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

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

(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

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.