From 0b0f056b329b028121edceacff107568a82f3356 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Wed, 11 Dec 2024 05:24:07 +0000 Subject: [PATCH] build based on edae822 --- dev/.documenter-siteinfo.json | 2 +- dev/API/index.html | 54 +++++++++--------- dev/CoulombTwoBody/a135f3ad.png | Bin 0 -> 42412 bytes dev/CoulombTwoBody/index.html | 40 ++++++++++--- dev/DeltaPotential/index.html | 4 +- dev/HarmonicOscillator/1ae58db3.png | Bin 128474 -> 0 bytes dev/HarmonicOscillator/aa28cafa.png | Bin 0 -> 128473 bytes dev/HarmonicOscillator/index.html | 10 ++-- dev/HydrogenAtom/2d213823.png | Bin 0 -> 30495 bytes dev/HydrogenAtom/371d120b.png | Bin 30504 -> 0 bytes dev/HydrogenAtom/5aa5fca7.png | Bin 0 -> 38087 bytes dev/HydrogenAtom/5e88a96a.png | Bin 91796 -> 0 bytes dev/HydrogenAtom/ba252e1f.png | Bin 0 -> 91927 bytes dev/HydrogenAtom/index.html | 42 +++++++++++--- dev/InfinitePotentialWell/9f36930f.png | Bin 144053 -> 0 bytes dev/InfinitePotentialWell/d8c30e0d.png | Bin 0 -> 144039 bytes dev/InfinitePotentialWell/index.html | 8 +-- dev/InfinitePotentialWell3D/2d3bc1c6.png | Bin 0 -> 119981 bytes dev/InfinitePotentialWell3D/b24a7c6c.png | Bin 119989 -> 0 bytes dev/InfinitePotentialWell3D/index.html | 12 ++-- .../{290a5eb4.png => 015776ce.png} | Bin 112148 -> 112148 bytes dev/MorsePotential/6e6d3fed.png | Bin 31561 -> 0 bytes dev/MorsePotential/9460b865.png | Bin 0 -> 31565 bytes dev/MorsePotential/index.html | 10 ++-- dev/PoschlTeller/index.html | 8 +-- dev/RigidRotor/01abb4db.png | Bin 91462 -> 0 bytes dev/RigidRotor/280ebe85.png | Bin 0 -> 91461 bytes dev/RigidRotor/index.html | 12 ++-- dev/SphericalOscillator/index.html | 12 ++-- dev/assets/fig/HarmonicOscillator.png | Bin 78529 -> 78528 bytes dev/assets/fig/InfinitePotentialWell.png | Bin 57989 -> 57633 bytes dev/assets/fig/MorsePotential.png | Bin 77372 -> 77372 bytes dev/assets/fig/PoschlTeller.png | Bin 67303 -> 67302 bytes dev/assets/fig/RigidRotor.png | Bin 96396 -> 96390 bytes dev/assets/fig/demonstration.png | Bin 35327 -> 35340 bytes dev/index.html | 2 +- dev/search_index.js | 2 +- 37 files changed, 135 insertions(+), 83 deletions(-) create mode 100644 dev/CoulombTwoBody/a135f3ad.png delete mode 100644 dev/HarmonicOscillator/1ae58db3.png create mode 100644 dev/HarmonicOscillator/aa28cafa.png create mode 100644 dev/HydrogenAtom/2d213823.png delete mode 100644 dev/HydrogenAtom/371d120b.png create mode 100644 dev/HydrogenAtom/5aa5fca7.png delete mode 100644 dev/HydrogenAtom/5e88a96a.png create mode 100644 dev/HydrogenAtom/ba252e1f.png delete mode 100644 dev/InfinitePotentialWell/9f36930f.png create mode 100644 dev/InfinitePotentialWell/d8c30e0d.png create mode 100644 dev/InfinitePotentialWell3D/2d3bc1c6.png delete mode 100644 dev/InfinitePotentialWell3D/b24a7c6c.png rename dev/MorsePotential/{290a5eb4.png => 015776ce.png} (88%) delete mode 100644 dev/MorsePotential/6e6d3fed.png create mode 100644 dev/MorsePotential/9460b865.png delete mode 100644 dev/RigidRotor/01abb4db.png create mode 100644 dev/RigidRotor/280ebe85.png diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 712fe12..18efa45 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-29T11:03:04","documenter_version":"1.8.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-11T05:24:01","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/dev/API/index.html b/dev/API/index.html index 2142c89..b62566a 100644 --- a/dev/API/index.html +++ b/dev/API/index.html @@ -1,19 +1,19 @@ -API reference · Antique.jl

API reference

Antique.CoulombTwoBodyType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{z_1 z_2}{r/a_0} E_\mathrm{h},\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2. 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:

CTB = 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, $z₂$ 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 (use the same unit as $m₁$ and $m₂$. For example of hydrogen atom, use $m_\mathrm{e}=9.1093837139\times10^{-31}\mathrm{kg}$, $m_1=9.1093837139\times10^{-31}\mathrm{kg}$ and $m_2=1.67262192595\times10^{-27}\mathrm{kg}$ in the IS unit system, use $~m_\mathrm{e}=1.0~m_\mathrm{e}$, $m_1=1.0~m_\mathrm{e}$ and $m_2=1836.152673426~m_\mathrm{e}$ in the atomic unit.), $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

References

source
Antique.DeltaPotentialType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

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

Parameters are specified with the following struct:

DP = 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).

References

source
Antique.HarmonicOscillatorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + \frac{1}{2} k x^2.\]

Parameters are specified with the following struct:

HO = 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).

References

Main:

Supplemental:

  • The Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18
  • L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965) p.595 (a.4), (a.6)
  • L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968) p.71 (13.12)
  • A. Messiah, Quanfum Mechanics (Dover Publications, 1999) p.491 (B.59)
  • W. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994) p.152 (7.22)
  • D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995) p.41 Table 2.1, p.43 (2.70)
  • D. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2
  • P. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008) p.293 Table 9.1
  • J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)
source
Antique.HydrogenAtomType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 - \frac{Z}{r/a_0} E_\mathrm{h},\]

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:

HA = 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).

References

Main:

  • The Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.5.17
  • cpprefjp, assoc_legendre, assoc_laguerre
  • A. Messiah, Quanfum Mechanics VOLUME Ⅰ (North-Holland Publishing Company, 1961), p.412 (XI.3), p.419 (XI.18) (XI.18a) (XI.18b), p.483 (B.12), p.493 (B.71) (B.72), p.494 (B.81), p495 (B.93)

Supplemental:

source
Antique.InfinitePotentialWellType

Model

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:

IPW = 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).

