Question

Why do we get information about position and momentum when we go to different representations. Why is momentum, which was related to time derivative of position in classical physics, now in QM just a different representation brought about by some unitary transformation. Is Ehrenfest's theorem the only link?

I just started studying QM. So please suggest some references explaining the structural aspects and different connections.I don't want to start with noncommutative geometry. I would like something of an introductory nature and motivating.

Answer 1

ou can get information for all observables in any representation. The reason to go to different ones is that it is easier to work with them depending on what you are doing. They are all equivalent by the Stone-von Neumann theorem, so it is a matter of convenience.

There is a theorem (mathematical) that roughly says that for any operator, that is of interest in QM, there is a representation of the operator as a multiplication operator, in it it acts as a multiplication by a function . In the coordinate space the position operators are multiplication by the coordinates. In momentum space it is the momentum that is represented as a multiplication. It is true for any of the QM observables. Unfortunately (or fortunately) since they do not commute there isn't a single representation for all of them. Hence people use more than one.

Edit: In response to the comment. This is probably written in many books, but here is a reference. Look at Folland's "Quantum Field Theory. A Tourist Guide for Mathematicians". The first section of chapter 3 gives a nice motivation for the use of self-adjoint operators for modeling QM observables.