Move documents into docstrings # 21

docs/src/CoulombTwoBody.md

@@ -8,17 +8,6 @@ This is the model of two particles interacting through Coulomb forces such as po
 
 ## Definitions
 
-This model is described with the time-independent Schrödinger equation
-```math
-  \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),
-```
-and the Hamiltonian
-```math
-  \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r),
-```
-where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.
-
-#### Parameters
 ```@docs; canonical=false
 Antique.CoulombTwoBody
 ```
@@ -58,13 +47,6 @@ Antique.Y(::CoulombTwoBody, ::Any, ::Any)
 Antique.P(::CoulombTwoBody, ::Any)
 ```
 
-#### Reference
-- _The Digital Library of Mathematical Functions_ (DLMF), [18.3 Table1](https://dlmf.nist.gov/18.3#T1), [18.5 Table1](https://dlmf.nist.gov/18.5#T1), [18.5.16](https://dlmf.nist.gov/18.5#E16), [18.5.17](https://dlmf.nist.gov/18.5#E17)
-- _cpprefjp_, [assoc_legendre](https://cpprefjp.github.io/reference/cmath/assoc_legendre.html), [assoc_laguerre](https://cpprefjp.github.io/reference/cmath/assoc_laguerre.html)
-- A. Messiah, _Quanfum Mechanics_ **VOLUME Ⅰ** (North-Holland Publishing Company, 1961), p.412 I. THE HYDROGEN ATOM
-- [D. J. Griffiths, D. F. Schroeter, _Introduction to Quantum Mechanics_ **Third Edition** (Cambridge University Press, 2018)](https://doi.org/10.1017/9781316995433) p.143 4.2 THE HYDROGEN ATOM, p.200 Problem 5.1, p.200 Problem 5.2
-- [W. Greiner, _Quantum Mechanics: An Introduction_ **Forth Edition** (Springer, 2001)](https://doi.org/10.1007/978-3-642-56826-8) p.217 The Hydrogen Atom, p.236 9.5 Spectrum of a Diatomic Molecule
-
 ## Usage & Examples
 
 [Install Antique.jl](@ref Install) 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 `ℏ` are set as optional arguments.

docs/src/DeltaPotential.md

@@ -4,22 +4,10 @@ CurrentModule = Antique
 
 # Delta Potential
 
-The Delta potential is one of the simplest models for quantum mechanical system in 1D.
-It always has one bound state and its wave function has a cusp at the origin.
+The Delta potential is one of the simplest models for quantum mechanical system in 1D. It always has one bound state and its wave function has a cusp at the origin.
 
 ## Definitions
 
-This model is described with the time-independent Schrödinger equation
-```math
-  \hat{H} \psi(x) = E \psi(x),
-```
-and the Hamiltonian
-```math
-  \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).
-```
-Parameters are specified with the following struct.
-
-#### Parameters
 ```@docs; canonical=false
 Antique.DeltaPotential
 ```
@@ -39,10 +27,6 @@ Antique.E(::DeltaPotential)
 Antique.ψ(::DeltaPotential, ::Any)
 ```
 
-#### Reference
-- [D. J. Griffiths, D. F. Schroeter, _Introduction to Quantum Mechanics_ **Third Edition** (Cambridge University Press, 2018)](https://doi.org/10.1017/9781316995433) p.63, 2.5.2 The Delta-Function Well
-- [UCSD Physics 130, Quantum Physics](https://quantummechanics.ucsd.edu/ph130a/130_notes/node154.html)
-
 ## Usage & Examples
 
 [Install Antique.jl](@ref Install) 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 `ℏ` are set as optional arguments.

docs/src/HarmonicOscillator.md

@@ -8,17 +8,6 @@ The harmonic oscillator is the most frequently used model in quantum physics.
 
