\nend\n\n#potential\nlines!(ax, 0..10, x -> V(MP, x), color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/MorsePotential.png\", f) # hide\n; # hide","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/","page":"Morse Potential","title":"Morse Potential","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/MorsePotential.log\"))","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 Coulomb 2-body system.","category":"page"},{"location":"HydrogenAtom/#Definitions","page":"Hydrogen Atom","title":"Definitions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"and the Hamiltonian","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH = - frachbar^22mu nabla^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. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.","category":"page"},{"location":"HydrogenAtom/#Parameters","page":"Hydrogen Atom","title":"Parameters","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.HydrogenAtom","category":"page"},{"location":"HydrogenAtom/#Antique.HydrogenAtom-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.HydrogenAtom","text":"HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nZ is the atomic number, m_mathrme is the electron mass, a_0is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"HydrogenAtom/#Potential","page":"Hydrogen Atom","title":"Potential","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.V(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.V-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.V","text":"V(model::HydrogenAtom, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Eigen-Values","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.E(::HydrogenAtom)","category":"page"},{"location":"HydrogenAtom/#Antique.E-Tuple{HydrogenAtom}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.E","text":"E(model::HydrogenAtom; n=1)\n\nE_n\n= -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2\n= -fracZ^22n^2 E_mathrmh\n\nwhere 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.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Eigen-Functions","page":"Hydrogen Atom","title":"Eigen Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.ψ(::HydrogenAtom, ::Any, ::Any, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.ψ-Tuple{HydrogenAtom, Any, Any, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.ψ","text":"ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Radial-Functions","page":"Hydrogen Atom","title":"Radial Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.R(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.R-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.R","text":"R(model::HydrogenAtom, r; n=1, l=0)\n\nR_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)\n\nwhere 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). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.L(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.L-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.L","text":"L(model::HydrogenAtom, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_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)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Spherical-Harmonics","page":"Hydrogen Atom","title":"Spherical Harmonics","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.Y(::HydrogenAtom, ::Any, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.Y-Tuple{HydrogenAtom, Any, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.Y","text":"Y(model::HydrogenAtom, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials","page":"Hydrogen Atom","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.P(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.P-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.P","text":"P(model::HydrogenAtom, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated 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 = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Parameters:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"H.Z\nH.Eₕ\nH.mₑ\nH.a₀\nH.ℏ","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eigen values:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"E(H, n=1)\nE(H, n=2)","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 = \", E(H,n=2) - E(H,n=1), \" Eₕ\")\nprintln(\"λ = \", ((E(H,n=2)-E(H,n=1))*Eₕ2nm⁻¹)^-1, \" 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":"# E. Tiesinga, et al., Rev. Mod. Phys. 93, 025010 (2021) 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# D. J. Griffiths, Am. J. Phys. 50, 698 (1982) https://doi.org/10.1119/1.12733\nδ = abs(ψ(H,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":"Potential energy curve:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using CairoMakie\n\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r~/~a_0$\", ylabel=L\"$V(r)~/~E_\\mathrm{h}$\", limits=(0.0,15.0,-2.0,0.2))\nlines!(ax, 0.1:0.01:20, r -> V(H, r))\nf","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 CairoMakie\nusing LaTeXStrings\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r~/~a_0$\", ylabel=L\"$r^2|R_{nl}(r)|^2~/~a_0^{-1}$\", limits=(0,20,0,0.58))\n\n# plot\nws = []\nls = []\nfor n in 1:3\n for l in 0:n-1\n w = lines!(\n ax,\n 0..20,\n r -> r^2 * R(H,r,n=n,l=l)^2,\n linewidth = 2,\n linestyle = [:solid,:dash,:dot,:dashdot,:dashdotdot][l+1],\n color = n,\n colormap = :tab10,\n colorrange = (1,10)\n )\n push!(ws, w)\n push!(ls, latexstring(\"n=$n, l=$l\"))\n end\nend\n\n# legend\naxislegend(ax, ws, ls, position=:rt)\n\nf","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Wave functions (electron density in n=5l=2m=1):","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Antique\nH = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)\nloop(x) = x<-1 ? loop(x+2) : (1 P(0,y,z), colorrange=(0.0,0.00001))\nf\nsave(\"assets/fig/HydrogenAtom.png\", f) # hide\n; # hide","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/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/HydrogenAtom.log\"))","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"CurrentModule = Antique","category":"page"},{"location":"InfinitePotentialWell3D/#Infinite-Potential-Well-3D-(Particle-in-a-3D-Box)","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D (Particle in a 3D-Box)","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"The three-dimensional infinite potential well (particle in a 3D-box) is one of the simplest models for a three-dimensional quantum mechanical system.","category":"page"},{"location":"InfinitePotentialWell3D/#Definitions","page":"Infinite Potential Well 3D","title":"Definitions","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":" hatH psi(xyz) = E psi(xyz)","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"and the Hamiltonian","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":" hatH = - frachbar^22m left(fracpartial^2partial x ^2 + fracpartial^2partial y ^2 + fracpartial^2partial z ^2right) + V(xyz)","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Parameters are specified with the following struct.","category":"page"},{"location":"InfinitePotentialWell3D/#Parameters","page":"Infinite Potential Well 3D","title":"Parameters","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Antique.InfinitePotentialWell3D","category":"page"},{"location":"InfinitePotentialWell3D/#Antique.InfinitePotentialWell3D-InfinitePotentialWell3D","page":"Infinite Potential Well 3D","title":"Antique.InfinitePotentialWell3D","text":"InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)\n\nL_xL_yL_z are the lengths of the box in x,y,z-direction, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"InfinitePotentialWell3D/#Potential","page":"Infinite Potential Well 3D","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Antique.V(::InfinitePotentialWell3D, ::Any, ::Any, ::Any)","category":"page"},{"location":"InfinitePotentialWell3D/#Antique.V-Tuple{InfinitePotentialWell3D, Any, Any, Any}-InfinitePotentialWell3D","page":"Infinite Potential Well 3D","title":"Antique.V","text":"V(model::InfinitePotentialWell3D, x,y,z)\n\nV(xyz) =\nleft\n beginarrayll\n 0 0 leq x leq L_x mathrmand 0 leq y leq L_y mathrmand 0 leq z leq L_z \n infty mathrmelsewhere\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell3D/#Eigen-Values","page":"Infinite Potential Well 3D","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Antique.E(::InfinitePotentialWell3D)","category":"page"},{"location":"InfinitePotentialWell3D/#Antique.E-Tuple{InfinitePotentialWell3D}-InfinitePotentialWell3D","page":"Infinite Potential Well 3D","title":"Antique.E","text":"E(model::InfinitePotentialWell3D; nx=1, ny=1, nz=1)\n\nE_n_xn_yn_z = frachbar^2 pi^22 m left(fracn_x^2L_x^2 + fracn_y^2L_y^2 + fracn_z^2L_z^2right)\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell3D/#Eigen-Functions","page":"Infinite Potential Well 3D","title":"Eigen Functions","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Antique.ψ(::InfinitePotentialWell3D, ::Any, ::Any, ::Any)","category":"page"},{"location":"InfinitePotentialWell3D/#Antique.ψ-Tuple{InfinitePotentialWell3D, Any, Any, Any}-InfinitePotentialWell3D","page":"Infinite Potential Well 3D","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell3D, x,y,z; nx=1, ny=1, nz=1)\n\nThe wave functions can be expressed as products of wave functions in a one-dimensional box.\n\npsi_n_xn_yn_z(xyz) = psi_n_x(x)psi_n_y(y)psi_n_z(z) = sqrtfrac8L_xL_yL_z sinleft(fracn_xpi xL_xright) sinleft(fracn_ypi yL_yright) sinleft(fracn_zpi zL_zright)\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell3D/#Usage-and-Examples","page":"Infinite Potential Well 3D","title":"Usage & Examples","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are supported. In this system, the model is generated by InfinitePotentialWell3D and several parameters Lx, Ly, Lz, m and ℏ are set as optional arguments.","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"using Antique\nIPW3D = InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Parameters:","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"IPW3D.Lx\nIPW3D.Ly\nIPW3D.Lz\nIPW3D.m\nIPW3D.ℏ","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Eigen values:","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"E(IPW3D, nx=1, ny=1, nz=1)\nE(IPW3D, nx=2, ny=1, nz=1)\nE(IPW3D, nx=1, ny=2, nz=1)\nE(IPW3D, nx=1, ny=1, nz=2)\nE(IPW3D, nx=2, ny=2, nz=1)\nE(IPW3D, nx=2, ny=1, nz=2)\nE(IPW3D, nx=1, ny=2, nz=2)\nE(IPW3D, nx=2, ny=2, nz=2)","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Wave functions:","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"The wave functions of the 3D infinite potential well can be expressed as a product of wave functions in 1D infinite potential wells for each of the three directions x,y,z. As the representation of 3D (non-spherical) wave functions easily becomes unclear, we refer to the documentation of the one-dimensional counterpart: InfinitePotentialWell (...)","category":"page"},{"location":"InfinitePotentialWell3D/#Testing","page":"Infinite Potential Well 3D","title":"Testing","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/InfinitePotentialWell3D.log\"))","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"CurrentModule = Antique","category":"page"},{"location":"RigidRotor/#Rigid-rotor","page":"Rigid Rotor","title":"Rigid rotor","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"The linear rigid rotor model can be used in quantum mechanics to predict the rotational energy of a diatomic molecule. The rotational energy depends on the moment of inertia for the system, I.","category":"page"},{"location":"RigidRotor/#Definitions","page":"Rigid Rotor","title":"Definitions","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"and the Hamiltonian","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"beginaligned\n hatH = - frachbar^22mu nabla^2 + V(r) \n = - frachbar^22I left frac1sintheta fracpartialpartialtheta left(sintheta fracpartialpartialthetaright) + frac1sin^2theta fracpartial^2partialphi^2 right\nendaligned","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"where I=mu R^2 is the moment of intertia, mu=left(frac1m_1+frac1m_2right)^-1 is the reduced mass of two particles and R is the distance between the two particles. Parameters are specified with the following struct.","category":"page"},{"location":"RigidRotor/#Parameters","page":"Rigid Rotor","title":"Parameters","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.RigidRotor","category":"page"},{"location":"RigidRotor/#Antique.RigidRotor-RigidRotor","page":"Rigid Rotor","title":"Antique.RigidRotor","text":"RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)\n\nm₁ and m₂ are mass of two particles, R is the distance, and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"RigidRotor/#Potential","page":"Rigid Rotor","title":"Potential","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.V(::RigidRotor, ::Any)","category":"page"},{"location":"RigidRotor/#Antique.V-Tuple{RigidRotor, Any}-RigidRotor","page":"Rigid Rotor","title":"Antique.V","text":"V(model::RigidRotor, r)\n\nbeginaligned\n V(r)\n = 0\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#Eigen-Values","page":"Rigid Rotor","title":"Eigen Values","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.E(::RigidRotor)","category":"page"},{"location":"RigidRotor/#Antique.E-Tuple{RigidRotor}-RigidRotor","page":"Rigid Rotor","title":"Antique.E","text":"E(model::RigidRotor; l=0)\n\nE_l\n= frachbar^22Il(l+1)\n\nwhere I=mu R^2 is the moment of inertia, R is the distance, and mu is the reduced mass of the two particles.