References

Proofs

source
Antique.InfinitePotentialWell3DType

Model

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:

IPW3D = InfinitePotentialWell3D(L=[1.0,1.0,1.0], m=1.0, ℏ=1.0)

$L$ is a vector of 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).

References

source
Antique.MorsePotentialType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. Parameters are specified with the following struct:

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).

References

source
Antique.PoschlTellerType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = +API reference · Antique.jl

API reference

Antique.CoulombTwoBodyType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{z_1 z_2}{r/a_0} E_\mathrm{h},\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2. 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:

CTB = 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, $z₂$ 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 (use the same unit as $m₁$ and $m₂$. For example of hydrogen atom, use $m_\mathrm{e}=9.1093837139\times10^{-31}\mathrm{kg}$, $m_1=9.1093837139\times10^{-31}\mathrm{kg}$ and $m_2=1.67262192595\times10^{-27}\mathrm{kg}$ in the IS unit system, use $~m_\mathrm{e}=1.0~m_\mathrm{e}$, $m_1=1.0~m_\mathrm{e}$ and $m_2=1836.152673426~m_\mathrm{e}$ in the atomic unit.), $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

References

source
Antique.DeltaPotentialType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

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

Parameters are specified with the following struct:

DP = 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).

References

source
Antique.HarmonicOscillatorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + \frac{1}{2} k x^2.\]

Parameters are specified with the following struct:

HO = 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).

References

Main:

Supplemental:

  • The Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18
  • L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965) p.595 (a.4), (a.6)
  • L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968) p.71 (13.12)
  • A. Messiah, Quanfum Mechanics (Dover Publications, 1999) p.491 (B.59)
  • W. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994) p.152 (7.22)
  • D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995) p.41 Table 2.1, p.43 (2.70)
  • D. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2
  • P. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008) p.293 Table 9.1
  • J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)
source
Antique.HydrogenAtomType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 - \frac{Z}{r/a_0} E_\mathrm{h},\]

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:

HA = 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).

References

Main:

  • The Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.5.17
  • cpprefjp, assoc_legendre, assoc_laguerre
  • A. Messiah, Quanfum Mechanics VOLUME Ⅰ (North-Holland Publishing Company, 1961), p.412 (XI.3), p.419 (XI.18) (XI.18a) (XI.18b), p.483 (B.12), p.493 (B.71) (B.72), p.494 (B.81), p495 (B.93)

Supplemental:

source
Antique.InfinitePotentialWellType

Model

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:

IPW = 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).

References

Proofs

source
Antique.InfinitePotentialWell3DType

Model

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:

IPW3D = InfinitePotentialWell3D(L=[1.0,1.0,1.0], m=1.0, ℏ=1.0)

$L$ is a vector of 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).

References

source
Antique.MorsePotentialType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. Parameters are specified with the following struct:

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).

References

source
Antique.PoschlTellerType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

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

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

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

Parameters are specified within the following struct:

PT = PoschlTeller(λ=1, m=1.0, ℏ=1.0, x₀=1.0)

$\lambda$ determines the potential strength. Currently only integer values for $\lambda$ are supported.

References

source
Antique.RigidRotorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[\begin{aligned} + - \frac{\hbar^2}{m x_0^2} \frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}^2(x/x_0)}.\]

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

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

Parameters are specified within the following struct:

PT = PoschlTeller(λ=1, m=1.0, ℏ=1.0, x₀=1.0)

$\lambda$ determines the potential strength. Currently only integer values for $\lambda$ are supported.

References

source
Antique.RigidRotorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

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

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:

RR = 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).

References

source
Antique.SphericalOscillatorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{1}{2} k r^2.\]

Parameters are specified with the following struct:

SO = SphericalOscillator(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).

References

source
Antique.EMethod

E(model::CoulombTwoBody; n::Int=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, $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ 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.EMethod

E(model::DeltaPotential)

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

source
Antique.EMethod

E(model::HarmonicOscillator; n::Int=0)

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

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

source
Antique.EMethod

E(model::HydrogenAtom; n::Int=1)

\[E_n +\end{aligned}\]

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:

RR = 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).

References

source
Antique.SphericalOscillatorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{1}{2} k r^2.\]

Parameters are specified with the following struct:

SO = SphericalOscillator(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).

References

source
Antique.EMethod

E(model::CoulombTwoBody; n::Int=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, $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ 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.

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Antique.EMethod

E(model::DeltaPotential)

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

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Antique.EMethod

E(model::HarmonicOscillator; n::Int=0)

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

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

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Antique.EMethod

E(model::HydrogenAtom; n::Int=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 $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ 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.

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Antique.EMethod

E(model::InfinitePotentialWell3D; n::Vector{Int64}=[1,1,1])

\[E_{n_x,n_y,n_z} += -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ 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.

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Antique.EMethod

E(model::InfinitePotentialWell3D; n::Vector{Int64}=[1,1,1])

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

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Antique.EMethod

E(model::InfinitePotentialWell; n::Int=1)

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

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Antique.EMethod

E(model::MorsePotential; n::Int=0, nocheck=false)

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

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

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Antique.EMethod

E(model::PoschlTeller; n::Int=0, nocheck=false)

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

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

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Antique.EMethod

E(model::RigidRotor; l::Int=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.

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Antique.EMethod

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

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

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

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Antique.HMethod

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

Rodrigues' formula & closed-form:

\[\begin{aligned} ++ \frac{\hbar^2 n_z^2 \pi^2}{2 m L_z^2}\]

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Antique.EMethod

E(model::InfinitePotentialWell; n::Int=1)

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

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Antique.EMethod

E(model::MorsePotential; n::Int=0, nocheck=false)

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

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

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Antique.EMethod

E(model::PoschlTeller; n::Int=0, nocheck=false)

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

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

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Antique.EMethod

E(model::RigidRotor; l::Int=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.

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Antique.EMethod

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

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

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

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Antique.HMethod

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

Rodrigues' formula & closed-form:

\[\begin{aligned} H_{n}(x) &:= (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2} \\ &= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}. @@ -29,7 +29,7 @@ H_{8}(x) &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}, \\ H_{9}(x) &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}, \\ &\vdots -\end{aligned}\]

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Antique.LMethod

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

Rodrigues' formula & closed-form:

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

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Antique.LMethod

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

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -52,7 +52,7 @@ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots -\end{aligned}\]

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Antique.LMethod

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

Rodrigues' formula & closed-form:

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

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Antique.LMethod

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

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -75,7 +75,7 @@ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots -\end{aligned}\]

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Antique.LMethod

L(model::MorsePotential, x; n=0, α=0)

Note

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

Rodrigues' formula & closed-form:

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

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Antique.LMethod

L(model::MorsePotential, x; n=0, α=0)

Note

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

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ @@ -97,7 +97,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots -\end{aligned}\]

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Antique.LMethod

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

Note

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

Rodrigues' formula & closed-form:

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

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Antique.LMethod

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

Note

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

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ @@ -119,7 +119,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots -\end{aligned}\]

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Antique.PMethod

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

Rodrigues' formula & closed-form:

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

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Antique.PMethod

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

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -141,7 +141,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

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Antique.PMethod

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

Rodrigues' formula & closed-form:

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

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Antique.PMethod

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

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -163,12 +163,12 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

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Antique.PMethod

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.