 ## Definitions
 
-This model is described with the time-independent Schrödinger equation
-```math
-  \hat{H} \psi(x) = E \psi(x),
-```
-and the Hamiltonian
-```math
-  \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).
-```
-Parameters are specified with the following struct.
-
-#### Parameters
 ```@docs; canonical=false
 Antique.HarmonicOscillator
 ```
@@ -43,24 +32,6 @@ Antique.ψ(::HarmonicOscillator, ::Any)
 Antique.H(::HarmonicOscillator, ::Any)
 ```
 
-#### Reference
-
-Main:
-- _The Digital Library of Mathematical Functions_ (DLMF) [18.5.18](https://dlmf.nist.gov/18.5#E18)
-- _cpprefjp_, [hermite](https://cpprefjp.github.io/reference/cmath/hermite.html)
-- [D. J. Griffiths, D. F. Schroeter, _Introduction to Quantum Mechanics_ **Third Edition** (Cambridge University Press, 2018)](https://doi.org/10.1017/9781316995433) p.48, 2.3.2 Analytic Method
-
-Supplemental:
-- The Digital Library of Mathematical Functions (DLMF)                                                    [18.3 Table1](https://dlmf.nist.gov/18.3#T1), [18.5 Table1](https://dlmf.nist.gov/18.5#T1), [18.5.13](https://dlmf.nist.gov/18.5#E13), [18.5.18](https://dlmf.nist.gov/18.5#E18)
-- L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965)                                  [p.595 (a.4), (a.6)](https://archive.org/details/ost-physics-landaulifshitz-quantummechanics/page/n607/mode/2up)
-- L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968)                                        [p.71 (13.12)](https://archive.org/details/ost-physics-schiff-quantummechanics/page/n87/mode/2up)
-- A. Messiah, Quanfum Mechanics (Dover Publications, 1999)                                                [p.491 (B.59)](https://archive.org/details/quantummechanics0000mess/page/491/mode/1up)
-- W. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994)                           [p.152 (7.22)](https://archive.org/details/quantummechanics0001grei_u4x0/page/152/mode/1up)
-- D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995)                                [p.41 Table 2.1](https://archive.org/details/griffiths-introduction-to-quantum-mechanics/page/41/mode/1up), [p.43 (2.70)](https://archive.org/details/griffiths-introduction-to-quantum-mechanics/page/43/mode/1up)
-- D. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) [p.170 Table 5.2](https://archive.org/details/McQuarrieSimonPhysicalChemistrySolutions/McQuarrie_Simon_Physical_Chemistry1997/page/n193/mode/1up)
-- P. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008)                [p.293 Table 9.1](https://archive.org/details/atkinsphysicalch00pwat/page/292/mode/2up)
-- J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) [p.524 (B.29)](https://doi.org/10.1017/9781108587280)
-
 ## Usage & Examples
 
 [Install Antique.jl](@ref Install) 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 `ℏ` are set as optional arguments.

docs/src/HydrogenAtom.md

@@ -8,17 +8,6 @@ The hydrogen atom is the simplest Coulomb 2-body system.
 
 ## Definitions
 
-This model is described with the time-independent Schrödinger equation
-```math
-  \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),
-```
-and the Hamiltonian
-```math
-  \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r),
-```
-where $\mu=\left(\frac{1}{m_\mathrm{e}}+\frac{1}{m_\mathrm{p}}\right)^{-1}$ is the reduced mass of electron $\mathrm{e}$ and proton $\mathrm{p}$. $\mu = m_\mathrm{e}$ holds in the limit $m_\mathrm{p}\rightarrow\infty$. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.
-
-#### Parameters
 ```@docs; canonical=false
 Antique.HydrogenAtom
 ```
@@ -58,25 +47,6 @@ Antique.Y(::HydrogenAtom, ::Any, ::Any)
 Antique.P(::HydrogenAtom, ::Any)
 ```
 
