# Connection between Quantum-Mechanical Hydrogen Atom and Harmonic Oscillator

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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 .

[more]
Contributed by: S. M. Blinder (March 2019)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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.

## Permanent Citation