\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#Eigen-Functions","page":"Rigid Rotor","title":"Eigen Functions","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.ψ(::RigidRotor, ::Any, ::Any)","category":"page"},{"location":"RigidRotor/#Antique.ψ-Tuple{RigidRotor, Any, Any}-RigidRotor","page":"Rigid Rotor","title":"Antique.ψ","text":"ψ(model::RigidRotor, θ, φ; l=0, m=0)\n\npsi_lm(thetavarphi) = Y_lm(thetavarphi)\n\nThe wave function is the spherical harmonics. The domain is 0leq theta lt pi and 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#Spherical-Harmonics","page":"Rigid Rotor","title":"Spherical Harmonics","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.Y(::RigidRotor, ::Any, ::Any)","category":"page"},{"location":"RigidRotor/#Antique.Y-Tuple{RigidRotor, Any, Any}-RigidRotor","page":"Rigid Rotor","title":"Antique.Y","text":"Y(model::RigidRotor, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#Associated-Legendre-Polynomials","page":"Rigid Rotor","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.P(::RigidRotor, ::Any)","category":"page"},{"location":"RigidRotor/#Antique.P-Tuple{RigidRotor, Any}-RigidRotor","page":"Rigid Rotor","title":"Antique.P","text":"P(model::RigidRotor, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#Usage-and-Examples","page":"Rigid Rotor","title":"Usage & Examples","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Install Antique.jl 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 RigidRotor and several parameters m₁, m₂, R and ℏ are set as optional arguments.","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"using Antique\nRR = RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Parameters:","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"RR.m₁\nRR.m₂\nRR.R\nRR.ℏ","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Eigen values:","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"E(RR, l=0)\nE(RR, l=1)\nE(RR, l=2)","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Wave functions:","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"ψ(RR, 0, 0, l=2, m=1)\nψ(RR, π/4, 0, l=2, m=1)\nψ(RR, π/4, π/2, l=2, m=1)","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"using CairoMakie\n\nf = Figure(size=(400,400))\nax = PolarAxis(f[1,1], title=L\"$\\theta\\mapsto|\\psi_{2,1}(\\theta,0)|^2$\", rticklabelsvisible=false)\nlines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=2,m=1))^2, linewidth=2)\n\nf","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"using CairoMakie\n\nf = Figure(size=(400,400))\nax = PolarAxis(f[1,1], title=L\"$\\theta\\mapsto|\\psi_{lm}(\\theta,0)|^2$\", rticklabelsvisible=false)\nl1 = lines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=0,m=0))^2, linewidth=2)\nl2 = lines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=1,m=0))^2, linewidth=2)\nl3 = lines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=1,m=1))^2, linewidth=2)\nl4 = lines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=2,m=1))^2, linewidth=2)\nLegend(f[2,1], [l1,l2,l3,l4], [L\"(l,m)=(0,0)\",L\"(1,0)\",L\"(1,1)\",L\"(2,1)\"], framevisible=false, orientation=:horizontal, tellwidth=false, tellheight=true)\n\nf\nsave(\"assets/fig/RigidRotor.png\", f) # hide\n; # hide","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"(Image: )","category":"page"},{"location":"RigidRotor/#Testing","page":"Rigid Rotor","title":"Testing","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","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":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/RigidRotor.log\"))","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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. 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 = HydrogenAtom(Z=1)\nE(H)\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":"He⁺ = HydrogenAtom(Z=2)\nE(He⁺)\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 \"PoschlTeller\"/\n \n PoschlTeller\n
\n
\n \n \"MorsePotential\"/\n \n MorsePotential\n
\n
","category":"page"},{"location":"","page":"Home","title":"Home","text":"Delta Potential DeltaPotential\nInfinite Potential Well InfinitePotentialWell\nHarmonic Oscillator HarmonicOscillator\nPoschlTeller PoschlTeller\nMorse Potential MorsePotential\nRigid Rotor RigidRotor\nInfinite PotentialWell 3D InfinitePotentialWell3D\nSpherical Oscillator SphericalOscillator\nHydrogen Atom HydrogenAtom\nCoulomb 2-Body System CoulombTwoBody","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":"#Developer's-Guide","page":"Home","title":"Developer's Guide","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Here is the guideline for adding new models.","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":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"CurrentModule = Antique","category":"page"},{"location":"PoschlTeller/#Pöschl-Teller-Potential","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"The Pöschl-Teller potential is one of the few potentials for which the quantum mechanical Schrödinger equation has an analytical solution. It has a finite number of bound states, which can be inferred easily from its potential strength parameter λ. It is defined for one-dimensional systems.","category":"page"},{"location":"PoschlTeller/#Definitions","page":"Pöschl-Teller Potential","title":"Definitions","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"and the Hamiltonian","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH = - frachbar^22 m fracmathrmd^2mathrmdx ^2 - frachbar^2m x_0^2 fraclambda(lambda+1)2 frac1mathrmcosh(xx_0)^2","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"After introducing the dimensionless variables","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" x^ast equiv xx_0qquad E^ast equiv frachbar^2m x_0^2 E","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"the Schrödinger equation reduces to","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH^ast psi(x^ast) = E^ast psi(x^ast)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"with","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH^ast = - frac12 fracmathrmd^2mathrmdx^ast^2 - fraclambda(lambda+1)2 frac1mathrmcosh(x^ast)^2","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Parameters are specified within the following struct.","category":"page"},{"location":"PoschlTeller/#Parameters","page":"Pöschl-Teller Potential","title":"Parameters","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.PoschlTeller","category":"page"},{"location":"PoschlTeller/#Antique.PoschlTeller-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.PoschlTeller","text":"PoschlTeller(λ=1.0, m=1.0, ℏ=1.0, x₀=1.0)\n\nlambda determines the potential strength.\n\n\n\n\n\n","category":"type"},{"location":"PoschlTeller/#Potential","page":"Pöschl-Teller Potential","title":"Potential","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.V(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.V-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.V","text":"V(model::PoschlTeller, x)\n\nbeginaligned\nV(x)\n= -frachbar^2m x_0^2 fraclambda(lambda+1)2 mathrmsech(x)^2\n= -frachbar^2m x_0^2 fraclambda(lambda+1)2 frac1mathrmcosh(x)^2\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Number-of-Bound-States","page":"Pöschl-Teller Potential","title":"Number of Bound States","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.nₘₐₓ(::PoschlTeller)","category":"page"},{"location":"PoschlTeller/#Antique.nₘₐₓ-Tuple{PoschlTeller}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::PoschlTeller)\n\nn_mathrmmax = leftlfloor lambda rightrfloor - 1\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Eigen-Values","page":"Pöschl-Teller Potential","title":"Eigen Values","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.E(::PoschlTeller)","category":"page"},{"location":"PoschlTeller/#Antique.E-Tuple{PoschlTeller}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.E","text":"E(model::PoschlTeller; n=0)\n\nE_n = -frachbar^2m x_0^2fracmu^22\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1.\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Eigen-Functions","page":"Pöschl-Teller Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.ψ(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.ψ-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.ψ","text":"ψ(model::PoschlTeller, x; n=0)\n\npsi_n(x) = P_lambda^mu(mathrmtanh(xx_0)) sqrtmufracGamma(lambda-mu+1)Gamma(lambda+mu+1)\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1 and P_lambda^mu are the associated Legendre functions.\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Associated-Legendre-Polynomials","page":"Pöschl-Teller Potential","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.P(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.P-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.P","text":"P(model::PoschlTeller, x; n=0, m=0)\n\nAssociated Legendre polynomials are the associated Legendre functions for integer indices. Here we use the same notation of the associated Legendre functions as in the model HydrogenAtom.\n\n```math \\begin{aligned} Pn^m(x) &= \\left( 1-x^2 \\right)^{m/2} \\frac{\\mathrm{d}^m}{\\mathrm{d}x^m} Pn(x) \\\n &= \\left( 1-x^2 \\right)^{m/2} \\frac{\\mathrm{d}^m}{\\mathrm{d}x^m} \\frac{1}{2^n n!} \\frac{\\mathrm{d}^n}{\\mathrm{d}x ^n} \\left[ \\left( x^2-1 \\right)^n \\right] \\\n &= \\frac{1}{2^n} (1-x^2)^{m/2} \\sum_{j=0}^{\\left\\lfloor\\frac{n-m}{2}\\right\\rfloor} (-1)^j \\frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \\end{aligned} ```\n\n\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Usage-and-Examples","page":"Pöschl-Teller Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Install Antique.jl 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 PoschlTeller and the parameters λ, m, ℏ, x₀.","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using Antique\nPT = PoschlTeller(λ=4.0)\n; # hide","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Parameters:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"PT.λ\nPT.m\nPT.ℏ\nPT.x₀","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Number of bound states:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"nₘₐₓ(PT)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Eigen values:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"E(PT, n=0)\nE(PT, n=1)\nE(PT, n=2)\nE(PT, n=3)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Potential energy curve:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using CairoMakie\n\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x)$\")\nlines!(ax, -6..6, x -> V(PT, x))\nf","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Wave functions:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using CairoMakie\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$\\psi(x)$\")\n\n# plot\nw0 = lines!(ax, -3..3, x -> ψ(PT, x, n=0))\nw1 = lines!(ax, -3..3, x -> ψ(PT, x, n=1))\nw2 = lines!(ax, -3..3, x -> ψ(PT, x, n=2))\nw3 = lines!