Connection between Quantum-Mechanical Hydrogen Atom and Harmonic Oscillator

The bound states of the hydrogen atom are governed by the geometrical symmetry (not considering the full dynamical symmetry ). Similarly, the two-dimensional isotropic harmonic oscillator exhibits the symmetry . To anyone versed in the theory of Lie groups, it would not be surprising that there might be an explicit connection between these two problems, in view of the local isomorphism between the corresponding Lie algebras and .

The radial Schrödinger equation for a hydrogen-like system, in atomic units, is given by

,

with the unnormalized solutions

,

where is an associated Laguerre polynomial. Consider now a two-dimensional isotropic harmonic oscillator, expressed in polar coordinates. The associated radial Schrödinger equation takes the form

Using DSolve, we find the unnormalized solutions:

.

The solutions of the two problems can be made equivalent by the substitutions:

, , .

For selected values of and , the graphic shows plots of the radial wavefunctions and , as well as the corresponding radial distribution functions (RDFs) and .

Julian Schwinger, in his quantum mechanics course [1], suggested a very clever method to solve the hydrogen-atom problem by converting it into the equation for a two-dimensional isotropic harmonic oscillator. To do this, let and . Then turns out to obey the radial equation for the oscillator.

The method is given as a problem in [1].

Reference

[1] G. Baym, Lectures on Quantum Mechanics, New York: W. A. Benjamin, 1969 p. 179.