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

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.

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