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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?

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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

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