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Is it possible that two different transitions in the non-relativistic quantummechanical model of the hydrogen atom give the same frequency? That is, candifferent energy level transitions in a hydrogen atom have the same photonradiation frequency? This question, which was asked during a Ph.D. oral exam in1997 at the University of Colorado Boulder, is well-known among physicsgraduate students. We show a general solution to this question, in which allequifrequency transition pairs can be obtained from the set of solutions of aDiophantine equation. This fun puzzle is a simple yet concrete example of hownumber theory can be relevant to quantum systems, a curious theme that emergesin theoretical physics but is usually inaccessible to the general audience.

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