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At first I thought that there is like a general wavefunction, that is something abstract that changes with time(like e.g. we know what rotation does, but the rotation matrix looks differently in different bases) so i thought a wave function is like a function of infinitely many variables, e.g. x-positing, p-momentum, spin, energy... and we need to "project" it on one of those bases <x|psi(t)> = psi (x, t) (so this step is like representing a rotation with a matrix in a basis). And then we are working in one of those bases (position), I can do other stuff now, like find probabilities amplitudes that it's in some state q using : <q | psi (x,t)> and do other find expected values etc... But after reading Griffiths introduction to quantum mechanics for a bit, I realized that my logic is probably wrong. So I started thinking again.

As I understand now, a general wavefunction is like a vector space of infinitely many square integrable functions (and their linear combinations and stuff are obviously in that vector space as well), so it's just like a database of infinitely many functions, which are all functions of x and t (in 1 dimension). And so now, we just take this wavefunction and take scalar product with state vectors that we are interested in straight away, <q | psi(x,t)> and we'll get the probability amplitude.

Is that correct? I'm a bit confused because states are vectors, can you give some examples of states? can a state be a function (e.g. q is some function, of what variables?) then obviously <q (...) | psi (x,t)> = integral q*psi(x,t)dx or something.

Thanks in advance

p.s. At our lectures, we were actually never told what a wavefunction is, our lecturer thought it made more sense to start with Probability Amplitudes and assume that everyone knows what a wavefunction means.