# energy difference uniqueness in hydrogen atom

Is the energy difference between two energy levels unique for that particular pair of levels for a hydrogen atom ? If so how can one prove it?

-
Good question, I came across this in past but never found an answer. Perhaps it would help others to understand the question if you formulated it as a problem from the theory of natural numbers. –  Ján Lalinský Jan 7 '14 at 23:37
If I understand right: I believe you are asking whether, using the Dirac formula for the energy levels $E_{n\,j} = \mu c^2\left(1+\left[\dfrac{Z\alpha}{n-|k|+\sqrt{k^2-Z^2\alpha^2}}\right]^2\right)^{‌​-1/2}$ and then proving $E_{n\,j}$ is unique for each pair$(n,j)\in \mathbb{Z}_0^+\times \mathbb{Z}_0^+$? If so, you might try Maths SE. This is a harder problem than it looks. –  WetSavannaAnimal aka Rod Vance Jan 8 '14 at 0:23
Why bother with the Dirac formula, it is messy and contains $\alpha$ which is not known to be simple number. More interesting is the Schroedinger non-relativistic case : for given $\Delta E$, find all $n,m$ such that $\Delta E = \frac{1}{n^2}- \frac{1}{m^2}$. –  Ján Lalinský Jan 8 '14 at 0:33

Richard, nice picture - what element is it? However, it does not answer the OQ - whether the $\omega$ of a transition corresponds to a unique pair of Hamiltonian eigenfunctions (perhaps there are more). –  Ján Lalinský Jan 7 '14 at 23:51
Otherwise, your post assumes that the atom can be only in discrete states with definite energy. This was working assumption years around 1912 and is still taught today (Bohr model, buck-shot model of light), but is not consistent with diff. equations and is not necessary to explain the spectrum today - the Schroedinger equation explains the spectral function with continuously changing $\psi$ functions, most of which do not imply that atom has one of the discrete set of energies. –  Ján Lalinský Jan 7 '14 at 23:52