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

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Antique.PMethod

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.

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

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Antique.PMethod

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

Rodrigues' formula & closed-form:

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

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Antique.PMethod

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

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -190,7 +190,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

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Antique.PMethod

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

Rodrigues' formula & closed-form:

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

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Antique.PMethod

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

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -212,42 +212,42 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

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Antique.RMethod

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 $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$, $a_\mu = a_0 \frac{m_\mathrm{e}}{\mu}$, $Z = - z_1 z_2$, 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.RMethod

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.RMethod

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.VMethod

V(model::CoulombTwoBody, r)

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

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Antique.RMethod

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 $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$, $a_\mu = a_0 \frac{m_\mathrm{e}}{\mu}$, $Z = - z_1 z_2$, 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.RMethod

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.RMethod

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.VMethod

V(model::CoulombTwoBody, r)

\[\begin{aligned} V(r) &= - \frac{z_1 z_2 e^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{z_1 z_2}{r/a_0} &= - \frac{z_1 z_2}{r/a_0} E_\mathrm{h}, -\end{aligned}\]

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ is the Hartree energy, one of atomic unit. The domain is $0\leq r \lt \infty$.

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Antique.VMethod

V(model::DeltaPotential, x)

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

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Antique.VMethod

V(model::HarmonicOscillator, x)

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

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ is the Hartree energy, one of atomic unit. The domain is $0\leq r \lt \infty$.

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Antique.VMethod

V(model::DeltaPotential, x)

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

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Antique.VMethod

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.VMethod

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.VMethod

V(model::HydrogenAtom, r)

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

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ is the Hartree energy, one of atomic unit. The domain is $0\leq r \lt \infty$.

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Antique.VMethod

V(model::InfinitePotentialWell; x)

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

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ is the Hartree energy, one of atomic unit. The domain is $0\leq r \lt \infty$.

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Antique.VMethod

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.VMethod

V(model::InfinitePotentialWell3D, x...)

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

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Antique.VMethod

V(model::InfinitePotentialWell3D, x...)

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

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Antique.VMethod

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.VMethod

V(model::PoschlTeller, x)

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

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Antique.VMethod

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.VMethod

V(model::PoschlTeller, x)

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

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Antique.VMethod

V(model::RigidRotor, r)

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

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Antique.VMethod

V(model::RigidRotor, r)

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

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Antique.VMethod

V(model::SphericalOscillator, r)

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

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Antique.VMethod

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.YMethod

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.YMethod

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.YMethod

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.YMethod

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.nₘₐₓMethod

nₘₐₓ(model::MorsePotential)

Note

Note that the number of bound states is equal to the maximum quantum number nₘₐₓ, since we count the ground state from n=1 in this model.

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

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

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Antique.nₘₐₓMethod

nₘₐₓ(model::PoschlTeller)

Note

Note that the number of bound states nₘₐₓ + 1 is not equal to the maximum quantum number nₘₐₓ, since we count the ground state from n=0 in this model.

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

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Antique.ψMethod

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

ψ(model::DeltaPotential, x)

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

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Antique.ψMethod

ψ(model::HarmonicOscillator, x; n::Int=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.ψMethod

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

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

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

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Antique.ψMethod

ψ(model::InfinitePotentialWell3D, x...; n::Vector{Int64}=[1,1,1])

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

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

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

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Antique.YMethod

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.YMethod

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.YMethod

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.YMethod

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.nₘₐₓMethod

nₘₐₓ(model::MorsePotential)

Note

Note that the number of bound states is equal to the maximum quantum number nₘₐₓ, since we count the ground state from n=1 in this model.

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

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

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Antique.nₘₐₓMethod

nₘₐₓ(model::PoschlTeller)

Note

Note that the number of bound states nₘₐₓ + 1 is not equal to the maximum quantum number nₘₐₓ, since we count the ground state from n=0 in this model.

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

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Antique.ψMethod

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

ψ(model::DeltaPotential, x)

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

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Antique.ψMethod

ψ(model::HarmonicOscillator, x; n::Int=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.ψMethod

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

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

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

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Antique.ψMethod

ψ(model::InfinitePotentialWell3D, x...; n::Vector{Int64}=[1,1,1])

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

\[\begin{aligned} \psi_{n_x,n_y,n_z}(x,y,z) &= \psi_{n_x}(x) \times \psi_{n_y}(y) @@ -255,4 +255,4 @@ &= \sqrt{\frac{2}{L_x}} \sin \frac{n_x \pi x}{L_x} \times \sqrt{\frac{2}{L_y}} \sin \frac{n_y \pi y}{L_y} \times \sqrt{\frac{2}{L_z}} \sin \frac{n_z \pi z}{L_z} -\end{aligned}\]

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Antique.ψMethod

ψ(model::MorsePotential, r; n::Int=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.ψMethod

ψ(model::PoschlTeller, x; n::Int=0)

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

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.ψMethod

ψ(model::RigidRotor, θ, φ; l::Int=0, m::Int=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.ψMethod

ψ(model::SphericalOscillator, r, θ, φ; n::Int=0, l::Int=0, m::Int=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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+\end{aligned}\]

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Antique.ψMethod

ψ(model::MorsePotential, r; n::Int=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.ψMethod

ψ(model::PoschlTeller, x; n::Int=0)

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

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.ψMethod

ψ(model::RigidRotor, θ, φ; l::Int=0, m::Int=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.ψMethod

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

Antique.CoulombTwoBodyType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{z_1 z_2}{r/a_0} E_\mathrm{h},\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2. 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:

CTB = 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, $z₂$ 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 (use the same unit as $m₁$ and $m₂$. For example of hydrogen atom, use $m_\mathrm{e}=9.1093837139\times10^{-31}\mathrm{kg}$, $m_1=9.1093837139\times10^{-31}\mathrm{kg}$ and $m_2=1.67262192595\times10^{-27}\mathrm{kg}$ in the IS unit system, use $~m_\mathrm{e}=1.0~m_\mathrm{e}$, $m_1=1.0~m_\mathrm{e}$ and $m_2=1836.152673426~m_\mathrm{e}$ in the atomic unit.), $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