(ax, -3..3, x -> ψ(PT, x, n=3))\n\n# legend\naxislegend(ax, [w0, w1, w2, w3], [L\"n=0\", L\"n=1\", L\"n=2\", L\"n=3\"], position=:lb)\n\nf","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x),~E_n,~\\psi_n(x) \\times 5 + E_n$\", aspect=1, limits=(-4,4,-10.5,1))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\nfor n in 0:3\n # classical turning point\n xE = acosh(sqrt(PT.λ*(PT.λ+1)/abs(E(PT,n=n))/2))\n # energy\n hlines!(ax, E(PT, n=n), color=:black, linewidth=1, linestyle=:dash)\n lines!(ax, [-xE,xE], fill(E(PT,n=n),2), color=:black, linewidth=2)\n # wave function\n lines!(ax, -4..4, x -> E(PT,n=n) + ψ(PT,x,n=n), linewidth=2)\nend\n\n#potential\nlines!(ax, -4..4, x -> V(PT,x), color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/PoschlTeller.png\", f) # hide\n; # hide","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"(Image: )","category":"page"},{"location":"PoschlTeller/#Testing","page":"Pöschl-Teller Potential","title":"Testing","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/PoschlTeller.log\"))","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":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"and the Hamiltonian","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters are specified with the following struct.","category":"page"},{"location":"InfinitePotentialWell/#Parameters","page":"Infinite Potential Well","title":"Parameters","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.InfinitePotentialWell","category":"page"},{"location":"InfinitePotentialWell/#Antique.InfinitePotentialWell-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.InfinitePotentialWell","text":"InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n\nL is the length of the box, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"InfinitePotentialWell/#Potential","page":"Infinite Potential Well","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.V(::InfinitePotentialWell, ::Any)","category":"page"},{"location":"InfinitePotentialWell/#Antique.V-Tuple{InfinitePotentialWell, Any}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.V","text":"V(model::InfinitePotentialWell; x)\n\nV(x) =\nleft\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\nright\n\n\n\n\n\n","category":"method"},{"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":"Antique.E(::InfinitePotentialWell)","category":"page"},{"location":"InfinitePotentialWell/#Antique.E-Tuple{InfinitePotentialWell}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.E","text":"E(model::InfinitePotentialWell; n=1)\n\nE_n = frachbar^2 n^2 pi^22 m L^2\n\n\n\n\n\n","category":"method"},{"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":"Antique.ψ(::InfinitePotentialWell, ::Any)","category":"page"},{"location":"InfinitePotentialWell/#Antique.ψ-Tuple{InfinitePotentialWell, Any}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell, x; n=1)\n\npsi_n(x) = sqrtfrac2L sin fracnpi xL\n\n\n\n\n\n","category":"method"},{"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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated 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 = InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n; #hide","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":"IPW.L\nIPW.m\nIPW.ℏ","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":"E(IPW, n=1)\nE(IPW, n=2)","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 CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$\\psi(x)$\")\n\n# plot\nw1 = lines!(ax, 0..1, x -> ψ(IPW, x, n=1))\nw2 = lines!(ax, 0..1, x -> ψ(IPW, x, n=2))\nw3 = lines!(ax, 0..1, x -> ψ(IPW, x, n=3))\nw4 = lines!(ax, 0..1, x -> ψ(IPW, x, n=4))\nw5 = lines!(ax, 0..1, x -> ψ(IPW, x, n=5))\n\n# legend\naxislegend(ax, [w1, w2, w3, w4, w5], [L\"n=1\", L\"n=2\", L\"n=3\", L\"n=4\", L\"n=5\"], position=:lb)\n\nf","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":"using CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x),~E_n,~\\psi_n(x) \\times 5 + E_n$\", aspect=1, limits=(-0.5,1.5,0,140))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\n# plot\nfor n in 1:5\n # energy\n lines!(ax, [0,IPW.L], fill(E(IPW,n=n),2), color=:black, linewidth=2)\n # wave function\n lines!(ax, 0..IPW.L, x -> E(IPW,n=n) + 5*ψ(IPW,x,n=n), linewidth=2)\nend\n\n#potential\nlines!(ax, [0,0,IPW.L,IPW.L], [140,0,0,140], color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/InfinitePotentialWell.png\", f) # hide\n; # hide","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/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/InfinitePotentialWell.log\"))","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"CurrentModule = Antique","category":"page"},{"location":"SphericalOscillator/#Spherical-Oscillator","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"The spherical oscillator (3D isotropic harmonic oscillator) is the most frequently used model in quantum physics. This model uses a spherical coordinate system.","category":"page"},{"location":"SphericalOscillator/#Definitions","page":"Spherical Oscillator","title":"Definitions","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":" hatH psi(r) = E psi(r)","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"and the Hamiltonian","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":" hatH = - frachbar^22mu nabla^2 + V(r)","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Parameters are specified with the following struct.","category":"page"},{"location":"SphericalOscillator/#Parameters","page":"Spherical Oscillator","title":"Parameters","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.SphericalOscillator","category":"page"},{"location":"SphericalOscillator/#Antique.SphericalOscillator-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.SphericalOscillator","text":"HarmonicOscillator(k=1.0, μ=1.0, ℏ=1.0)\n\nk is the force constant, μ is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"SphericalOscillator/#Potential","page":"Spherical Oscillator","title":"Potential","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.V(::SphericalOscillator, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.V-Tuple{SphericalOscillator, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.V","text":"V(model::SphericalOscillator, r)\n\nV(r)\n= frac12 k r^2\n= frac12 mu omega^2 r^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkmu is the angular frequency and xi = sqrtfracmuomegahbarr.\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Eigen-Values","page":"Spherical Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.E(::SphericalOscillator)","category":"page"},{"location":"SphericalOscillator/#Antique.E-Tuple{SphericalOscillator}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.E","text":"E(model::SphericalOscillator; n=0, l=0)\n\nE_nl\n= left(2n + l + frac32right)hbar omega\n\nwhere omega = sqrtkmu.\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Eigen-Functions","page":"Spherical Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.ψ(::SphericalOscillator, ::Any, ::Any, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.ψ-Tuple{SphericalOscillator, Any, Any, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.ψ","text":"ψ(model::SphericalOscillator, r, θ, φ; n=0, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Radial-Functions","page":"Spherical Oscillator","title":"Radial Functions","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.R(::SphericalOscillator, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.R-Tuple{SphericalOscillator, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.R","text":"R(model::SphericalOscillator, r; n=0, l=0)\n\nR_nl(r) = sqrt fracgamma^322sqrtpi sqrtfrac2^n+l+3 n(2n+2l+1) xi^l expleft(-xi^22right)L_n^(l+frac12) left(xi^2right)\n\nwhere gamma = muomegahbar and xi = sqrtgammar = sqrtmuomegahbarr are defined. The generalized Laguerre polynomials are defined as L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Generalized-Laguerre-Polynomials","page":"Spherical Oscillator","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"note: Note\nThe generalized Laguerre polynomials L_n^(alpha)(x), not the associated Laguerre polynomials L_n^k(x), are used in this model. ","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.L(::SphericalOscillator, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.L-Tuple{SphericalOscillator, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.L","text":"L(model::SphericalOscillator, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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 \nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Spherical-Harmonics","page":"Spherical Oscillator","title":"Spherical Harmonics","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.Y(::SphericalOscillator, ::Any, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.Y-Tuple{SphericalOscillator, Any, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.Y","text":"Y(model::SphericalOscillator, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Associated-Legendre-Polynomials","page":"Spherical Oscillator","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Antique.P(::SphericalOscillator, ::Any)","category":"page"},{"location":"SphericalOscillator/#Antique.P-Tuple{SphericalOscillator, Any}-SphericalOscillator","page":"Spherical Oscillator","title":"Antique.P","text":"P(model::SphericalOscillator, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"SphericalOscillator/#Usage-and-Examples","page":"Spherical Oscillator","title":"Usage & Examples","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Install Antique.jl 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 SphericalOscillator and several parameters k, μ and ℏ are set as optional arguments.","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"using Antique\nSO = SphericalOscillator(k=1.0, μ=1.0, ℏ=1.0)\n; # hide","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Parameters:","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"SO.k\nSO.μ\nSO.ℏ","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"Eigen values:","category":"page"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"E(SO, n=0)\nE(SO, n=1)\nE(SO, n=2)\nE(SO, n=0, l=1)\nE(SO, n=1, l=1)\nE(SO, n=2, l=1)","category":"page"},{"location":"SphericalOscillator/#Testing","page":"Spherical Oscillator","title":"Testing","text":"","category":"section"},{"location":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical 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":"SphericalOscillator/","page":"Spherical Oscillator","title":"Spherical Oscillator","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/SphericalOscillator.log\"))","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"CurrentModule = Antique","category":"page"},{"location":"DeltaPotential/#Delta-Potential","page":"Delta Potential","title":"Delta Potential","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"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.","category":"page"},{"location":"DeltaPotential/#Definitions","page":"Delta Potential","title":"Definitions","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"and the Hamiltonian","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Parameters are specified with the following struct.","category":"page"},{"location":"DeltaPotential/#Parameters","page":"Delta Potential","title":"Parameters","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.DeltaPotential","category":"page"},{"location":"DeltaPotential/#Antique.DeltaPotential-DeltaPotential","page":"Delta Potential","title":"Antique.DeltaPotential","text":"DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n\nalpha is the potential strength, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"DeltaPotential/#Potential","page":"Delta Potential","title":"Potential","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.V(::DeltaPotential, ::Any)","category":"page"},{"location":"DeltaPotential/#Antique.V-Tuple{DeltaPotential, Any}-DeltaPotential","page":"Delta Potential","title":"Antique.V","text":"V(model::DeltaPotential, x)\n\nV(x) = -alpha delta(x)\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Eigen-Values","page":"Delta Potential","title":"Eigen Values","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.E(::DeltaPotential)","category":"page"},{"location":"DeltaPotential/#Antique.E-Tuple{DeltaPotential}-DeltaPotential","page":"Delta Potential","title":"Antique.E","text":"E(model::DeltaPotential)\n\nE = - fracmalpha^22hbar^2\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Eigen-Functions","page":"Delta Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.