Equal Radiation Frequencies from Different Transitions in the Non-Relativistic Quantum Mechanical Hydrogen Atom

TL;DR

This paper explores whether different energy level transitions in the non-relativistic hydrogen atom can produce identical radiation frequencies, revealing a connection between quantum physics and number theory through solutions of a Diophantine equation.

Contribution

It provides a general solution to the problem of equifrequency transitions in hydrogen, linking quantum energy levels to number theory and Diophantine equations.

Findings

All equifrequency transition pairs can be derived from Diophantine solutions.

Number theory plays a role in understanding quantum transition frequencies.

The problem illustrates a novel intersection of physics and mathematics.

Abstract

Is it possible that two different transitions in the non-relativistic quantum mechanical model of the hydrogen atom give the same frequency? That is, can different energy level transitions in a hydrogen atom have the same photon radiation frequency? This question, which was asked during a Ph.D. oral exam in 1997 at the University of Colorado Boulder, is well-known among physics graduate students. We show a general solution to this question, in which all equifrequency transition pairs can be obtained from the set of solutions of a Diophantine equation. This fun puzzle is a simple yet concrete example of how number theory can be relevant to quantum systems, a curious theme that emerges in theoretical physics but is usually inaccessible to the general audience.