References

source

Potential

Antique.VMethod

V(model::CoulombTwoBody, r)

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

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

Antique.CoulombTwoBodyType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{z_1 z_2}{r/a_0} E_\mathrm{h},\]

where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2. 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:

CTB = 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, $z₂$ 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 (use the same unit as $m₁$ and $m₂$. For example of hydrogen atom, use $m_\mathrm{e}=9.1093837139\times10^{-31}\mathrm{kg}$, $m_1=9.1093837139\times10^{-31}\mathrm{kg}$ and $m_2=1.67262192595\times10^{-27}\mathrm{kg}$ in the IS unit system, use $~m_\mathrm{e}=1.0~m_\mathrm{e}$, $m_1=1.0~m_\mathrm{e}$ and $m_2=1836.152673426~m_\mathrm{e}$ in the atomic unit.), $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

References

source

Potential

Antique.VMethod

V(model::CoulombTwoBody, r)

\[\begin{aligned} V(r) &= - \frac{z_1 z_2 e^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{z_1 z_2}{r/a_0} &= - \frac{z_1 z_2}{r/a_0} E_\mathrm{h}, -\end{aligned}\]

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ is the Hartree energy, one of atomic unit. The domain is $0\leq r \lt \infty$.

source

Eigenvalues

Antique.EMethod

E(model::CoulombTwoBody; n::Int=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, $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ 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

Eigenfunctions

Antique.ψMethod

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

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 $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$, $a_\mu = a_0 \frac{m_\mathrm{e}}{\mu}$, $Z = - z_1 z_2$, 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.LMethod

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

Rodrigues' formula & closed-form:

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

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ is the Hartree energy, one of atomic unit. The domain is $0\leq r \lt \infty$.

source

Eigenvalues

Antique.EMethod

E(model::CoulombTwoBody; n::Int=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, $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ 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

Eigenfunctions

Antique.ψMethod

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

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 $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$, $a_\mu = a_0 \frac{m_\mathrm{e}}{\mu}$, $Z = - z_1 z_2$, 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.LMethod

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

Rodrigues' formula & closed-form:

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

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Spherical Harmonics

Antique.YMethod

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.PMethod

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

Rodrigues' formula & closed-form:

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

source

Spherical Harmonics

Antique.YMethod

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.PMethod

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() and potential V() will be exported. In this system, the model is generated by CoulombTwoBody and several parameters z₁, z₂, m₁, m₂, mₑ, a₀, Eₕ and 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() and potential V() will be exported. In this system, the model is generated by CoulombTwoBody and several parameters z₁, z₂, m₁, m₂, mₑ, a₀, Eₕ and 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

Eigenvalues

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
@@ -219,7 +219,33 @@
 ³He⁺	-8666.566269930268 MHz	  Antique.jl + CODATA2018
   	-8665.649867(10)   MHz	  Karshenboim(2001)
 µp	44.16603467817586  THz	  Antique.jl + CODATA2018
-  	44.18270842599667  THz	  Griffiths(1982), Adamczak(2012)

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} + 44.18270842599667 THz Griffiths(1982), Adamczak(2012)

1S wave function of Ps:

import Antique
+Ps = Antique.CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)
+@show Antique.E(Ps)
+
+using CairoMakie
+
+fig = Figure(
+    size = (420,300),
+    fontsize = 11.5,
+    backgroundcolor = :transparent
+)
+
+ax = Axis(
+    fig[1,1],
+    xlabel = L"$r / a_0$",
+    ylabel = L"$\psi(r) / a_0^{-3/2}$",
+    ylabelsize = 16.5,
+    xlabelsize = 16.5,
+)
+
+lines!(ax, 0..10, r -> exp(-r/2)/sqrt(8π), label="exp(-r/2)/sqrt(8π)")
+lines!(ax, 0..2, r -> (1-r/2)/sqrt(8π), label="(1-r/2)/sqrt(8π)")
+lines!(ax, 0..10, r -> abs(Antique.ψ(Ps,r,0,0)), linestyle=:dash, color=:black, label="Antique.jl")
+
+axislegend(ax, position=:rt, framevisible=false)
+
+fig
Example block output

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) \\ &= \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] \\ @@ -1990,4 +2016,4 @@ 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 cba0c1e..367e7a4 100644 --- a/dev/DeltaPotential/index.html +++ b/dev/DeltaPotential/index.html @@ -1,5 +1,5 @@ -Delta Potential · Antique.jl

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

Antique.DeltaPotentialType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

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

Parameters are specified with the following struct:

DP = 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).

References

source

Potential

Antique.VMethod

V(model::DeltaPotential, x)

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

source

Eigenvalues

Antique.EMethod

E(model::DeltaPotential)

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

source

Eigenfunctions

Antique.ψMethod

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() and potential V() will be exported. In this system, the model is generated by DeltaPotential and several parameters α, m and are set as optional arguments.

using Antique
+Delta Potential · Antique.jl

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

Antique.DeltaPotentialType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

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

Parameters are specified with the following struct:

DP = 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).

References

source

Potential

Antique.VMethod

V(model::DeltaPotential, x)

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

source

Eigenvalues

Antique.EMethod

E(model::DeltaPotential)

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

source

Eigenfunctions

Antique.ψMethod

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() and potential V() will be exported. In this system, the model is generated by DeltaPotential and several parameters α, m and 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

Eigenvalues:

julia> E(DP)-0.5

Wave functions:

using CairoMakie
 
 # setting
@@ -88,4 +88,4 @@
 2.0 | 0.5 | 2.0 |   -0.250000000000 |   -0.250000000000 ✔
 2.0 | 2.0 | 0.1 | -400.000000000000 | -400.000000000000 ✔
 2.0 | 2.0 | 0.5 |  -16.000000000000 |  -16.000000000000 ✔
-2.0 | 2.0 | 2.0 |   -1.000000000000 |   -1.000000000000 ✔
+2.0 | 2.0 | 2.0 | -1.000000000000 | -1.000000000000 ✔
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z$1(8{0#zeY003a70l(`T(C$f)%5O8S1ckd2sF{UD(H9yA=7g3{dm=yq@qb}Y7hksG zerXk?72y&&4RJs$7y;QG3gC9w|520fQ$IgHa52fe&ZV|3^U<^l7?_xj%!l)hoSt`A zIEN`<5F+@0WgpC`W~lSWkKS={abQnSeMwG90Y<(3d#)pfUIl+s1KbM=?JN}NAOt2F zO#b$*?r^3sh)E*1q2uG@pe$8cSxL{#O!7`&$cOs}IJeY)nth|;ou8kVCi;6py#GE0gBv3SfmN{nZt=AL&u4!T5NzHFf`9D~Y`VGs i4glW$|8lpT$0wARjon0!^zMJ9AucQ -Hydrogen Atom · Antique.jl

Hydrogen Atom

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

Definitions

Antique.HydrogenAtomType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 - \frac{Z}{r/a_0} E_\mathrm{h},\]

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:

HA = 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).