-#### Reference
-
-Main:
-- _The Digital Library of Mathematical Functions_ (DLMF), [18.3 Table1](https://dlmf.nist.gov/18.3#T1), [18.5 Table1](https://dlmf.nist.gov/18.5#T1), [18.5.16](https://dlmf.nist.gov/18.5#E16), [18.5.17](https://dlmf.nist.gov/18.5#E17)
-- _cpprefjp_, [assoc_legendre](https://cpprefjp.github.io/reference/cmath/assoc_legendre.html), [assoc_laguerre](https://cpprefjp.github.io/reference/cmath/assoc_laguerre.html)
-- 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:
-- cpprefjp, [legendre](https://cpprefjp.github.io/reference/cmath/legendre.html), [assoc_legendre](https://cpprefjp.github.io/reference/cmath/assoc_legendre.html), [laguerre](https://cpprefjp.github.io/reference/cmath/laguerre.html), [assoc_laguerre](https://cpprefjp.github.io/reference/cmath/assoc_laguerre.html)
-- The Digital Library of Mathematical Functions (DLMF), [18.3 Table1](https://dlmf.nist.gov/18.3#T1), [18.5 Table1](https://dlmf.nist.gov/18.5#T1), [18.5.16](https://dlmf.nist.gov/18.5#E16), [18.5.17](https://dlmf.nist.gov/18.5#E17), [18.5.12](https://dlmf.nist.gov/18.5#E12)
-- L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965), [p.598 (c.1)](https://archive.org/details/ost-physics-landaulifshitz-quantummechanics/page/n611/mode/2up), [p.598 (c.4)](https://archive.org/details/ost-physics-landaulifshitz-quantummechanics/page/n611/mode/2up), [p.603 (d.13)](https://archive.org/details/ost-physics-landaulifshitz-quantummechanics/page/n615/mode/2up), [p.603 (d.13)](https://archive.org/details/ost-physics-landaulifshitz-quantummechanics/page/n615/mode/2up)
-- L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968), [p.79 (14.12)](https://archive.org/details/ost-physics-schiff-quantummechanics/page/n95/mode/1up), [p.93 (16.19)](https://archive.org/details/ost-physics-schiff-quantummechanics/page/n109/mode/1up)
-- A. Messiah, Quanfum Mechanics (Dover Publications, 1999), [p.493 (B.72)](https://archive.org/details/quantummechanics0000mess/page/491/mode/1up), [p.494 Table](https://archive.org/details/quantummechanics0000mess/page/494/mode/1up), [p.493 (B.72)](https://archive.org/details/quantummechanics0000mess/page/491/mode/1up), [p.483 (B.12)](https://archive.org/details/quantummechanics0000mess/page/483/mode/1up), [p.483 (B.12)](https://archive.org/details/quantummechanics0000mess/page/483/mode/1up)
-- W. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994), [p.83 (4)](https://archive.org/details/quantummechanics0001grei_u4x0/page/83/mode/1up), [p.83 (5)](https://archive.org/details/quantummechanics0001grei_u4x0/page/83/mode/1up), [p.149 (21)](https://archive.org/details/quantummechanics0001grei_u4x0/page/149/mode/1up)
-- D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995), [p.126 (4.28)](https://archive.org/details/griffiths-introduction-to-quantum-mechanics/page/126/mode/1up), [p.96 Table3.1](https://archive.org/details/griffiths-introduction-to-quantum-mechanics/page/95/mode/1up), [p.126 (4.27)](https://archive.org/details/griffiths-introduction-to-quantum-mechanics/page/126/mode/1up), [p.139 (4.88)](https://archive.org/details/griffiths-introduction-to-quantum-mechanics/page/139/mode/1up), [p.140 Table4.4](https://archive.org/details/griffiths-introduction-to-quantum-mechanics/page/140/mode/1up), [p.139 (4.87)](https://archive.org/details/griffiths-introduction-to-quantum-mechanics/page/139/mode/1up), [p.140 Table4.5](https://archive.org/details/griffiths-introduction-to-quantum-mechanics/page/140/mode/1up)
-- D. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997), [p.195 Table6.1](https://archive.org/details/McQuarrieSimonPhysicalChemistrySolutions/McQuarrie_Simon_Physical_Chemistry1997/page/n218/mode/1up), [p.196 (6.26)](https://archive.org/details/McQuarrieSimonPhysicalChemistrySolutions/McQuarrie_Simon_Physical_Chemistry1997/page/n219/mode/1up), [p.196 Table6.2](https://archive.org/details/McQuarrieSimonPhysicalChemistrySolutions/McQuarrie_Simon_Physical_Chemistry1997/page/n220/mode/1up), [p.207 Table6.4](https://archive.org/details/McQuarrieSimonPhysicalChemistrySolutions/McQuarrie_Simon_Physical_Chemistry1997/page/n230/mode/1up)
-- P. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008), [p.234](https://archive.org/details/atkinsphysicalch00pwat/page/324/mode/2up?q=Laguerre)
-- [J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021)](https://doi.org/10.1017/9781108587280), p.245 Problem 3.30.b, 
-
 ## Usage & Examples
 
