My question is about position measurement in non relativistic quantum mechanics. I’ve been taught that when you measure the value of an observable for some state of a system described by $|psirangle$ then the state of the sistem “collapses” to the eigenvector associated with the eigenvalue measured.
Following this logic, when you measure the position of a particle described by $|psirangle$ and you get a value $x_0$ then the state should collapse to an eigenvector $|x_0rangle$. However, as is obvious, this eigenvector is the dirac delta function in position basis, $langle x|x_0rangle = delta (x-x_0)$. This seems to me unacceptable since this state is non normalizable and furthermore simple things, such as $langle hat{x}rangle$ are undefined for this state.
The first thing I tried when I came up with this is to try to describe the state with a density matrix $hat{rho} = sum_{i} p_i vert psi rangle langle psi vert$ but I couldn’t manage to find a way in which to do this satisfactorily. On the one hand, if I try to do this in the obvious way I get something like $mathrm{Tr}(hat{rho} hat{x})=sum_i p_i langle hat{x} rangle_{x_i}$ which is not helpful because I’ve got no way to individually evaluate the $langle hat{x} rangle_{x_i}$ terms. On the other hand I think I would need some way to assign a probability distribution to the different eigenstates, it seems to me that in a real world scenario you would assign, for example, a gaussian centered in some value $x_0$ to the value you measure (I apologize if this seems vague but I hope the point comes across) and not probabilties to the individual eigenstates in a discrete way, as is usual when one defines the density matrix operator.
In short, how is this usually done in practice? I believe there must be a straightforeward way of going about this that I’m not seeing, or at a generally agreed upon manner to handle it. I’m sure experimentalists make measurements of position all the time, and I guess its only natural to describe the state after measurement in some way. How is this done? It bugs me that this seems such a simple, basic question, yet I haven’t been able to find the answer anywhere.
Note: I’ve been taught the Copenhagen interpretation (or wharever you want to call the fact that the state “collapses” after measurement) and understand quantum mechanics based on this. I would really appreciate answers to stick with this idea unless for some reason its unavoidable to talk about other interpretations in this context. My guess is that, if its true that all interpretations (or most) are actually “equal” in the sense that they have the same underlying mathematics and give the same answer then this question must have some satisfactory answer in the context of this interpretation. If this is not the case for some reason please explain why, and please take into account that I’m not really familiar with other interpretations of QM. Thanks in advance!!