ψ(::DeltaPotential, ::Any)","category":"page"},{"location":"DeltaPotential/#Antique.ψ-Tuple{DeltaPotential, Any}-DeltaPotential","page":"Delta Potential","title":"Antique.ψ","text":"ψ(model::DeltaPotential, x)\n\npsi(x) = fracsqrtmalphahbar mathrme^-malpha xhbar^2\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Usage-and-Examples","page":"Delta Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Install Antique.jl 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.","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using Antique\nDP = DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Parameters:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"DP.α\nDP.m\nDP.ℏ","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Eigen values:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"E(DP)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Wave functions:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using CairoMakie\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$\\psi(x)$\")\n\n# plot\nw = lines!(ax, -5..5, x -> ψ(DP, x))\n\nf","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x),~E_n,~\\psi_n(x) \\times 5 + E_n$\", aspect=1, limits=(-5,5,-0.6,0.6))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\n# energy\nhlines!(ax, E(DP), color=:black, linewidth=1, linestyle=:dash)\n\n# wave function\nlines!(ax, -5..5, x -> E(DP) + ψ(DP,x), linewidth=2)\n\n#potential\nlines!(ax, [-5,0,0,0,5], [0,0,-1,0,0], color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/DeltaPotential.png\", f) # hide\n; # hide","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"(Image: )","category":"page"},{"location":"DeltaPotential/#Testing","page":"Delta Potential","title":"Testing","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/DeltaPotential.log\"))","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"CurrentModule = Antique","category":"page"},{"location":"CoulombTwoBody/#Coulomb-2-Body-System","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"This is the model of two particles interacting through Coulomb forces such as positronium, muonium, hydrogen atoms, deuterium atoms, etc.","category":"page"},{"location":"CoulombTwoBody/#Definitions","page":"Coulomb 2-Body System","title":"Definitions","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"and the Hamiltonian","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":" hatH = - frachbar^22mu nabla^2 + V(r)","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"where mu=left(frac1m_1+frac1m_2right)^-1 is the reduced mass of particle 1 and particle 2. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.","category":"page"},{"location":"CoulombTwoBody/#Parameters","page":"Coulomb 2-Body System","title":"Parameters","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.CoulombTwoBody","category":"page"},{"location":"CoulombTwoBody/#Antique.CoulombTwoBody-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.CoulombTwoBody","text":"CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nz₁ is the charge number of particle 1, z₁ is the charge number of particle 2, m₁ is the mass of particle 1, m₂ is the mass of particle 2, m_mathrme is the electron mass (the unit of m₁ and m₂), a_0 is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"CoulombTwoBody/#Potential","page":"Coulomb 2-Body System","title":"Potential","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.V(::CoulombTwoBody, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.V-Tuple{CoulombTwoBody, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.V","text":"V(model::CoulombTwoBody, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Eigen-Values","page":"Coulomb 2-Body System","title":"Eigen Values","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.E(::CoulombTwoBody)","category":"page"},{"location":"CoulombTwoBody/#Antique.E-Tuple{CoulombTwoBody}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.E","text":"E(model::CoulombTwoBody; n=1)\n\nE_n\n= -frac(z_1 z_2)^22n^2 fracmum_mathrme E_mathrmh\n\nwhere mu=left(frac1m_1+frac1m_2right)^-1 is the reduced mass of particle 1 and particle 2, 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.\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Eigen-Functions","page":"Coulomb 2-Body System","title":"Eigen Functions","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.ψ(::CoulombTwoBody, ::Any, ::Any, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.ψ-Tuple{CoulombTwoBody, Any, Any, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.ψ","text":"ψ(model::CoulombTwoBody, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Radial-Functions","page":"Coulomb 2-Body System","title":"Radial Functions","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.R(::CoulombTwoBody, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.R-Tuple{CoulombTwoBody, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.R","text":"R(model::CoulombTwoBody, r; n=1, l=0)\n\nR_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_muright)^3 left(frac2Zrn a_muright)^l exp left(-fracZrn a_muright) L_n+l^2l+1 left(frac2Zrn a_muright)\n\nwhere frac1mu = frac1m_1+frac1m_2, a_mu = a_0 fracm_mathrmemu, 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). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Associated-Laguerre-Polynomials","page":"Coulomb 2-Body System","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.L(::CoulombTwoBody, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.L-Tuple{CoulombTwoBody, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.L","text":"L(model::CoulombTwoBody, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_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)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Spherical-Harmonics","page":"Coulomb 2-Body System","title":"Spherical Harmonics","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.Y(::CoulombTwoBody, ::Any, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.Y-Tuple{CoulombTwoBody, Any, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.Y","text":"Y(model::CoulombTwoBody, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Associated-Legendre-Polynomials","page":"Coulomb 2-Body System","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Antique.P(::CoulombTwoBody, ::Any)","category":"page"},{"location":"CoulombTwoBody/#Antique.P-Tuple{CoulombTwoBody, Any}-CoulombTwoBody","page":"Coulomb 2-Body System","title":"Antique.P","text":"P(model::CoulombTwoBody, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"CoulombTwoBody/#Usage-and-Examples","page":"Coulomb 2-Body System","title":"Usage & Examples","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Install Antique.jl 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 CoulombTwoBody and several parameters z₁, z₂, m₁, m₂, mₑ, a₀, Eₕ and ℏ are set as optional arguments.","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"using Antique\nPs = CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"CoulombTwoBody/#Parameters-2","page":"Coulomb 2-Body System","title":"Parameters","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Ps.z₁\nPs.z₂\nPs.m₁\nPs.m₂\nPs.mₑ\nPs.a₀\nPs.Eₕ\nPs.ℏ","category":"page"},{"location":"CoulombTwoBody/#Eigen-Values-2","page":"Coulomb 2-Body System","title":"Eigen Values","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Examples of positronium:","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"E(Ps, n=1)\nE(Ps, n=2)","category":"page"},{"location":"CoulombTwoBody/#Mass-and-Charge-Dependence","page":"Coulomb 2-Body System","title":"Mass and Charge Dependence","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"The values of masses are cited from the 2018 CODATA recommended values, E. Tiesinga, et al., Rev. Mod. Phys. 93, 025010 (2021).","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"me = 1.0 # me # \nmµ = 206.7682830 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mmusme\nmp = 1836.15267343 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mpsme\nmd = 3670.48296788 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mdsme\nmt = 5496.92153573 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mtsme\nmh = 5495.88528007 # me # https://physics.nist.gov/cgi-bin/cuu/Value?mhsme\nma = 7294.29954142 # me # https://physics.nist.gov/cgi-bin/cuu/Value?malsme\n\nPs = CoulombTwoBody(m₁=me, m₂=me)\nMu = CoulombTwoBody(m₁=me, m₂=mµ)\nH = CoulombTwoBody(m₁=me, m₂=mp)\nD = CoulombTwoBody(m₁=me, m₂=md)\nT = CoulombTwoBody(m₁=me, m₂=mt)\nBO = CoulombTwoBody(m₁=me, m₂=Inf)\n\nHe3⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=mh)\nHe4⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=ma)\nHe∞⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=Inf)\n\npμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mp)\ndμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=md)\ntμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mt)\nbμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=Inf)\nhμ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=mµ, m₂=mh)\nαμ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=mµ, m₂=ma)\n\nprintln(\" \\tE / Eₕ\")\nprintln(\"Ps \\t\", E(Ps))\nprintln(\"Mu \\t\", E(Mu))\nprintln(\"H \\t\", E(H))\nprintln(\"D \\t\", E(D))\nprintln(\"T \\t\", E(T))\nprintln(\"∞H \\t\", E(BO))\nprintln(\"³He⁺\\t\", E(He3⁺))\nprintln(\"⁴He⁺\\t\", E(He4⁺))\nprintln(\"∞He⁺\\t\", E(He∞⁺))\nprintln(\"pμ \\t\", E(pμ))\nprintln(\"dμ \\t\", E(dμ))\nprintln(\"tμ \\t\", E(tμ))\nprintln(\"bμ \\t\", E(bμ))\nprintln(\"hμ \\t\", E(hμ))\nprintln(\"αμ \\t\", E(αμ))","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"println(\" \\t<δ³(r)> / a₀⁻³\")\nprintln(\"1/8π =\\t\", 1/8/π)\nprintln(\"Ps \\t\", abs(ψ(Ps,0,0,0))^2)\nprintln(\"Mu \\t\", abs(ψ(Mu,0,0,0))^2)\nprintln(\"H \\t\", abs(ψ(H ,0,0,0))^2)\nprintln(\"D \\t\", abs(ψ(D ,0,0,0))^2)\nprintln(\"T \\t\", abs(ψ(T ,0,0,0))^2)\nprintln(\"∞H \\t\", abs(ψ(BO,0,0,0))^2)\nprintln(\"1/π = \\t\", 1/π)","category":"page"},{"location":"CoulombTwoBody/#Lifetime-of-Positronium","page":"Coulomb 2-Body System","title":"Lifetime of Positronium","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"The lifetime tau of positronium (Ps, mathrme^+mathrme^-) is written as","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"tau = frac1Gamma","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Gamma = 4 pi alpha^4 c a_0^2 langledelta^3(pmbr)rangle","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"where langledelta^3(pmbr)rangle = langlepsidelta^3(pmbr)psirangle = psi(pmb0)^2 = frac18pi a_0^-3 simeq 2685times10^29mathrmm^-3 is the value of probability density at the origin (r=0). Reference:","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"(7.169) in D. J. Griffiths, Introduction to Elementary Particles (John Wiley & Sons, Inc. 1987) ISBN 0-471-60386-4\nS. Berko, H. N. Pendleton, Annual Review of Nuclear and Particle Science, 30, 543 (1980)\nA. M. Frolov, S. I. Kryuchkov, and V. H. Smith, Jr., Phys. Rev. A, 51, 4514 (1995)","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"α = 7.2973525693e-3 # # https://physics.nist.gov/cgi-bin/cuu/Value?alph\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\n\nPs = CoulombTwoBody(z₁=1, z₂=-1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\nδ = abs(ψ(Ps,0,0,0))^2 * a₀^(-3)\nΓ = 4 * π * α^4 * c * a₀^2 * δ\nτ = 1/Γ\nprintln(\"<δ> = \", abs(ψ(Ps,0,0,0))^2, \" a₀⁻³\")\nprintln(\" = \", δ, \" m⁻³\")\nprintln(\"Γ = \", Γ / 1e9, \" GHz\")\nprintln(\"τ = \", τ / 1e-12, \" ps\")","category":"page"},{"location":"CoulombTwoBody/#Hyperfine-Splitting","page":"Coulomb 2-Body System","title":"Hyperfine Splitting","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"The hyperfine splitting of hydrogen atoms is given as","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"Delta E (mathrmH) = -frac23 mu_0 gamma_mathrmp gamma_mathrme hbar^2 langledelta^3(pmbr)rangle","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"in Griffiths(1982). This fomula is not available for deuterium (D) and positronium (Ps). Because of the different spin between the proton and the deuteron for D, the contribution of positron-electron pair annihilation for Ps. Note the definition of