 [Install Antique.jl](@ref Install) 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 `ℏ` are set as optional arguments.

docs/src/InfinitePotentialWell.md

@@ -8,17 +8,6 @@ The infinite potential well (particle in a box) is the simplest model for quantu
 
 ## Definitions
 
-This model is described with the time-independent Schrödinger equation
-```math
-  \hat{H} \psi(x) = E \psi(x),
-```
-and the Hamiltonian
-```math
-  \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).
-```
-Parameters are specified with the following struct.
-
-#### Parameters
 ```@docs; canonical=false
 Antique.InfinitePotentialWell
 ```
@@ -38,13 +27,6 @@ Antique.E(::InfinitePotentialWell)
 Antique.ψ(::InfinitePotentialWell, ::Any)
 ```
 
-#### Reference
-- [D. J. Griffiths, D. F. Schroeter, _Introduction to Quantum Mechanics_ **Third Edition** (Cambridge University Press, 2018)](https://doi.org/10.1017/9781316995433) p.31, 2.2 THE INFINITE SQUARE WELL
-
-#### Proofs
-- [Eigen Functions & Eigen Values](https://ja.wolframalpha.com/input?i2d=true&i=D%5B%5C%2840%29Sqrt%5BDivide%5B2%2Ca%5D%5Dsin%5C%2840%29Divide%5Bn%CF%80x%2Ca%5D%5C%2841%29%5C%2841%29%2C%7Bx%2C2%7D%5D)
-- [Normalization](https://ja.wolframalpha.com/input?i=Integrate%5B%28%28Sqrt%5B2%2Fa%5Dsin%28%CF%80x%2Fa%29%29%29%5E2%2C+%7Bx%2C0%2Ca%7D%5D)
-
 ## Usage & Examples
 
 [Install Antique.jl](@ref Install) 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 `ℏ` are set as optional arguments.

docs/src/InfinitePotentialWell3D.md

@@ -8,17 +8,6 @@ The three-dimensional infinite potential well (particle in a 3D-box) is one of t
 
 ## Definitions
 
-This model is described with the time-independent Schrödinger equation
-```math
-  \hat{H} \psi(x,y,z) = E \psi(x,y,z),
-```
-and the Hamiltonian
-```math
-  \hat{H} = - \frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial x ^2} + \frac{\partial^2}{\partial y ^2} + \frac{\partial^2}{\partial z ^2}\right) + V(x,y,z).
-```
-Parameters are specified with the following struct.
-
-#### Parameters
 ```@docs; canonical=false
 Antique.InfinitePotentialWell3D
 ```
@@ -38,9 +27,6 @@ Antique.E(::InfinitePotentialWell3D)
 Antique.ψ(::InfinitePotentialWell3D, ::Any, ::Any, ::Any)
 ```
 
-#### Reference
-- [D. A. McQuarrie, J. D. Simon, _Physical chemistry : a molecular approach_ (University Science Books, 1997)](https://uscibooks.aip.org/books/physical-chemistry-a-molecular-approach/) p.90, 3-9. The Problem of a Particle in a Three-Dimensional Box Is a Simple Extension of the One-Dimensional Case
-
 ## Usage & Examples
 