References

Main:

  • The Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.5.17
  • cpprefjp, assoc_legendre, assoc_laguerre
  • A. Messiah, Quanfum Mechanics VOLUME Ⅰ (North-Holland Publishing Company, 1961), p.412 (XI.3), p.419 (XI.18) (XI.18a) (XI.18b), p.483 (B.12), p.493 (B.71) (B.72), p.494 (B.81), p495 (B.93)

Supplemental:

source

Potential

Antique.VMethod

V(model::HydrogenAtom, r)

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

Hydrogen Atom

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

Definitions

Antique.HydrogenAtomType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 - \frac{Z}{r/a_0} E_\mathrm{h},\]

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:

HA = 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).

References

Main:

  • The Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.5.17
  • cpprefjp, assoc_legendre, assoc_laguerre
  • A. Messiah, Quanfum Mechanics VOLUME Ⅰ (North-Holland Publishing Company, 1961), p.412 (XI.3), p.419 (XI.18) (XI.18a) (XI.18b), p.483 (B.12), p.493 (B.71) (B.72), p.494 (B.81), p495 (B.93)

Supplemental:

source

Potential

Antique.VMethod

V(model::HydrogenAtom, r)

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

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ is the Hartree energy, one of atomic unit. The domain is $0\leq r \lt \infty$.

source

Eigenvalues

Antique.EMethod

E(model::HydrogenAtom; n::Int=1)

\[E_n +\end{aligned}\]

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ is the Hartree energy, one of atomic unit. The domain is $0\leq r \lt \infty$.

source

Eigenvalues

Antique.EMethod

E(model::HydrogenAtom; n::Int=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 $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ 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

Eigenfunctions

Antique.ψMethod

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

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.LMethod

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

Rodrigues' formula & closed-form:

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

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ 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

Eigenfunctions

Antique.ψMethod

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

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.LMethod

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.YMethod

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.PMethod

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

Rodrigues' formula & closed-form:

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

source

Spherical Harmonics

Antique.YMethod

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.PMethod

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

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. In this system, the model is generated by HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and 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() and potential V() will be exported. In this system, the model is generated by HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and 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

Eigenvalues:

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
@@ -84,7 +84,33 @@
 f = Figure()
 ax = 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))
 lines!(ax, 0.1:0.01:20, r -> V(H, r))
-f
Example block output

Radial functions:

using CairoMakie
+f
Example block output

1S wave function:

import Antique
+H = Antique.HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)
+@show Antique.E(H)
+
+using CairoMakie
+
+fig = Figure(
+    size = (420,300),
+    fontsize = 11.5,
+    backgroundcolor = :transparent
+)
+
+ax = Axis(
+    fig[1,1],
+    xlabel = L"$r / a_0$",
+    ylabel = L"$\psi(r) / a_0^{-3/2}$",
+    ylabelsize = 16.5,
+    xlabelsize = 16.5,
+)
+
+lines!(ax, 0..5, r -> exp(-r)/sqrt(π), label="exp(-r)/sqrt(π)")
+lines!(ax, 0..1, r -> (1-r)/sqrt(π), label="(1-r)/sqrt(π)")
+lines!(ax, 0..5, r -> abs(Antique.ψ(H,r,0,0)), linestyle=:dash, color=:black, label="Antique.jl")
+
+axislegend(ax, position=:rt, framevisible=false)
+
+fig
Example block output

Radial functions:

using CairoMakie
 using LaTeXStrings
 
 # setting
@@ -114,7 +140,7 @@
 # legend
 axislegend(ax, ws, ls, position=:rt)
 
-f
Example block output

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} +fExample block output

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) \\ &= \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] \\ @@ -1349,4 +1375,4 @@ 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 ✔

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zjSLJthO52p>^?Wg^0V#Qrxd(I?aWs7gvR?@o9oriEViP)ZBj^+ePCc9fk3dewQWs2 zz%YM)Ym==>kIYNI&CC#1etd$&h-tCe@NwqHjR!cQaKp!8Nc+MnAvAOoq9hh&moHs* z_4I5Ar-xyaic3r5a;e}cgWiM{Bm@{DdA`7br|2Z(Aw28VW6=P{>`68UYf6f*;9LS)K z*Zk*Z5DV~#CAe=vw;q0B6_wB>nAicM^{k#XP3(gc^ iH&n6+%m3^5KN}3Iq(f*t*hvr} -Infinite Potential Well · Antique.jl

Infinite Potential Well (Particle in a Box)

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

Definitions

Antique.InfinitePotentialWellType

Model

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:

IPW = 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).

References

Proofs

source

Potential

Antique.VMethod

V(model::InfinitePotentialWell; x)

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

Infinite Potential Well (Particle in a Box)

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

Definitions

Antique.InfinitePotentialWellType

Model

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:

IPW = 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).

References

Proofs

source

Potential

Antique.VMethod

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

Eigenvalues

Antique.EMethod

E(model::InfinitePotentialWell; n::Int=1)

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

source

Eigenfunctions

Antique.ψMethod

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

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

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. In this system, the model is generated by InfinitePotentialWell and several parameters L, m and are set as optional arguments.

using Antique
+\right.\]

source

Eigenvalues

Antique.EMethod

E(model::InfinitePotentialWell; n::Int=1)

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

source

Eigenfunctions

Antique.ψMethod

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

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

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. In this system, the model is generated by InfinitePotentialWell and several parameters L, m and 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

Eigenvalues:

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

Wave functions:

using CairoMakie
 
 # settings
@@ -22,7 +22,7 @@
 # legend
 axislegend(ax, [w1, w2, w3, w4, w5], [L"n=1", L"n=2", L"n=3", L"n=4", L"n=5"], position=:lb)
 
