quantum mechanics - Why is a proton assumed to be always at the center while applying the Schrödinger equation?

Why is a proton assumed to be always at the center while applying the Schrödinger equation? Isn't it a quantum particle?

Answer

There is a rigorous formal analysis which lets you do this. The true problem, of course allows both the proton and the electron to move. The corresponding Schrödinger equation thus has the coordinates of both as variables. To simplify things, one usually transforms those variables to the relative separation and the centre-of-mass position. It turns out that the problem then separates (for a central force) into a "stationary proton" equation and a free particle equation for the COM.

There is a small price to pay for this: the mass for the centre of mass motion is the total mass - as you'd expect - but the radial equation has a mass given by the reduced mass $$\mu=\frac {Mm}{M+m}=\frac{m}{1+m/M} ,$$ which is close to the electron mass $m$ since the proton mass $M$ is much greater.

It's important to note that an exactly analogous separation holds for the classical treatment of the Kepler problem.

Regarding self-interactions, these are very hard to deal with without invoking the full machinery of quantum electrodynamics. Fortunately, in the low-energy limits where hydrogen atoms can form, it turns out you can completely neglect them.