>>10059901“interactions” are hairy in QM. if you’re referring to e.g. electromagnetic interactions, say e.g. between a proton and electron in a hydrogen atom, then textbook QM says you go to the reduced mass picture and solve for a single particle in a potential well. but really we know that the real thing going on has to do with the exchange of photons, so now you’re talking QED. so when your interactions are mediated by photons (like in high energy physics) or phonons or plasmons (like in quantum computing) you can’t easily use QM intuition
however you could think about non-interacting particles, take e.g. non-interacting spinless fermions (and even if that sounds wrong, that’s only because you must already know a little QFT or particle physics—only there can you derive the spin statistics theorem)
for noninteracting spinless identical particles, we know we have to write down the wave function in a way that makes them indistinguishable:
|psi> = |a>|b> + |b>|a>
or
|psi> = |a>|b> - |b>|a>
(just ignore normalizations)
by definition a fermion is the second version
now if you calculate the inner product of that state you get
(<a|<b| - <b|<a|)(|a>|b> - |b>|a>)
= <a|a><b|b> - <b|a><a|b> - <a|b><b|a> + <b|b><a|a>
if a and b are orthogonal eigenbasis vectors of your hilbert space, your answer comes out nicely, you get a positive amplitude
however if you let them be the same basis vector, (set b=a) then you get 1-1-1+1=0. so there is 0 probability of them being in the same state
so that’s the pauli exclusion principle—the (degeneracy pressure) “interaction” of the two fermions prohibits them from being in the same state.
that’s the best QM example of how to think of “interactions” in pure QM in terms of gilbert spaces and bra-kets