-f
Example block output

Potential energy curve, Energy levels, Wave functions:

using CairoMakie
+f
Example block output

Potential energy curve, Energy levels, Wave functions:

using CairoMakie
 
 # settings
 f = Figure()
@@ -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/2d3bc1c6.png b/dev/InfinitePotentialWell3D/2d3bc1c6.png new file mode 100644 index 0000000000000000000000000000000000000000..9f745e60bf94420cc4dd364895d15cb7cd32fa2e GIT binary patch literal 119981 zcmeFZ^;267)HQmL;_mLQ#fp0=?!_g=-HU6{;w|o4pg@7*?pEC0y|}x5r|~8>o2}Xbh?-4j@kz-O{`6tm@Neh51Ml~7KepJy{6{d*mzuQP7?Vl)5H6h&b`kQ z9>ydq!tLimu-$J1FeDHtgE1r+9=>g$0sn4+f2w$VlKA@HKU9&hCI0tbj0m+9466Sg zkJYSq2MWGQW8`2s%LXDS zf5rYDX?#TXVUXT~=H;Vm4nbP30oki$;1H!}#@oIil>o3-7%Yzr3*9;l@?GZ88Xk`l zUWLTP3Nr*Xy0eM8$FweCGbmPK-hMje(bjt;6{xo9*qlxk#@jm+-HUkBMUYCNNf-4U z*~GLGO(x`8ua9z0_=({Kbu!p#F<65K8u98c0vwpcY3@>46kj zu|DYBFgT#UgH!IzDV>2+-VU8MFN9N8-FWRbM)3$~_8#9Qt7(R}BWqo;b5++&p4%x< zREnLF6ckEW*eS5w9X@qGO6Yf{OqRs>cY(zTl|1y4EGc(SQkz0fztcYe%Yo$=6->^K z+aq^hsrKcYVAe}?7+SBGYj$bp`hJZ^vt7cdQl+RnX7ac50Rl{m4=nzgffV9_x95NL z2^_BoSy_E=?1P3ZU5IVTs)O#&MNQizSiXKorI3b8ucQJMBb0L<^eLrUKUL)| 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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

Antique.InfinitePotentialWell3DType

Model

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:

IPW3D = InfinitePotentialWell3D(L=[1.0,1.0,1.0], m=1.0, ℏ=1.0)

$L$ is a vector of 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).

References

source

Potential

Antique.VMethod

V(model::InfinitePotentialWell3D, x...)

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

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

Antique.InfinitePotentialWell3DType

Model

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:

IPW3D = InfinitePotentialWell3D(L=[1.0,1.0,1.0], m=1.0, ℏ=1.0)

$L$ is a vector of 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).

References

source

Potential

Antique.VMethod

V(model::InfinitePotentialWell3D, x...)

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

source

Eigenvalues

Antique.EMethod

E(model::InfinitePotentialWell3D; n::Vector{Int64}=[1,1,1])

\[E_{n_x,n_y,n_z} +\right.\]

source

Eigenvalues

Antique.EMethod

E(model::InfinitePotentialWell3D; n::Vector{Int64}=[1,1,1])

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

source

Eigenfunctions

Antique.ψMethod

ψ(model::InfinitePotentialWell3D, x...; n::Vector{Int64}=[1,1,1])

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

\[\begin{aligned} ++ \frac{\hbar^2 n_z^2 \pi^2}{2 m L_z^2}\]

source

Eigenfunctions

Antique.ψMethod

ψ(model::InfinitePotentialWell3D, x...; n::Vector{Int64}=[1,1,1])

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

\[\begin{aligned} \psi_{n_x,n_y,n_z}(x,y,z) &= \psi_{n_x}(x) \times \psi_{n_y}(y) @@ -16,7 +16,7 @@ &= \sqrt{\frac{2}{L_x}} \sin \frac{n_x \pi x}{L_x} \times \sqrt{\frac{2}{L_y}} \sin \frac{n_y \pi y}{L_y} \times \sqrt{\frac{2}{L_z}} \sin \frac{n_z \pi z}{L_z} -\end{aligned}\]

source

Usage & Examples

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

using Antique
 IPW3D = InfinitePotentialWell3D(L=[1.0,1.0,1.0], m=1.0, ℏ=1.0)

Parameters:

julia> IPW3D.L3-element Vector{Float64}:
  1.0
  1.0
@@ -36,7 +36,7 @@
 # legend
 axislegend(ax, [w1, w2, w3, w4, w5], [L"n=[1,1,1]", L"n=[2,1,1]", L"n=[1,2,1]", L"n=[3,1,1]", L"n=[4,1,1]"], position=:lb)
 
-f
Example block output

Testing

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

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

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

IPW3D = InfinitePotentialWell3D([1.0, 1.0, 1.0], 1.0, 1.0)
+f
Example block output

Testing

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

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

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

IPW3D = InfinitePotentialWell3D([1.0, 1.0, 1.0], 1.0, 1.0)
 ix | iy | iz | jx | jy | jz |        analytical |         numerical 
 -- | -- | -- | -- | -- | -- | ----------------- | ----------------- 
  1 |  1 |  1 |  1 |  1 |  1 |    1.000000000000 |    1.000000000000 ✔
@@ -398,4 +398,4 @@
   2 |  1 |  2 |   60.437780565964 |   60.437782548907 ✔
   2 |  2 |  1 |   62.636085293552 |   62.636087361042 ✔
   2 |  2 |  2 |   65.560412402708 |   65.560414590995 ✔
-
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g&$Z`nfoKd z!l0K27b=_?d!Xo;q4p?*V4%D@WvVW_#JPdKPD9EE@Z~57UXdL=HYfJ%558?^gO;Zc zs9FH54VY9oQ{TRM!>o0$zlr?QG{=v#@*3drkTe*1t_=V;IC;bH0cVABzX;u!ZP%8D0KF zuI9ao36rfpJ})ZZ5iw`6|_hHviQtmRrD07E5*gdfus#G9m=MrlK|;nh9g-cqbpS@*#ija z%(`iAwMp!4G=q7y z^5-rJ3Ko9h_>OP|y0{egc2Wij0wRx!S_-g@d$7)Cs?@i>K?lcB77-z#jmLH1ea?5PWb{dI32JngWo}QLupy+VNE?J=srdAkl?(Qf`|8>jwlC z@c%5X`7cIQV{m%~svLgc-T}ql-Yt-FP;baL=qM@WLATCWkInk#=6N8K?{ND4qS$7I%Up~5dcN&fNLjNj@}d$IChBn;%@9W_kSG@cEImxy6S_MAbDG}Xc?ya^r0-hXO>DXYNHi+fcJvNCnEgzgs=NF4yyu=< zCot4c1VXTqstu{9B4%(buW@lTG&c)Iy1=ze8~$}7{J%EPJ2`7;^a8s0KJx)cneRel zJjw8`EZu7;@Dzc~ZZmI0XeR`}MP^5#(SW2U5FzK~<-viRgbay8 uCWOStH;_gQ|KG>Y|3x|bKMek-_Vz~=aSa`({uzs`g{;g2Os3S6=l=uEL*|(P literal 0 HcmV?d00001 diff --git a/dev/MorsePotential/index.html b/dev/MorsePotential/index.html index dcd0f8c..90efcf3 100644 --- a/dev/MorsePotential/index.html +++ b/dev/MorsePotential/index.html @@ -1,5 +1,5 @@ -Morse Potential · Antique.jl