gyromagnetic ratio. The mass of protons is used for all nucleons and nuclei:","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"beginaligned\ngamma_mathrme = frac-e2 m_mathrme g_mathrme \ngamma_mathrme^+ = frac+e2 m_mathrme g_mathrme \ngamma_mathrmmu = frac-e2 m_mathrmmu g_mathrmmu \ngamma_mathrmp = frac+e2 m_mathrmp g_mathrmp \ngamma_mathrmd = frac+e2 m_mathrmp g_mathrmd \ngamma_mathrmt = frac+e2 m_mathrmp g_mathrmt \ngamma_mathrmh = frac+2e2 m_mathrmp g_mathrmh \nendaligned","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"The value of probability density at the origin is langledelta^3(pmbr)rangle = langlepsidelta^3(pmbr)psirangle = psi(pmb0)^2 simeq frac1pi a_0^-3 simeq 2148times10^30mathrmm^-3 in Mu, H, D and T. This values are very different in Ps, ^3mathrmHe^+ and muonic hydrogen (mathrmpmu) due to the difference of reduced masses and charges. The energy can be converted to frequency (Hz) by v = Delta E h.","category":"page"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"a₀ = 5.29177210903e-11 # m # https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0\nEₕ = 4.3597447222071e-18 # J # https://physics.nist.gov/cgi-bin/cuu/Value?hr\nℏ = 1.054571817e-34 # J s # https://physics.nist.gov/cgi-bin/cuu/Value?hbar\n\nme = 9.1093837015e-31 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?me\nmµ = 1.883531627e-28 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mmu\nmp = 1.67262192369e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mp\nmd = 3.3435837724e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?md\nmt = 5.0073567446e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mt\nmh = 5.0064127796e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mh\n\ne = 1.602176634e-19 # C # https://physics.nist.gov/cgi-bin/cuu/Value?e\nµ₀ = 1.25663706212e-6 # N A-2 # https://physics.nist.gov/cgi-bin/cuu/Value?mu0\nh = 6.62607015e-34 # J Hz-1 # https://physics.nist.gov/cgi-bin/cuu/Value?h\neV = 1.602176634e-19 # J # https://physics.nist.gov/cgi-bin/cuu/Value?evj\n\nge = 2.00231930436256 # https://physics.nist.gov/cgi-bin/cuu/Value?gem\ngµ = 2.0023318418 # https://physics.nist.gov/cgi-bin/cuu/Value?gmum\ngp = 5.5856946893 # https://physics.nist.gov/cgi-bin/cuu/Value?gp\ngd = 0.8574382338 # https://physics.nist.gov/cgi-bin/cuu/Value?gdn\ngt = 5.957924931 # https://physics.nist.gov/cgi-bin/cuu/Value?gtn\ngh = -4.255250615 # https://physics.nist.gov/cgi-bin/cuu/Value?ghn\n\nPs = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=me, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nMu = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mµ, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nH = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mp, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nD = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=md, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nT = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mt, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\nhe = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=mh, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\npμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mp, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)\n\nΔE_H = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gp / mp * abs(ψ(H,0,0,0))^2\nΔE_D = 1 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gd / mp * abs(ψ(D,0,0,0))^2\nΔE_T = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gt / mp * abs(ψ(H,0,0,0))^2\nΔE_Ps = 7/6 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * ge / me * abs(ψ(Ps,0,0,0))^2\nΔE_Mu = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gµ / mµ * abs(ψ(Mu,0,0,0))^2\nΔE_he = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gh / mp * abs(ψ(he,0,0,0))^2\nΔE_pµ = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * gµ / mµ * gp / mp * abs(ψ(pµ,0,0,0))^2\n\nprintln(\"H \\t\", ΔE_H / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"1420.405751768(1) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"D \\t\", ΔE_D / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"327.384352522(2) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"T \\t\", ΔE_T / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"1516.701470773(8) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"Ps\\t\", ΔE_Ps / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"203391.7(6) MHz (https://doi.org/10.48550/arXiv.hep-ph/0310099)\")\nprintln(\"Mu\\t\", ΔE_Mu / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"4463.30278(5) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"³He⁺\\t\", ΔE_he / h / 1e6, \" MHz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", \"-8665.649867(10) MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)\")\nprintln(\"µp\\t\", ΔE_pµ / h / 1e12, \" THz (Antique.jl + CODATA2018)\")\nprintln(\" \\t\", 0.182725*eV / h / 1e12 ,\" THz (https://doi.org/10.1119/1.12733, https://doi.org/10.1016/j.nimb.2012.04.001)\")","category":"page"},{"location":"CoulombTwoBody/#Testing","page":"Coulomb 2-Body System","title":"Testing","text":"","category":"section"},{"location":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","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":"CoulombTwoBody/","page":"Coulomb 2-Body System","title":"Coulomb 2-Body System","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/CoulombTwoBody.log\"))","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":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"and the Hamiltonian","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters are specified with the following struct.","category":"page"},{"location":"HarmonicOscillator/#Parameters","page":"Harmonic Oscillator","title":"Parameters","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.HarmonicOscillator","category":"page"},{"location":"HarmonicOscillator/#Antique.HarmonicOscillator-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.HarmonicOscillator","text":"HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)\n\nk is the force constant, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"HarmonicOscillator/#Potential","page":"Harmonic Oscillator","title":"Potential","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.V(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.V-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.V","text":"V(model::HarmonicOscillator, x)\n\nV(x)\n= frac12 k x^2\n= frac12 m omega^2 x^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkm is the angular frequency and xi = sqrtfracmomegahbarx.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Eigen-Values","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.E(::HarmonicOscillator)","category":"page"},{"location":"HarmonicOscillator/#Antique.E-Tuple{HarmonicOscillator}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.E","text":"E(model::HarmonicOscillator; n=0)\n\nE_n = hbar omega left( n + frac12 right)\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Eigen-Functions","page":"Harmonic Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.ψ(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.ψ-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.ψ","text":"ψ(model::HarmonicOscillator, x; n=0)\n\npsi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)\n\nwhere 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 defined.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Hermite-Polynomials","page":"Harmonic Oscillator","title":"Hermite Polynomials","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.H(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.H-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.H","text":"H(model::HarmonicOscillator, x; n=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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\nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated 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 = HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"HO.k\nHO.m\nHO.ℏ","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Eigen values:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"E(HO, n=0)\nE(HO, n=1)","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 CairoMakie\n\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x)$\")\nlines!(ax, -5..5, x -> V(HO, x))\nf","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 CairoMakie\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$\\psi(x)$\")\n\n# plot\nw0 = lines!(ax, -5..5, x -> ψ(HO, x, n=0))\nw1 = lines!(ax, -5..5, x -> ψ(HO, x, n=1))\nw2 = lines!(ax, -5..5, x -> ψ(HO, x, n=2))\nw3 = lines!(ax, -5..5, x -> ψ(HO, x, n=3))\nw4 = lines!(ax, -5..5, x -> ψ(HO, x, n=4))\n\n# legend\naxislegend(ax, [w0, w1, w2, w3, w4], [L\"n=0\", L\"n=1\", L\"n=2\", L\"n=3\", L\"n=4\"], position=:lb)\n\nf","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 CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$x$\", ylabel=L\"$V(x),~E_n,~\\psi_n(x) \\times 5 + E_n$\", aspect=1, limits=(-5,5,0,5.2))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\nfor n in 0:4\n # classical turning point\n xE = sqrt(2*HO.k*E(HO, n=n))\n # energy\n lines!(ax, [-xE,xE], fill(E(HO,n=n),2), color=:black, linewidth=2)\n hlines!(ax, E(HO, n=n), color=:black, linewidth=1, linestyle=:dash)\n # wave function\n lines!(ax, -5..5, x -> E(HO,n=n) + 0.5*ψ(HO,x,n=n), linewidth=2)\nend\n\n#potential\nlines!(ax, -5..5, x -> V(HO, x), color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/HarmonicOscillator.png\", f) # hide\n; # hide","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/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/HarmonicOscillator.log\"))","category":"page"},{"location":"API/","page":"API reference","title":"API reference","text":"CurrentModule = Antique","category":"page"},{"location":"API/#API-reference","page":"API reference","title":"API reference","text":"","category":"section"},{"location":"API/","page":"API reference","title":"API reference","text":"Order = [:type, :function]\nModules = [Antique]","category":"page"},{"location":"API/#Antique.CoulombTwoBody","page":"API reference","title":"Antique.CoulombTwoBody","text":"CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nz₁ is the charge number of particle 1, z₁ is the charge number of particle 2, m₁ is the mass of particle 1, m₂ is the mass of particle 2, m_mathrme is the electron mass (the unit of m₁ and m₂), a_0 is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.DeltaPotential","page":"API reference","title":"Antique.DeltaPotential","text":"DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n\nalpha is the potential strength, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.HarmonicOscillator","page":"API reference","title":"Antique.HarmonicOscillator","text":"HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)\n\nk is the force constant, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.HydrogenAtom","page":"API reference","title":"Antique.HydrogenAtom","text":"HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nZ is the atomic number, m_mathrme is the electron mass, a_0is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.InfinitePotentialWell","page":"API reference","title":"Antique.InfinitePotentialWell","text":"InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n\nL is the length of the box, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.InfinitePotentialWell3D","page":"API reference","title":"Antique.InfinitePotentialWell3D","text":"InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)\n\nL_xL_yL_z are the lengths of the box in x,y,z-direction, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.MorsePotential","page":"API reference","title":"Antique.MorsePotential","text":"MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)\n\nr_mathrme is the equilibrium bond distance, D_mathrme is the the well depth , k is the force constant, mu is the reduced mass and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.PoschlTeller","page":"API reference","title":"Antique.PoschlTeller","text":"PoschlTeller(λ=1.0, m=1.0, ℏ=1.0, x₀=1.0)\n\nlambda determines the potential strength.