Equal Radiation Frequencies from Different Transitions in Hydrogen Atoms

Trung Phan, Kim Tuan Do

Published 2020-10-13Version 1

Disregarding degeneracies and considering only different atomic levels, is it possible that two different transitions in hydrogen atom give the same frequency of radiation? That is, can different energy level transitions in a hydrogen atom have the same photon radiation frequency? The answer is definitely yes, and infinitely many transitions have been found, but to our knowledge a generalization is still lacking. In this note, we will show a general solution, i.e., how all equifrequency transition pairs can be obtained. This puzzle is a simple yet concrete example of how number theory can help understanding quantum systems, a curious theme that emerges in theoretical physics3 but usually inaccessible to high school and college students.