+23 -0.004570 -0.002505 ΔE < 0 ✔ diff --git a/dev/PoschlTeller/index.html b/dev/PoschlTeller/index.html index 164f368..89c703a 100644 --- a/dev/PoschlTeller/index.html +++ b/dev/PoschlTeller/index.html @@ -1,16 +1,16 @@ Pöschl-Teller Potential · Antique.jl

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

Antique.PoschlTellerType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

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

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

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

Parameters are specified within the following struct:

PT = PoschlTeller(λ=1, m=1.0, ℏ=1.0, x₀=1.0)

$\lambda$ determines the potential strength. Currently only integer values for $\lambda$ are supported.

References

source

Potential

Antique.VMethod

V(model::PoschlTeller, x)

\[\begin{aligned} + - \frac{\hbar^2}{m x_0^2} \frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}^2(x/x_0)}.\]

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

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

Parameters are specified within the following struct:

PT = PoschlTeller(λ=1, m=1.0, ℏ=1.0, x₀=1.0)

$\lambda$ determines the potential strength. Currently only integer values for $\lambda$ are supported.

References

source

Potential

Antique.VMethod

V(model::PoschlTeller, x)

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

source

Maximum Quantum Number

Antique.nₘₐₓMethod

nₘₐₓ(model::PoschlTeller)

Note

Note that the number of bound states nₘₐₓ + 1 is not equal to the maximum quantum number nₘₐₓ, since we count the ground state from n=0 in this model.

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

source

Eigenvalues

Antique.EMethod

E(model::PoschlTeller; n::Int=0, nocheck=false)

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

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

source

Eigenfunctions

Antique.ψMethod

ψ(model::PoschlTeller, x; n::Int=0)

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

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.PMethod

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.

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

source

Maximum Quantum Number

Antique.nₘₐₓMethod

nₘₐₓ(model::PoschlTeller)

Note

Note that the number of bound states nₘₐₓ + 1 is not equal to the maximum quantum number nₘₐₓ, since we count the ground state from n=0 in this model.

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

source

Eigenvalues

Antique.EMethod

E(model::PoschlTeller; n::Int=0, nocheck=false)

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

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

source

Eigenfunctions

Antique.ψMethod

ψ(model::PoschlTeller, x; n::Int=0)

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

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.PMethod

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.

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

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 nₘₐₓ() will be exported. In this system, the model is generated by PoschlTeller and the parameters λ, m, , x₀.

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 nₘₐₓ() will be exported. In this system, the model is generated by PoschlTeller and the parameters λ, m, , x₀.

using Antique
 PT = PoschlTeller(λ=4, m=1.0, ℏ=1.0, x₀=1.0)

Parameters:

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

Maximum quantum number:

julia> nₘₐₓ(PT)3

Eigenvalues:

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()
@@ -779,4 +779,4 @@
 12  -2.000000  -2.500000  ΔE < 0  ✔
 13  -4.500000  -3.500000  ΔE < 0  ✔
 14  -8.000000  -4.500000  ΔE < 0  ✔
-
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\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], \\ &= \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:

RR = 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).

References

source

Potential

Antique.VMethod

V(model::RigidRotor, r)

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

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:

RR = 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).

References

source

Potential

Antique.VMethod

V(model::RigidRotor, r)

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

source

Eigenvalues

Antique.EMethod

E(model::RigidRotor; l::Int=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

Eigenfunctions

Antique.ψMethod

ψ(model::RigidRotor, θ, φ; l::Int=0, m::Int=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.YMethod

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.PMethod

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

Rodrigues' formula & closed-form:

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

source

Eigenvalues

Antique.EMethod

E(model::RigidRotor; l::Int=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

Eigenfunctions

Antique.ψMethod

ψ(model::RigidRotor, θ, φ; l::Int=0, m::Int=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.YMethod

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.PMethod

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

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -29,14 +29,14 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. In this system, the model is generated by RigidRotor and several parameters m₁, m₂, R and 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() and potential V() will be exported. In this system, the model is generated by RigidRotor and several parameters m₁, m₂, R and 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

Eigenvalues:

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=(420,300))
 ax = PolarAxis(f[1,1], title=L"$\theta\mapsto|\psi_{2,1}(\theta,0)|^2$", rticklabelsvisible=false, thetalimits=(0,pi))
 lines!(ax, 0..2pi, θ->abs(ψ(RR,θ,0,l=2,m=1))^2, linewidth=2)
 
-f
Example block output
using CairoMakie
+f
Example block output
using CairoMakie
 
 f = Figure(size=(420,300))
 ax = PolarAxis(f[1,1], title=L"$\theta\mapsto|\psi_{lm}(\theta,0)|^2$", rticklabelsvisible=false, thetalimits=(0,pi))
@@ -599,4 +599,4 @@
 2.0 | 2.0 | 0.5 | 3.0 |   24.000000000000 |   23.999999694843 ✔
 2.0 | 2.0 | 2.0 | 1.0 |    0.250000000000 |    0.250000000000 ✔
 2.0 | 2.0 | 2.0 | 2.0 |    0.750000000000 |    0.750000000006 ✔
-2.0 | 2.0 | 2.0 | 3.0 |    1.500000000000 |    1.499999980928 ✔
+2.0 | 2.0 | 2.0 | 3.0 | 1.500000000000 | 1.499999980928 ✔
diff --git a/dev/SphericalOscillator/index.html b/dev/SphericalOscillator/index.html index 7bc46e8..a5d94df 100644 --- a/dev/SphericalOscillator/index.html +++ b/dev/SphericalOscillator/index.html @@ -1,9 +1,9 @@ -Spherical Oscillator · Antique.jl

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

Antique.SphericalOscillatorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{1}{2} k r^2.\]

Parameters are specified with the following struct:

SO = SphericalOscillator(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).

References

source

Potential

Antique.VMethod

V(model::SphericalOscillator, r)

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

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

Antique.SphericalOscillatorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{1}{2} k r^2.\]

Parameters are specified with the following struct:

SO = SphericalOscillator(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).