 [Install Antique.jl](@ref Install) 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 `ℏ` are set as optional arguments.
@@ -74,13 +60,6 @@ E(IPW3D, nx=1, ny=2, nz=2)
 E(IPW3D, nx=2, ny=2, nz=2)
 ```
 
-Wave functions:
-
-The wave functions of the 3D infinite potential well can be expressed as a product of wave functions in 1D infinite potential wells for each of the three directions ``x``,``y``,``z``.
-As the representation of 3D (non-spherical) wave functions easily becomes unclear, we refer to the documentation of the one-dimensional counterpart: InfinitePotentialWell (...)
-
-
-
 ## Testing
 
 Unit testing and Integration testing were done using numerical integration ([QuadGK.jl](https://juliamath.github.io/QuadGK.jl/stable/)). The test script is [here](https://github.com/ohno/Antique.jl/blob/main/test/InfinitePotentialWell3D.jl).

docs/src/MorsePotential.md

@@ -8,17 +8,6 @@ The Morse potential is a model for inter-nuclear anharmonic vibration in a diato
 
 ## Definitions
 
-This model is described with the time-independent Schrödinger equation
-```math
-  \hat{H} \psi(r) = E \psi(r),
-```
-and the Hamiltonian
-```math
-  \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + V(r)
-```
-Parameters are specified with the following struct.
-
-#### Parameters
 ```@docs; canonical=false
 Antique.MorsePotential
 ```
@@ -49,12 +38,6 @@ Antique.ψ(::MorsePotential, ::Any)
 Antique.L(::MorsePotential, ::Any)
 ```
 
-#### Reference
-- [P. M. Morse, _Phys. Rev._, **34**, 57 (1929)](https://doi.org/10.1103/PhysRev.34.57)
-- [J. P. Dahl, M. Springborg, _J. Chem. Phys._, **88**, 4535 (1988). (62), (63)](https://doi.org/10.1063/1.453761)
-- [W. K. Shao, Y. He, J. Pan, _J. Nonlinear Sci. Appl._, **9**, 5, 3388 (2016). (1.6)](http://dx.doi.org/10.22436/jnsa.009.05.124) 
-- The Digital Library of Mathematical Functions (DLMF) [18.3 Table1](https://dlmf.nist.gov/18.3#T1), [18.5 Table1](https://dlmf.nist.gov/18.5#T1), [18.5.12](https://dlmf.nist.gov/18.5#E12), [18.5.17_5](https://dlmf.nist.gov/18.5#E17_5)
-
 ## Usage & Examples
 
 [Install Antique.jl](@ref Install) for the first use and run `using Antique` before each use. The energy `E()`, wavefunction `ψ()`, potential `V()` and `nₘₐₓ()` will be exported. In this system, the model is generated by `MorsePotential` and several parameters `rₑ`, `Dₑ`, `k`, `µ` and `ℏ` are set as optional arguments.

docs/src/PoschlTeller.md

@@ -8,32 +8,6 @@ The Pöschl-Teller potential is one of the few potentials for which the quantum
 
 ## Definitions
 
-This model is described with the time-independent Schrödinger equation
-```math
-  \hat{H} \psi(x) = E \psi(x),
-```
-and the Hamiltonian
-```math
-  \hat{H} =
-  - \frac{\hbar^2}{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
-```math
-  x^\ast \equiv x/x_0,\qquad E^\ast \equiv \frac{\hbar^2}{m x_0^2} E
-```
-the Schrödinger equation reduces to
-```math
-  \hat{H}^\ast \psi(x^\ast) = E^\ast \psi(x^\ast),
-```
-with
-```math
-  \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.
-
-#### Parameters
 ```@docs; canonical=false
 Antique.PoschlTeller
 ```
@@ -63,10 +37,6 @@ Antique.ψ(::PoschlTeller, ::Any)
 Antique.P(::PoschlTeller, ::Any)
 ```
 
-#### Reference
-- [G. Pöschl, E. Teller, _Zeitschrift für Physik_, **83** (3–4), 143 (1933)](https://doi.org/10.1007%2FBF01331132): More general definitions are given as (2a), (2b) or (11).
-- [S. Flügge, Practical Quantum Mechanics (Springer Berlin Heidelberg, 1999)](https://doi.org/10.1007/978-3-642-61995-3) [p.94 Problem 39. Potential hole of modified Poschl-Teller type](https://archive.org/details/PracticalQuantumMechanicsS.Flgge/page/n111/mode/2up).
-
 ## Usage & Examples
 
 [Install Antique.jl](@ref Install) for the first use and run `using Antique` before each use. The energy `E()`, wavefunction `ψ()`, potential `V()` and `nₘₐₓ()` will be exported. In this system, the model is generated by `PoschlTeller` and the parameters `λ`, `m`, `ℏ`, `x₀`.