Hydrogen simulation using Quantum Mechanics from quantumhydrogen import plot_hydrogen_orbitals
## Quantum numbers
n = 3
l = 2
m = 0
# 1D plot (Radial wavefunction)
plot_hydrogen_orbitals (n ,l ,m ,posy = None )
# 2D plot
plot_hydrogen_orbitals (n ,l ,m )
# 3d plot (UPCOMING)
plot_hydrogen_orbitals (n ,l ,m ,posz = (- 12 ,12 ))Radial (1D) with n=2,l=0,m=0 :
2D with n=4,l=1,m=-1 :
Hydrogen Wave Function Equations The hydrogen wavefunction equals to :
$$\psi_{n \ell m}(r, \theta, \phi)=R_{n \ell}(r) Y_{\ell}^{m}(\theta, \phi)$$
With n,l,m the first 3 quantum numbers , $R_{n \ell}$ the radial wavefunction and $Y_{\ell}^{m}$ the angular wavefunction.
Angular wavefunction / Spherical Harmonics The regularized angular wavefunction (called spherical harmonics) equals to :
$$Y_{\ell}^{m}(\theta, \phi)=\sqrt{\frac{(2 \ell+1)}{4 \pi} \frac{(\ell-m) !}{(\ell+m) !}} e^{i m \phi} P_{\ell}^{m}(\cos \theta)$$
Where $P_{\ell}^{m}$ is an associated Legendre polynomial , defined by the diffential equation :
$$ {\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}P_{\ell }^{m}(x)\right]+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right]P_{\ell }^{m}(x)=0$$
The Radial wavefunction equals to :
$$ R_{n \ell}(r) = \sqrt{\left(\frac{2}{n a}\right)^{3} \frac{(n-\ell-1) !}{2 n(n+\ell) !}} e^{-r / n a}\left(\frac{2 r}{n a}\right)^{\ell}\left[L_{n-\ell-1}^{2 \ell+1}(2 r / n a)\right] $$
Where $L_{n}^{\alpha}$ is an associated Laguerre polynomial , defined to be the solutions to the differential equations :
$$ x,y'' + (\alpha +1 - x),y' + n,y = 0 $$