\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.RigidRotor","page":"API reference","title":"Antique.RigidRotor","text":"RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)\n\nm₁ and m₂ are mass of two particles, R is the distance, and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.SphericalOscillator","page":"API reference","title":"Antique.SphericalOscillator","text":"HarmonicOscillator(k=1.0, μ=1.0, ℏ=1.0)\n\nk is the force constant, μ is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.E-Tuple{CoulombTwoBody}","page":"API reference","title":"Antique.E","text":"E(model::CoulombTwoBody; n=1)\n\nE_n\n= -frac(z_1 z_2)^22n^2 fracmum_mathrme E_mathrmh\n\nwhere mu=left(frac1m_1+frac1m_2right)^-1 is the reduced mass of particle 1 and particle 2, 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.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{DeltaPotential}","page":"API reference","title":"Antique.E","text":"E(model::DeltaPotential)\n\nE = - fracmalpha^22hbar^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{HarmonicOscillator}","page":"API reference","title":"Antique.E","text":"E(model::HarmonicOscillator; n=0)\n\nE_n = hbar omega left( n + frac12 right)\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{HydrogenAtom}","page":"API reference","title":"Antique.E","text":"E(model::HydrogenAtom; n=1)\n\nE_n\n= -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2\n= -fracZ^22n^2 E_mathrmh\n\nwhere 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.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{InfinitePotentialWell3D}","page":"API reference","title":"Antique.E","text":"E(model::InfinitePotentialWell3D; nx=1, ny=1, nz=1)\n\nE_n_xn_yn_z = frachbar^2 pi^22 m left(fracn_x^2L_x^2 + fracn_y^2L_y^2 + fracn_z^2L_z^2right)\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{InfinitePotentialWell}","page":"API reference","title":"Antique.E","text":"E(model::InfinitePotentialWell; n=1)\n\nE_n = frachbar^2 n^2 pi^22 m L^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{MorsePotential}","page":"API reference","title":"Antique.E","text":"E(model::MorsePotential; n=0)\n\nE_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2\n\nwhere omega = sqrtkµ and chi = frachbaromega4D_mathrme are defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{PoschlTeller}","page":"API reference","title":"Antique.E","text":"E(model::PoschlTeller; n=0)\n\nE_n = -frachbar^2m x_0^2fracmu^22\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{RigidRotor}","page":"API reference","title":"Antique.E","text":"E(model::RigidRotor; l=0)\n\nE_l\n= frachbar^22Il(l+1)\n\nwhere I=mu R^2 is the moment of inertia, R is the distance, and mu is the reduced mass of the two particles.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{SphericalOscillator}","page":"API reference","title":"Antique.E","text":"E(model::SphericalOscillator; n=0, l=0)\n\nE_nl\n= left(2n + l + frac32right)hbar omega\n\nwhere omega = sqrtkmu.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.H-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.H","text":"H(model::HarmonicOscillator, x; n=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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\nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{CoulombTwoBody, Any}","page":"API reference","title":"Antique.L","text":"L(model::CoulombTwoBody, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_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)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.L","text":"L(model::HydrogenAtom, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_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)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.L","text":"L(model::MorsePotential, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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 \nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{SphericalOscillator, Any}","page":"API reference","title":"Antique.L","text":"L(model::SphericalOscillator, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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 \nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{CoulombTwoBody, Any}","page":"API reference","title":"Antique.P","text":"P(model::CoulombTwoBody, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.P","text":"P(model::HydrogenAtom, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.P","text":"P(model::PoschlTeller, x; n=0, m=0)\n\nAssociated Legendre polynomials are the associated Legendre functions for integer indices. Here we use the same notation of the associated Legendre functions as in the model HydrogenAtom.\n\n```math \\begin{aligned} Pn^m(x) &= \\left( 1-x^2 \\right)^{m/2} \\frac{\\mathrm{d}^m}{\\mathrm{d}x^m} Pn(x) \\\n &= \\left( 1-x^2 \\right)^{m/2} \\frac{\\mathrm{d}^m}{\\mathrm{d}x^m} \\frac{1}{2^n n!} \\frac{\\mathrm{d}^n}{\\mathrm{d}x ^n} \\left[ \\left( x^2-1 \\right)^n \\right] \\\n &= \\frac{1}{2^n} (1-x^2)^{m/2} \\sum_{j=0}^{\\left\\lfloor\\frac{n-m}{2}\\right\\rfloor} (-1)^j \\frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \\end{aligned} ```\n\n\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{RigidRotor, Any}","page":"API reference","title":"Antique.P","text":"P(model::RigidRotor, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{SphericalOscillator, Any}","page":"API reference","title":"Antique.P","text":"P(model::SphericalOscillator, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.R-Tuple{CoulombTwoBody, Any}","page":"API reference","title":"Antique.R","text":"R(model::CoulombTwoBody, r; n=1, l=0)\n\nR_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_muright)^3 left(frac2Zrn a_muright)^l exp left(-fracZrn a_muright) L_n+l^2l+1 left(frac2Zrn a_muright)\n\nwhere frac1mu = frac1m_1+frac1m_2, a_mu = a_0 fracm_mathrmemu, 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). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.R-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.R","text":"R(model::HydrogenAtom, r; n=1, l=0)\n\nR_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)\n\nwhere 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). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.R-Tuple{SphericalOscillator, Any}","page":"API reference","title":"Antique.R","text":"R(model::SphericalOscillator, r; n=0, l=0)\n\nR_nl(r) = sqrt fracgamma^322sqrtpi sqrtfrac2^n+l+3 n(2n+2l+1) xi^l expleft(-xi^22right)L_n^(l+frac12) left(xi^2right)\n\nwhere gamma = muomegahbar and xi = sqrtgammar = sqrtmuomegahbarr are defined. The generalized Laguerre polynomials are defined as L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{CoulombTwoBody, Any}","page":"API reference","title":"Antique.V","text":"V(model::CoulombTwoBody, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{DeltaPotential, Any}","page":"API reference","title":"Antique.V","text":"V(model::DeltaPotential, x)\n\nV(x) = -alpha delta(x)\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.V","text":"V(model::HarmonicOscillator, x)\n\nV(x)\n= frac12 k x^2\n= frac12 m omega^2 x^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkm is the angular frequency and xi = sqrtfracmomegahbarx.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.V","text":"V(model::HydrogenAtom, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{InfinitePotentialWell, Any}","page":"API reference","title":"Antique.V","text":"V(model::InfinitePotentialWell; x)\n\nV(x) =\nleft\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{InfinitePotentialWell3D, Any, Any, Any}","page":"API reference","title":"Antique.V","text":"V(model::InfinitePotentialWell3D, x,y,z)\n\nV(xyz) =\nleft\n beginarrayll\n 0 0 leq x leq L_x mathrmand 0 leq y leq L_y mathrmand 0 leq z leq L_z \n infty mathrmelsewhere\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.V","text":"V(model::MorsePotential, r)\n\nV(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)\n\nwhere a = sqrtfrack2Dₑ is defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.V","text":"V(model::PoschlTeller, x)\n\nbeginaligned\nV(x)\n= -frachbar^2m x_0^2 fraclambda(lambda+1)2 mathrmsech(x)^2\n= -frachbar^2m x_0^2 fraclambda(lambda+1)2 frac1mathrmcosh(x)^2\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{RigidRotor, Any}","page":"API reference","title":"Antique.V","text":"V(model::RigidRotor, r)\n\nbeginaligned\n V(r)\n = 0\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{SphericalOscillator, Any}","page":"API reference","title":"Antique.V","text":"V(model::SphericalOscillator, r)\n\nV(r)\n= frac12 k r^2\n= frac12 mu omega^2 r^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkmu is the angular frequency and xi = sqrtfracmuomegahbarr.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{CoulombTwoBody, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::CoulombTwoBody, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{HydrogenAtom, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::HydrogenAtom, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{RigidRotor, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::RigidRotor, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{SphericalOscillator, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::SphericalOscillator, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.nₘₐₓ-Tuple{MorsePotential}","page":"API reference","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::MorsePotential)\n\nn_mathrmmax = leftlfloor frac2 D_e - omegaomega rightrfloor\n\nwhere omega = sqrtkµ is defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.nₘₐₓ-Tuple{PoschlTeller}","page":"API reference","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::PoschlTeller)\n\nn_mathrmmax = leftlfloor lambda rightrfloor - 1\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{CoulombTwoBody, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::CoulombTwoBody, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{DeltaPotential, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::DeltaPotential, x)\n\npsi(x) = fracsqrtmalphahbar mathrme^-malpha xhbar^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::HarmonicOscillator, x; n=0)\n\npsi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)\n\nwhere 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 defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{HydrogenAtom, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{InfinitePotentialWell, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell, x; n=1)\n\npsi_n(x) = sqrtfrac2L sin fracnpi xL\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{InfinitePotentialWell3D, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell3D, x,y,z; nx=1, ny=1, nz=1)\n\nThe wave functions can be expressed as products of wave functions in a one-dimensional box.\n\npsi_n_xn_yn_z(xyz) = psi_n_x(x)psi_n_y(y)psi_n_z(z) = sqrtfrac8L_xL_yL_z sinleft(fracn_xpi xL_xright) sinleft(fracn_ypi yL_yright) sinleft(fracn_zpi zL_zright)\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::MorsePotential, r; n=0)\n\npsi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)\n\nN_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), lambda = fracsqrt2mu D_mathrmeahbar, a = sqrtfrack2Dₑ, L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight), xi = 2lambdamathrme^-a(r-r_e) are defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::PoschlTeller, x; n=0)\n\npsi_n(x) = P_lambda^mu(mathrmtanh(xx_0)) sqrtmufracGamma(lambda-mu+1)Gamma(lambda+mu+1)\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1 and P_lambda^mu are the associated Legendre functions.