# Energy level transition questions

1. Aug 8, 2013

### Masonn

Say for a simple hydrogen atom, an electron absorbs just the right amount of energy such that it jumps up from energy levels n = 1 to n = 5. When it comes back down, is it possible to say, do n = 5 to n =3 to n = 2 to n =1, releasing 3 photons with their respective specific frequencies/energies? Or is it only able to go from n = 5 to n = 1? And why?

It's not a homework question, it's a question that came out of a classroom discussion.

2. Aug 8, 2013

3. Aug 8, 2013

### fzero

Emission and absorption of photons from an electron is described by quantum electrodynamics, called QED for short. This is an example of a quantum field theory and to properly describe the transitions in hydrogen would require a lot of background and technology. But we can get some qualitative information out using some information that requires a bit of quantum mechanics knowledge and some intuition.

Basically, the important idea is that the strength of the electromagnetic interaction is proportional to the electric charge, $e$. The amplitude for a single photon transition is then proportional to $e$, so the probability of a single photon transition is proportional to $e^2$. Since the value of the electric charge depends on the system of units that we use, it turns out to be a much simpler to express this in terms of the dimensionless fine structure constant

$$\alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c} \sim \frac{1}{137}.$$

So we can as well say that the probability for the emission of a single photon carries a factor of $\alpha$.

Now suppose there is a particular transition from n=5 to n=1 that satisfies the angular momentum selection rules that Jolb pointed out. This transition can indeed proceed by the emission of 2 photons, but the probability of this happening is now proportional to $\alpha^2$, so it is roughly 100 times less likely than the single photon transition. Similarly, the probability for the 3 photon transition will be proportional to $\alpha^3$, so it is 10000 times less likely than the single photon transition.

Multiphoton transitions have indeed been observed and in fact allow certain transitions that are forbidden from occurring via a single photon transition by the selection rules. An important example is the 2s to 1s transition in hydrogen.