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I took the course of quantum mechanics long time ago. From there I learn how to describe an atom with wave function. For example, Hydrogen has the wave function in (spherical coordinates) space. In the book they consider a reduced mass for the nucleus and the only external electron so the Hamiltonian simplified to a single term for momentum and potential. My first question is is this momentum refer to the motion of the center of mass of the system? If I solve the Schrodinger equation and get the momentum of the center of mass, how can I figure out what the momentum of the electron and/or nucleus separately?

Honestly, solving the Hydrogen problem is not trivial for me though it is more or less understandable. But if we consider a more complicate case like an atom with more than 1 electron. What's the form of the wave function for that kind of atom look like? I know in that case, we cannot use the reduce mass to simplify the Hamiltonian, so it contains one term for the nucleus momentum, n momentum terms for n electrons, n potential terms for potential energy between each electron to the nucleus, C(n,2) terms for potential between any 2 electrons. I am not sure if this system is solvable but it is very complicate. My question is if it is possible to separate the system into two: one for nucleus alone and one for electrons, we solve the two systems separately to get two wave functions and then multiply the results to get the complete wave function?

My next question is about an even complicate system. Let's say I have n atoms instead of one. How generally people solve these systems if each of them have more than 1 electron? Or let's make it simple, consider n hydrogen atoms, each has 1 electron. If we could mix those n atoms but ignore interaction between atoms. Does it mean I can solve the problem for 1 atom to get wave function ##|\psi_1\rangle## so the complete wave function for the whole system is ##\prod_n|\psi_n\rangle##

My last question is about including the spin for electron, P electrons in each atom and K atoms without interactions between them. Is it correct to write down the wave function as

##\prod_{k, p}|\Psi_n\rangle |\Phi_{kp}\rangle##

with electronic wave function for the ##k##th atom as

##

|\Phi_{kp}\rangle = |s_pn_pl_pm_p\rangle_k

##

where ##s, n, l, m## stands for quantum number of spin, principal, angular momentum and magnetic.

I apology for the long question. I am trying to read other material so to learn how multiple particles system is dealed. It is very hard to understand most of the content. Above statements are based on my understanding so far, they might be wrong. I am looking for a general picture how complicate system is tackled.

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# I Help to understand the wave function for atom (gas)

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