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{RigidRotor, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::RigidRotor, θ, φ; l=0, m=0)\n\npsi_lm(thetavarphi) = Y_lm(thetavarphi)\n\nThe wave function is the spherical harmonics. The domain is 0leq theta lt pi and 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{SphericalOscillator, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::SphericalOscillator, r, θ, φ; n=0, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"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":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH psi(r) = E psi(r)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"and the Hamiltonian","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters are specified with the following struct.","category":"page"},{"location":"MorsePotential/#Parameters","page":"Morse Potential","title":"Parameters","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.MorsePotential","category":"page"},{"location":"MorsePotential/#Antique.MorsePotential-MorsePotential","page":"Morse Potential","title":"Antique.MorsePotential","text":"MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)\n\nr_mathrme is the equilibrium bond distance, D_mathrme is the the well depth , k is the force constant, mu is the reduced mass and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"MorsePotential/#Potential","page":"Morse Potential","title":"Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.V(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.V-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.V","text":"V(model::MorsePotential, r)\n\nV(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)\n\nwhere a = sqrtfrack2Dₑ is defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Eigen-Values","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.E(::MorsePotential)","category":"page"},{"location":"MorsePotential/#Antique.E-Tuple{MorsePotential}-MorsePotential","page":"Morse Potential","title":"Antique.E","text":"E(model::MorsePotential; n=0)\n\nE_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2\n\nwhere omega = sqrtkµ and chi = frachbaromega4D_mathrme are defined.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Number-of-Bound-States","page":"Morse Potential","title":"Number of Bound States","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.nₘₐₓ(::MorsePotential)","category":"page"},{"location":"MorsePotential/#Antique.nₘₐₓ-Tuple{MorsePotential}-MorsePotential","page":"Morse Potential","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::MorsePotential)\n\nn_mathrmmax = leftlfloor frac2 D_e - omegaomega rightrfloor\n\nwhere omega = sqrtkµ is defined.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Eigen-Functions","page":"Morse Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.ψ(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.ψ-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.ψ","text":"ψ(model::MorsePotential, r; n=0)\n\npsi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)\n\nN_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), lambda = fracsqrt2mu D_mathrmeahbar, a = sqrtfrack2Dₑ, L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight), xi = 2lambdamathrme^-a(r-r_e) are defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials","page":"Morse Potential","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"note: Note\nThe generalized Laguerre polynomials L_n^(alpha)(x), not the associated Laguerre polynomials L_n^k(x), are used in this model. ","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.L(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.L-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.L","text":"L(model::MorsePotential, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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 \nendaligned\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated 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":"# 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\nDₑ = - 0.5 - (-0.602634619106539878727562156289)\nk = 2*((-1.1026342144949464615+1/2.00) - (-0.602634619106539878727562156289)) / (2.00 - rₑ)^2\nµ = 1/(1/1836.15267343 + 1/1836.15267343)\nℏ = 1.0\n\nusing Antique\nMP = MorsePotential(rₑ=rₑ, Dₑ=Dₑ, k=k, µ=µ, ℏ=ℏ)\n; #hide","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"MP.rₑ\nMP.Dₑ\nMP.k\nMP.µ\nMP.ℏ","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Number of bound states:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"nₘₐₓ(MP)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Eigen values:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"E(MP, n=0)\nE(MP, n=1)","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 CairoMakie\n\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r$\", ylabel=L\"$V(r)$\", limits=(0.0,20.0,-0.11,0.1))\nlines!(ax, 0.1:0.01:20, r -> V(MP, r))\nf","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 CairoMakie\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r$\", ylabel=L\"$\\psi(r)$\")\n\n# plot\nw0 = lines!(ax, 0..5, x -> ψ(MP, x, n=0))\nw1 = lines!(ax, 0..5, x -> ψ(MP, x, n=1))\nw2 = lines!(ax, 0..5, x -> ψ(MP, x, n=2))\nw3 = lines!(ax, 0..5, x -> ψ(MP, x, n=3))\nw4 = lines!(ax, 0..5, x -> ψ(MP, x, n=4))\n\n# legend\naxislegend(ax, [w0, w1, w2, w3, w4], [L\"n=0\", L\"n=1\", L\"n=2\", L\"n=3\", L\"n=4\"], position=:lb)\n\nf","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":"using Antique\n\n# https://physics.nist.gov/cgi-bin/cuu/Value?mmusme\nm = 206.7682830\nμ = 1 / (1/m + 1/m)\nMP = MorsePotential(μ=μ)\n# @show nₘₐₓ(MP)\n\nusing CairoMakie\n\n# settings\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r$\", ylabel=L\"$V(r),~E_n,~\\psi_n(r) \\times 5 + E_n$\", aspect=1, limits=(0.5,10,-0.105,0.007))\n# hidespines!(ax)\n# hidedecorations!(ax)\n\nprintln(nₘₐₓ(MP))\n\nfor n in 0:nₘₐₓ(MP)\n # energy\n EMP = E(MP, n=n)\n lines!(ax, 0.1:0.01:15, r -> EMP > V(MP, r) ? EMP : NaN, color=:black, linewidth=2)\n hlines!(ax, E(MP, n=n), color=:black, linewidth=1, linestyle=:dash)\n # wave function\n lines!(ax, 0..10, x -> E(MP,n=n) + 0.0065*ψ(MP,x,n=n), linewidth=2)\nend\n\n#potential\nlines!(ax, 0..10, x -> V(MP, x), color=:black, linewidth=2)\n\nf\nsave(\"assets/fig/MorsePotential.png\", f) # hide\n; # hide","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/","page":"Morse Potential","title":"Morse Potential","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/MorsePotential.log\"))","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 Coulomb 2-body system.","category":"page"},{"location":"HydrogenAtom/#Definitions","page":"Hydrogen Atom","title":"Definitions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"and the Hamiltonian","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH = - frachbar^22mu nabla^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. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.","category":"page"},{"location":"HydrogenAtom/#Parameters","page":"Hydrogen Atom","title":"Parameters","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.HydrogenAtom","category":"page"},{"location":"HydrogenAtom/#Antique.HydrogenAtom-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.HydrogenAtom","text":"HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nZ is the atomic number, m_mathrme is the electron mass, a_0is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"HydrogenAtom/#Potential","page":"Hydrogen Atom","title":"Potential","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.V(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.V-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.V","text":"V(model::HydrogenAtom, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Eigen-Values","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.E(::HydrogenAtom)","category":"page"},{"location":"HydrogenAtom/#Antique.E-Tuple{HydrogenAtom}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.E","text":"E(model::HydrogenAtom; n=1)\n\nE_n\n= -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2\n= -fracZ^22n^2 E_mathrmh\n\nwhere 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.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Eigen-Functions","page":"Hydrogen Atom","title":"Eigen Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.ψ(::HydrogenAtom, ::Any, ::Any, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.ψ-Tuple{HydrogenAtom, Any, Any, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.ψ","text":"ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Radial-Functions","page":"Hydrogen Atom","title":"Radial Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.R(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.R-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.R","text":"R(model::HydrogenAtom, r; n=1, l=0)\n\nR_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)\n\nwhere 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). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.L(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.L-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.L","text":"L(model::HydrogenAtom, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_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)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Spherical-Harmonics","page":"Hydrogen Atom","title":"Spherical Harmonics","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.Y(::HydrogenAtom, ::Any, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.Y-Tuple{HydrogenAtom, Any, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.Y","text":"Y(model::HydrogenAtom, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials","page":"Hydrogen Atom","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.P(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.P-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.P","text":"P(model::HydrogenAtom, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated 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 = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Parameters:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"H.Z\nH.Eₕ\nH.mₑ\nH.a₀\nH.ℏ","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eigen values:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"E(H, n=1)\nE(H, n=2)","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 = \", E(H,n=2) - E(H,n=1), \" Eₕ\")\nprintln(\"λ = \", ((E(H,n=2)-E(H,n=1))*Eₕ2nm⁻¹)^-1, \" 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":"# E. Tiesinga, et al., Rev. Mod. Phys. 93, 025010 (2021) 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# D. J. Griffiths, Am. J. Phys. 50, 698 (1982) https://doi.org/10.1119/1.12733\nδ = abs(ψ(H,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":"Potential energy curve:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using CairoMakie\n\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r~/~a_0$\", ylabel=L\"$V(r)~/~E_\\mathrm{h}$\", limits=(0.0,15.0,-2.0,0.2))\nlines!(ax, 0.1:0.01:20, r -> V(H, r))\nf","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 CairoMakie\nusing LaTeXStrings\n\n# setting\nf = Figure()\nax = Axis(f[1,1], xlabel=L\"$r~/~a_0$\", ylabel=L\"$r^2|R_{nl}(r)|^2~/~a_0^{-1}$\", limits=(0,20,0,0.58))\n\n# plot\nws = []\nls = []\nfor n in 1:3\n for l in 0:n-1\n w = lines!(\n ax,\n 0..20,\n r -> r^2 * R(H,r,n=n,l=l)^2,\n linewidth = 2,\n linestyle = [:solid,:dash,:dot,:dashdot,:dashdotdot][l+1],\n color = n,\n colormap = :tab10,\n colorrange = (1,10)\n )\n push!(ws, w)\n push!