How to Solve the (Radial) Schrödinger Equation II

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import mmf_setup;mmf_setup.nbinit()
import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

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How to Solve the (Radial) Schrödinger Equation II#

Here we continue our previous explorations of numerical methods for solving the radial Schrödinger Equation, trying to overcome some of the shortcomings seen there. We will focus on problems with a Coulomb potential that has both the mild singularity at \(r=0\) and the long tail \(V \sim 1/r\).

For testing we use the exact energies for the non-relativistic hydrogen atom:

\[\begin{gather*} V(r) = \frac{-\alpha}{r}, \qquad E_{l,n} = \frac{-m\alpha^2/2\hbar^2}{2(1+l+n)^2}. \end{gather*}\]

The eigenstates can also be expressed analytically:

\[\begin{gather*} \psi_{n,l,m}(r, \theta, \phi) \propto e^{-r/na}\left(\frac{2r}{na}\right)^{l} L_{n-l-1}^{2l+1}\left(\frac{2r}{na}\right)Y_{l}^{m}(\theta, \phi), \qquad a = \frac{\hbar^2}{m \alpha}, \end{gather*}\]

where \(m\) is the reduced mass \(m = m_em_p/(m_e+m_p)\), \(a\) is the reduced Bohr radius, \(L_{n-l-1}^{2l+1}(\rho)\) is a generalized Laguerre polynomial of degree \(n-l-1\), and \(Y_{l}^{m}(\theta, \phi)\) is a spherical harmonic function of degree \(l\) and order \(m\).

These exact solutions might be used to deal with the singular properties of the Coulomb potential, but we postpone discussing this for now in favour of more general techniques.

See Also#

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