What's is a perturbation Hamiltonian

I'm not sure to understand what's is the perturbation Hamiltonian exactly.

It seems like in the case of the proton-electron the perturbation Hamiltonian is a correction of the potential energy, a term that we add to the potential where we consider the proton as a point and not a sphere. Am I correct ?

Is the perturbation Hamiltonian always a correction? Is it a way to make the problem more realistic?

In the example above, why not just write a Hamiltonian for the proton-electron as if they where spheres and not points and not just add a correction to the existing one.

Maybe I'm wrong and what I say doesn't make any sense, but again I don't really understand the difference between perturbation Hamiltonian and the correct.

• Are you talking about the proton radius correction to the hydrogen energy levels? In which case the "perturbation hamiltonian" is the difference between the potential of a sphere of charge and a point charge. We do it this way because we can exactly solve the Schrodinger equation when the proton is a point charge, but it would be a lot more difficult to solve it for the potential you get when the proton is a sphere of uniform charge density. But the correction is VERY small, so treating the difference between the two potentials as a correction is extremely accurate Commented Mar 31, 2023 at 9:24

In physics we often have to deal with problems that are not solvable exactly. E.g., we may want to find the eigenstates and eigenfucntions of a Hamiltonian $$H$$, but it is impossible or difficult to do mathematically. One rather general approach to bypass this problem is by splitting the Hamiltonian into a part that we can solve/diagonalize exactly, $$H_0$$ and the other part, which we call perturbation $$V$$: $$H= H_0 + V.$$ We then can try to approximate the solution of $$H$$ by calculating the corrections to the solution for $$H_0$$.
This, the choice $$H_0$$ and $$V$$ is first of all a matter of convenience, but, depending on the problem and the method used to calculate the corrections, there might be additional constraints on what can be used as $$V$$.