References

source

Potential

Antique.VMethod

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

Eigenvalues

Antique.EMethod

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

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

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

source

Eigenfunctions

Antique.ψMethod

ψ(model::SphericalOscillator, r, θ, φ; n::Int=0, l::Int=0, m::Int=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.RMethod

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

Antique.LMethod

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

Note

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

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

Eigenvalues

Antique.EMethod

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

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

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

source

Eigenfunctions

Antique.ψMethod

ψ(model::SphericalOscillator, r, θ, φ; n::Int=0, l::Int=0, m::Int=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.RMethod

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

Antique.LMethod

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

Note

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

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.YMethod

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.PMethod

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

Rodrigues' formula & closed-form:

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

source

Spherical Harmonics

Antique.YMethod

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.PMethod

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() and potential V() will be exported. In this system, the model is generated by SphericalOscillator and several parameters k, μ and 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() and potential V() will be exported. In this system, the model is generated by SphericalOscillator and several parameters k, μ and 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

Eigenvalues:

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 ✔

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Morse Potential

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

Definitions

Antique.MorsePotentialType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. Parameters are specified with the following struct:

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).

References

source

Potential

Antique.VMethod

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

Eigenvalues

Antique.EMethod

E(model::MorsePotential; n::Int=0, nocheck=false)

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

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

source

Maximum Quantum Number

Antique.nₘₐₓMethod

nₘₐₓ(model::MorsePotential)

Note

Note that the number of bound states is equal to the maximum quantum number nₘₐₓ, since we count the ground state from n=1 in this model.

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

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

source

Eigenfunctions

Antique.ψMethod

ψ(model::MorsePotential, r; n::Int=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

Antique.LMethod

L(model::MorsePotential, x; n=0, α=0)

Note

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

Rodrigues' formula & closed-form:

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

Morse Potential

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

Definitions

Antique.MorsePotentialType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. Parameters are specified with the following struct:

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).

References

source

Potential

Antique.VMethod

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

Eigenvalues

Antique.EMethod

E(model::MorsePotential; n::Int=0, nocheck=false)

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

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

source

Maximum Quantum Number

Antique.nₘₐₓMethod

nₘₐₓ(model::MorsePotential)

Note

Note that the number of bound states is equal to the maximum quantum number nₘₐₓ, since we count the ground state from n=1 in this model.

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

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

source

Eigenfunctions

Antique.ψMethod

ψ(model::MorsePotential, r; n::Int=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

Antique.LMethod

L(model::MorsePotential, x; n=0, α=0)

Note

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

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

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and nₘₐₓ() will be exported. In this system, the model is generated by MorsePotential and several parameters rₑ, Dₑ, k, µ and are set as optional arguments.

# Parameters for H₂⁺
+\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 nₘₐₓ() will be exported. In this system, the model is generated by MorsePotential and several parameters rₑ, Dₑ, k, µ and 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
@@ -37,7 +37,7 @@
 f = Figure()
 ax = Axis(f[1,1], xlabel=L"$r$", ylabel=L"$V(r)$",  limits=(0.0,20.0,-0.11,0.1))
 lines!(ax, 0.1:0.01:20, r -> V(MP, r))
-f
Example block output

Wave functions:

using CairoMakie
+f
Example block output

Wave functions:

using CairoMakie
 
 # setting
 f = Figure()
@@ -53,7 +53,7 @@
 # legend
 axislegend(ax, [w0, w1, w2, w3, w4], [L"n=0", L"n=1", L"n=2", L"n=3", L"n=4"], position=:lb)
 
-f
Example block output

Potential energy curve, Energy levels, Comparison with harmonic oscillator:

using Antique
+f
Example block output

Potential energy curve, Energy levels, Comparison with harmonic oscillator:

using Antique
 
 # https://physics.nist.gov/cgi-bin/cuu/Value?mmusme
 m = 206.7682830
@@ -881,4 +881,4 @@
 20  -0.000327  -0.000869  ΔE < 0  ✔
 21  -0.001196  -0.001414  ΔE < 0  ✔
 22  -0.002610  -0.001959  ΔE < 0  ✔
-23  -0.004570  -0.002505  ΔE < 0  ✔

Harmonic Oscillator

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

Definitions

Antique.HarmonicOscillatorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + \frac{1}{2} k x^2.\]

Parameters are specified with the following struct:

HO = 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).

References

Main:

Supplemental:

  • The Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18
  • L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965) p.595 (a.4), (a.6)
  • L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968) p.71 (13.12)
  • A. Messiah, Quanfum Mechanics (Dover Publications, 1999) p.491 (B.59)
  • W. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994) p.152 (7.22)
  • D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995) p.41 Table 2.1, p.43 (2.70)
  • D. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2
  • P. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008) p.293 Table 9.1
  • J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)
source

Potential

Antique.VMethod

V(model::HarmonicOscillator, x)

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

Harmonic Oscillator

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

Definitions

Antique.HarmonicOscillatorType

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + \frac{1}{2} k x^2.\]

Parameters are specified with the following struct:

HO = 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).

References

Main:

Supplemental:

  • The Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18
  • L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965) p.595 (a.4), (a.6)
  • L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968) p.71 (13.12)
  • A. Messiah, Quanfum Mechanics (Dover Publications, 1999) p.491 (B.59)
  • W. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994) p.152 (7.22)
  • D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995) p.41 Table 2.1, p.43 (2.70)
  • D. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2
  • P. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008) p.293 Table 9.1
  • J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)
source

Potential

Antique.VMethod

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

Eigenvalues

Antique.EMethod

E(model::HarmonicOscillator; n::Int=0)

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

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

source

Eigenfunctions

Antique.ψMethod

ψ(model::HarmonicOscillator, x; n::Int=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.HMethod

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

Eigenvalues

Antique.EMethod

E(model::HarmonicOscillator; n::Int=0)

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

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

source

Eigenfunctions

Antique.ψMethod

ψ(model::HarmonicOscillator, x; n::Int=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.HMethod

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

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. In this system, the model is generated by HarmonicOscillator and several parameters k, m and 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() and potential V() will be exported. In this system, the model is generated by HarmonicOscillator and several parameters k, m and 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

Eigenvalues:

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

Potential energy curve:

using CairoMakie
 
 f = Figure()
@@ -40,7 +40,7 @@
 # legend
 axislegend(ax, [w0, w1, w2, w3, w4], [L"n=0", L"n=1", L"n=2", L"n=3", L"n=4"], position=:lb)
 
-f
Example block output

Potential energy curve, Energy levels, Wave functions:

using CairoMakie
+f
Example block output

Potential energy curve, Energy levels, Wave functions:

using CairoMakie
 
 # settings
 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 ✔