(ls, latexstring(\"n=$n, l=$l\"))\n end\nend\n\n# legend\naxislegend(ax, ws, ls, position=:rt)\n\nf","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Wave functions (electron density in n=5l=2m=1):","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Antique\nH = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)\nloop(x) = x<-1 ? loop(x+2) : (1 P(0,y,z), colorrange=(0.0,0.00001))\nf\nsave(\"assets/fig/HydrogenAtom.png\", f) # hide\n; # hide","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/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/HydrogenAtom.log\"))","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"CurrentModule = Antique","category":"page"},{"location":"InfinitePotentialWell3D/#Infinite-Potential-Well-3D-(Particle-in-a-3D-Box)","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D (Particle in a 3D-Box)","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"The three-dimensional infinite potential well (particle in a 3D-box) is one of the simplest models for a three-dimensional quantum mechanical system.","category":"page"},{"location":"InfinitePotentialWell3D/#Definitions","page":"Infinite Potential Well 3D","title":"Definitions","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":" hatH psi(xyz) = E psi(xyz)","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"and the Hamiltonian","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":" hatH = - frachbar^22m left(fracpartial^2partial x ^2 + fracpartial^2partial y ^2 + fracpartial^2partial z ^2right) + V(xyz)","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Parameters are specified with the following struct.","category":"page"},{"location":"InfinitePotentialWell3D/#Parameters","page":"Infinite Potential Well 3D","title":"Parameters","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Antique.InfinitePotentialWell3D","category":"page"},{"location":"InfinitePotentialWell3D/#Antique.InfinitePotentialWell3D-InfinitePotentialWell3D","page":"Infinite Potential Well 3D","title":"Antique.InfinitePotentialWell3D","text":"InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)\n\nL_xL_yL_z are the lengths of the box in x,y,z-direction, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"InfinitePotentialWell3D/#Potential","page":"Infinite Potential Well 3D","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Antique.V(::InfinitePotentialWell3D, ::Any, ::Any, ::Any)","category":"page"},{"location":"InfinitePotentialWell3D/#Antique.V-Tuple{InfinitePotentialWell3D, Any, Any, Any}-InfinitePotentialWell3D","page":"Infinite Potential Well 3D","title":"Antique.V","text":"V(model::InfinitePotentialWell3D, x,y,z)\n\nV(xyz) =\nleft\n beginarrayll\n 0 0 leq x leq L_x mathrmand 0 leq y leq L_y mathrmand 0 leq z leq L_z \n infty mathrmelsewhere\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell3D/#Eigen-Values","page":"Infinite Potential Well 3D","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Antique.E(::InfinitePotentialWell3D)","category":"page"},{"location":"InfinitePotentialWell3D/#Antique.E-Tuple{InfinitePotentialWell3D}-InfinitePotentialWell3D","page":"Infinite Potential Well 3D","title":"Antique.E","text":"E(model::InfinitePotentialWell3D; nx=1, ny=1, nz=1)\n\nE_n_xn_yn_z = frachbar^2 pi^22 m left(fracn_x^2L_x^2 + fracn_y^2L_y^2 + fracn_z^2L_z^2right)\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell3D/#Eigen-Functions","page":"Infinite Potential Well 3D","title":"Eigen Functions","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Antique.ψ(::InfinitePotentialWell3D, ::Any, ::Any, ::Any)","category":"page"},{"location":"InfinitePotentialWell3D/#Antique.ψ-Tuple{InfinitePotentialWell3D, Any, Any, Any}-InfinitePotentialWell3D","page":"Infinite Potential Well 3D","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell3D, x,y,z; nx=1, ny=1, nz=1)\n\nThe wave functions can be expressed as products of wave functions in a one-dimensional box.\n\npsi_n_xn_yn_z(xyz) = psi_n_x(x)psi_n_y(y)psi_n_z(z) = sqrtfrac8L_xL_yL_z sinleft(fracn_xpi xL_xright) sinleft(fracn_ypi yL_yright) sinleft(fracn_zpi zL_zright)\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell3D/#Usage-and-Examples","page":"Infinite Potential Well 3D","title":"Usage & Examples","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are supported. In this system, the model is generated by InfinitePotentialWell3D and several parameters Lx, Ly, Lz, m and ℏ are set as optional arguments.","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"using Antique\nIPW3D = InfinitePotentialWell3D(Lx=1.0, Ly=1.0, Lz=1.0, m=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Parameters:","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"IPW3D.Lx\nIPW3D.Ly\nIPW3D.Lz\nIPW3D.m\nIPW3D.ℏ","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Eigen values:","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"E(IPW3D, nx=1, ny=1, nz=1)\nE(IPW3D, nx=2, ny=1, nz=1)\nE(IPW3D, nx=1, ny=2, nz=1)\nE(IPW3D, nx=1, ny=1, nz=2)\nE(IPW3D, nx=2, ny=2, nz=1)\nE(IPW3D, nx=2, ny=1, nz=2)\nE(IPW3D, nx=1, ny=2, nz=2)\nE(IPW3D, nx=2, ny=2, nz=2)","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Wave functions:","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"The wave functions of the 3D infinite potential well can be expressed as a product of wave functions in 1D infinite potential wells for each of the three directions x,y,z. As the representation of 3D (non-spherical) wave functions easily becomes unclear, we refer to the documentation of the one-dimensional counterpart: InfinitePotentialWell (...)","category":"page"},{"location":"InfinitePotentialWell3D/#Testing","page":"Infinite Potential Well 3D","title":"Testing","text":"","category":"section"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"InfinitePotentialWell3D/","page":"Infinite Potential Well 3D","title":"Infinite Potential Well 3D","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/InfinitePotentialWell3D.log\"))","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"CurrentModule = Antique","category":"page"},{"location":"RigidRotor/#Rigid-rotor","page":"Rigid Rotor","title":"Rigid rotor","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"The linear rigid rotor model can be used in quantum mechanics to predict the rotational energy of a diatomic molecule. The rotational energy depends on the moment of inertia for the system, I.","category":"page"},{"location":"RigidRotor/#Definitions","page":"Rigid Rotor","title":"Definitions","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"and the Hamiltonian","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"beginaligned\n hatH = - frachbar^22mu nabla^2 + V(r) \n = - frachbar^22I left frac1sintheta fracpartialpartialtheta left(sintheta fracpartialpartialthetaright) + frac1sin^2theta fracpartial^2partialphi^2 right \n = fracL^22I\nendaligned","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"where I=mu R^2 is the moment of intertia, mu=left(frac1m_1+frac1m_2right)^-1 is the reduced mass of two particles, R is the distance between the two particles, and L^2 is the angular momentum operator. Parameters are specified with the following struct.","category":"page"},{"location":"RigidRotor/#Parameters","page":"Rigid Rotor","title":"Parameters","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.RigidRotor","category":"page"},{"location":"RigidRotor/#Antique.RigidRotor-RigidRotor","page":"Rigid Rotor","title":"Antique.RigidRotor","text":"RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)\n\nm₁ and m₂ are mass of two particles, R is the distance, and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"RigidRotor/#Potential","page":"Rigid Rotor","title":"Potential","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.V(::RigidRotor, ::Any)","category":"page"},{"location":"RigidRotor/#Antique.V-Tuple{RigidRotor, Any}-RigidRotor","page":"Rigid Rotor","title":"Antique.V","text":"V(model::RigidRotor, r)\n\nbeginaligned\n V(r)\n = 0\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#Eigen-Values","page":"Rigid Rotor","title":"Eigen Values","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.E(::RigidRotor)","category":"page"},{"location":"RigidRotor/#Antique.E-Tuple{RigidRotor}-RigidRotor","page":"Rigid Rotor","title":"Antique.E","text":"E(model::RigidRotor; l=0)\n\nE_l\n= frachbar^22Il(l+1)\n\nwhere I=mu R^2 is the moment of inertia, R is the distance, and mu is the reduced mass of the two particles.\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#Eigen-Functions","page":"Rigid Rotor","title":"Eigen Functions","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.ψ(::RigidRotor, ::Any, ::Any)","category":"page"},{"location":"RigidRotor/#Antique.ψ-Tuple{RigidRotor, Any, Any}-RigidRotor","page":"Rigid Rotor","title":"Antique.ψ","text":"ψ(model::RigidRotor, θ, φ; l=0, m=0)\n\npsi_lm(thetavarphi) = Y_lm(thetavarphi)\n\nThe wave function is the spherical harmonics. The domain is 0leq theta lt pi and 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#Spherical-Harmonics","page":"Rigid Rotor","title":"Spherical Harmonics","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.Y(::RigidRotor, ::Any, ::Any)","category":"page"},{"location":"RigidRotor/#Antique.Y-Tuple{RigidRotor, Any, Any}-RigidRotor","page":"Rigid Rotor","title":"Antique.Y","text":"Y(model::RigidRotor, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^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\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#Associated-Legendre-Polynomials","page":"Rigid Rotor","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Antique.P(::RigidRotor, ::Any)","category":"page"},{"location":"RigidRotor/#Antique.P-Tuple{RigidRotor, Any}-RigidRotor","page":"Rigid Rotor","title":"Antique.P","text":"P(model::RigidRotor, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\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)\nendaligned\n\nwhere 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.\n\nExamples:\n\nbeginaligned\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\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"RigidRotor/#References","page":"Rigid Rotor","title":"References","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Anderson, J.M. Introduction to Quantum Chemistry, 1969, W.A. Benjamin, Inc, p.91-100\nChemistry Libre Texts","category":"page"},{"location":"RigidRotor/#Usage-and-Examples","page":"Rigid Rotor","title":"Usage & Examples","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Install Antique.jl 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 RigidRotor and several parameters m₁, m₂, R and ℏ are set as optional arguments.","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"using Antique\nRR = RigidRotor(m₁=1.0, m₂=1.0, R=1.0, ℏ=1.0)\n; #hide","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Parameters:","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"RR.m₁\nRR.m₂\nRR.R\nRR.ℏ","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Eigen values:","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"E(RR, l=0)\nE(RR, l=1)\nE(RR, l=2)","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"Wave functions:","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"ψ(RR, 0, 0, l=2, m=1)\nψ(RR, π/4, 0, l=2, m=1)\nψ(RR, π/4, π/2, l=2, m=1)","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"using CairoMakie\n\nf = Figure(size=(400,400))\nax = PolarAxis(f[1,1], title=L\"$\\theta\\mapsto|\\psi_{2,1}(\\theta,0)|^2$\", rticklabelsvisible=false)\nlines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=2,m=1))^2, linewidth=2)\n\nf","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"using CairoMakie\n\nf = Figure(size=(400,400))\nax = PolarAxis(f[1,1], title=L\"$\\theta\\mapsto|\\psi_{lm}(\\theta,0)|^2$\", rticklabelsvisible=false)\nl1 = lines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=0,m=0))^2, linewidth=2)\nl2 = lines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=1,m=0))^2, linewidth=2)\nl3 = lines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=1,m=1))^2, linewidth=2)\nl4 = lines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=2,m=1))^2, linewidth=2)\nLegend(f[2,1], [l1,l2,l3,l4], [L\"(l,m)=(0,0)\",L\"(1,0)\",L\"(1,1)\",L\"(2,1)\"], framevisible=false, orientation=:horizontal, tellwidth=false, tellheight=true)\n\nf\nsave(\"assets/fig/RigidRotor.png\", f) # hide\n; # hide","category":"page"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"(Image: )","category":"page"},{"location":"RigidRotor/#Testing","page":"Rigid Rotor","title":"Testing","text":"","category":"section"},{"location":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","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":"RigidRotor/","page":"Rigid Rotor","title":"Rigid Rotor","text":"using Markdown\nusing Antique\nMarkdown.parse(Antique.load(\"../../test/result/RigidRotor.log\"))","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 energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. 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 = HydrogenAtom(Z=1)\nE(H)\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":"He⁺ = HydrogenAtom(Z=2)\nE(He⁺)\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 \"PoschlTeller\"/\n \n PoschlTeller\n
\n
\n \n \"MorsePotential\"/\n \n MorsePotential\n
\n
","category":"page"},{"location":"","page":"Home","title":"Home","text":"Delta Potential DeltaPotential\nInfinite Potential Well InfinitePotentialWell\nHarmonic Oscillator HarmonicOscillator\nPoschlTeller PoschlTeller\nMorse Potential MorsePotential\nRigid Rotor RigidRotor\nInfinite PotentialWell 3D InfinitePotentialWell3D\nSpherical Oscillator SphericalOscillator\nHydrogen Atom HydrogenAtom\nCoulomb 2-Body System CoulombTwoBody","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":"#Developer's-Guide","page":"Home","title":"Developer's Guide","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Here is the guideline for adding new models.","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"}] }