# Measuring the size of the proton from the hydrogen atom spectrum?

I was reading that besides measuring the angle of ricocheted electrons bouncing off the proton to pin down its size, it is also possible to excite the electron and then measure the frequency of the light emitted by the excited electron. Why would the gap between ground state and excited state tell us the size of a proton? Is there something I have missed?

• On a related note, measurements of the proton size in muonic hydrogen (i.e., a hydrogen atom where the electron is replaced by a muon) yield a slightly smaller size than in normal hydrogen. See journals.aps.org/prd/abstract/10.1103/PhysRevD.98.013002 Dec 21, 2019 at 11:52
• @PM2Ring: I read the muon orbits much closer than electron and they can measure the proton charge radius better but muon is too short lived despite being heavy a bad choice no? Dec 21, 2019 at 12:29
• True, it is a problem that the muon has a short mean lifetime (around 2.2 microseconds), but we don't have much choice about that. Most subatomic particles have short lifetimes, so particle physicists are used to doing experiments that operate on short timescales. A couple of microseconds is actually quite long when you're used to working with processes that happen in nanoseconds and shorter. ;) Dec 21, 2019 at 12:43
• @PM 2Ring, I think that discrepancy was resolved just this past year. See science.sciencemag.org/content/365/6457/1007 Dec 21, 2019 at 16:26
• Do you really mean size, or do you mean mass? You can get the mass because the energy levels depend on the reduced mass.
– user4552
Dec 22, 2019 at 3:14

This is an interesting and non-trivial problem. Basically the Coulomb potential assumes a point particle but, if the proton is modelled as a solid sphere of finite radius, part of the electron wave function would be "inside" the proton, where the assumption of point charge no longer holds.

To account for this one must modify the Coulomb potential from $$1/r$$ outside the proton to (basically) $$C r^2$$ inside, where $$C$$ is some constant. The simplest model is to think of the proton as a uniformly charged sphere (constant volume charge density) so the $$Cr^2$$ term comes from Gauss's law for the potential inside this type of sphere.

This small perturbation in the potential will affect slightly the energy level. Since for small distances the radial probability density generally goes like $$r^{2(\ell+1)}$$, the smaller values of $$\ell$$ will produce wave functions with larger probabilities of having the electron "inside" the proton, so experiments were done measuring the energy difference between $$2S_{1/2}$$ and $$2P_{1/2}$$ which have $$\ell=0$$ and $$\ell=1$$ respectively. These states would normally have the same energy under the pure Coulomb potential since both are $$n=2$$ states, but they are affected differently under the assumption that the proton has a non-zero volume.

The story of the "proton problem" goes back 10 years or so, when a group in Geneva made extremely accurate measurements of the size of the nucleus. Basically, they deduced what value of the radius of the proton (assumed as a uniform spherical charge distribution) was needed to reproduce their experimental measurements of energy levels, and it didn't agree with the accepted value. There's a good synopsis of this

The proton -- smaller than thought: Scientists measure charge radius of hydrogen nucleus and stumble across physics mysteries https://www.sciencedaily.com/releases/2010/07/100712103339.htm

(They used muonic hydrogen since the Bohr radius of this system is smaller than the usual electron-proton system, thus enhancing the portion of the wavefunction inside the nucleus.)

The unexpected result was only confirmed this year. A summary of new results can be found here and the actual paper of the experiment

Bezginov, N., Valdez, T., Horbatsch, M., Marsman, A., Vutha, A.C. and Hessels, E.A., 2019. A measurement of the atomic hydrogen Lamb shift and the proton charge radius. Science, 365(6457), pp.1007-1012

appears to be available online from this link provided courtesy of GoogleScholar.

Note there are other perturbations in hydrogen - the fine and hyperfine structure - which have to be accounted for as well, making this volume effect non-trivial to isolate.

I love this stuff. It shows that the hydrogen atom is not completely archeological but there's still some interesting surprises to be found in this canonical example of undergraduate level quantum mechanics

• It might be simplest to analyze, but it seems somewhat unphysical (at least to me, who doesn't know much about proton structure) that all (or most of) the charge would reside on some particular distance from the center. Dec 21, 2019 at 16:53
• @Ruslan yes Zero means a spherical volume with uniform 3D charge density, and not a spherical shell or a solid conducting ball where the charge would migrate to the surface.
– uhoh
Dec 21, 2019 at 17:01
• @ZeroTheHero +1 fantastic summary of the problem, thanks! btw is my comment above correct?
– uhoh
Dec 21, 2019 at 17:05
• @uhoh yes you are correct. Dec 21, 2019 at 17:18
• @EricTowers right. That would be a spherical shell, not a sphere. Dec 23, 2019 at 0:42

Usually when determining the energies of the hydrogen atom we assume the the proton is a point charge. By changing that to a finite charge distribution the potential is altered for small electron-proton distances. The energies are sensitive to the extent of the distribution. By comparing simulations to very accurate measurements of excitation energies information is found on the proton's charge distribution.