Exact electron energy levels in hydrogen

Question

I was wondering, if you wanted to write down the exact expression for electron energies in a hydrogen atom, what would you have to include?

There's principal splitting, fine structure and hyperfine splitting if we don't include external fields, that should be it, right?

For external magnetic/electric fields, there's the Zeeman/Stark effect and for gravity, presumably gravitational tidal splitting.

Would that make it exact? Where would you start from if you wanted to make it as exact as possible, maybe even including some low-order quantum gravity corrections?

Of course, I understand that this is useless for all practical purposes and the scales you'd have to include would be wildly different, but how would we do it in principle? Are there any known results?

Answer 1

There's no such thing as "exact" in physics - there's only "exact enough to match the measurements". Fortunately, the precision spectroscopy of hydrogen is one of the longest-running series of experiments in quantum mechanics, and it shows no sign of slowing down.

In essence, the precise details of the energy levels are governed by a whole load of quantum electrodynamics, which go way deeper than just fine and hyperfine structures. Few of these are likely to have catchy names: it's just a long and involved QED calculation, and you only do it when you need to and to the order of perturbation that you need to before you stop and neglect higher orders of QED interactions.

In addition to these contributions, there are likely to be weak-interaction contributions to the energy, coming from the exchange of W and Z bosons between the electron and the proton's component quarks. Similarly, the radius of the proton is known to play a role (i.e. the fact that the electron is actually interacting with a whirling ball of quarks rather than a point charge); this occasionally makes the news as the so-called proton radius puzzle.

In principle, there are also gravitational corrections somewhere in there, at about the fortieth significant figure and onwards, but we're nowhere close to that kind of accuracy in experiment.

We have achieved, though, the measurement of energy differences (like the 1s-2s transition) at upwards of fifteen significant figures, to the point where the uncertainty in the frequency of that radiation is exclusively due to the fundamental uncertainty in the definition of the SI second, which itself comes from the fundamental linewidth of the relevant hyperfine transition in caesium. All-optical methods of frequency comparison, based essentially on frequency combs and related advances, can go one or two significant figures beyond that in the measurement of ratios of energy level differences.

If you want to know more details about the theory, the NIST atomic levels database (enter H I to get the data) sends the reader to this paper, and its references are probably a good entrance to the rabbit hole.

Answer 2

To include in the list:

• the spin-orbit coupling term (or is this your principal splitting), there is also a term due to the Thomas precession,
• and there is also a gaggle of other terms put in by hand, such as the Darwin term, usually included in the fine structure.
• There are also relativistic corrections,
• and corrections due to the finite size of the proton. Actually, the calculations are sufficiently accurate to do the reverse, i.e. infer the size of the proton from the details of the transitions: see for instance here for details.
• Then of course there are QED corrections etc.