When 1 proton and 1 electron become an hydrogen atom?

Question

To solve the Schrödinger equation we construct the Hamiltonian of the system that is: $$\hat{H}=\hat{K}+V$$ So far so good. What bothers me is the conceptual approach. Do we assume that nothing else exists in the universe except 1 proton and 1 electron? That is as the electron and proton approach each other they wouldn't bump into anything else. If this is the case then it doesn't make sense because it isn't realistic. In a real world scenario around an hydrogen atom there are $N_A$ other molecules. Or we assume that a proton and an electron are close enough that we can neglect their interactions with everything else? Or we start from the assumption that we have an hydrogen atom and we apply Quantum mechanical postulates? I think it is a stupid question but I can't find an answer.

Answer 1

It is not a stupid question, and the answer is quite interesting. The key concept here is scale. Think about the Hydrogen atom, the diameter of the nucleus is close to the femtometer scale $10^{-15}$m while the full size of the atom is around the Amstrong scale $10^{-10}m$. The ratio is, roughly speaking, 5 orders of magnitude. Huge. Now, we know that the nucleus is made of a proton, and the proton is made of two up quarks and a down quark, gluons and other stuff. Now go back to the electron, who is living on the Amstrong scale spining, possibly absorbing and emiting photons, oscillating etc around the very very very tiny little nucleus, that he can barely see. The point is that the electron does not see the quarks, nor the gluons, but can only feel an atracting force coming from the nuclei, and this is what suffices to describe the physics of the Hydrogen atom at the Amstrong scale. The same happens with space-time for example, at our daily life scale the Newton theory of flat space is good approximation, but if we go to bigger scales we can start to feel the curvature of space-time and we must use Einstein theory.