3D isotropic harmonic oscillator

[ohno]

Implement the module of :HarmonicOscillator3D. The analytical solutions are written in Wikipedia or Messiah's Quantum Mechanics. The code will be almost similar to the code of hydrogen atom.

The spherical harmonics are used in the hydrogen atom.

https://github.com/ohno/Antique.jl/blob/2994e17ee8dd233a6d47c0cdeeefc1c8d4bfe413/src/HydrogenAtom.jl#L32-L39

The generalized Laguerre polynomials are used in Morse potential.

https://github.com/ohno/Antique.jl/blob/2994e17ee8dd233a6d47c0cdeeefc1c8d4bfe413/src/MorsePotential.jl#L41-L42

[ajarifi]

Hi, I think this is good for the quark model in hadron physics. I will do this.

[ohno]

Thank you for your pull request. I marged it. You can see the hidden document page here: https://ohno.github.io/Antique.jl/dev/HarmonicOscillator3D/

[ohno]

By the way, the minimum value of $n$ is 1 in hydrogen atom. Is not the minimum value in 3D harmonic oscillator 0? I'm not sure.

[ohno]

@ajarifi

There are 2 typos in the docstring of R(). Please fix them when you update the code.

before:

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),

after:

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),

The generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, not the associated Laguerre polynomials $L_n^{k}(x)$, are used in the 3-dimensional harmonic oscillator as I said in #30. Please see codes of Morse potential:

https://github.com/ohno/Antique.jl/blob/3361355de0cc7987d1735fa74d813fe74e14c869/src/MorsePotential.jl#L61-L67

[ajarifi]

By the way, the minimum value of n is 1 in hydrogen atom. Is not the minimum value in 3D harmonic oscillator 0? I'm not sure.

I think it is from n=0 for harmonic oscillator.

[ohno]

According to S. Flügge, Practical Quantum Mechanics (Springer Berlin Heidelberg, 1999), this model seems to be called a Spherical oscilator. It seems better to use :SphericalOscillator for the isotropic three-dimensional harmonic oscillator and reserve :HarmonicOscillator3D for the anisotropic (general) 3D harmonic oscillator.

By the way, the Kummer's (confluent hypergeometric) function $M(a, b, z) =_1 F_1(a ; b ; z)$ is used for the eigen functions.

u(r, \vartheta, \varphi) = C r^{\prime} \mathrm{e}^{-\frac{\lambda}{2} r^2}{ }_1 F_1\left(-n_r, l+\frac{3}{2} ; \lambda r^2\right) Y_{l, m}(\vartheta, \varphi)

This is consistent to the solution using the generalized Laguerre polynomials:

L_n^{(\alpha)}(x) = \left(\begin{array}{c}
n+\alpha \\
n
\end{array}\right) M(-n, \alpha+1, x)

Please see Wikipedia.