# Momentum-Representations in Quantum Mechanics

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.

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You 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.

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For the sake of completeness, the theorem is called spectral theorem and it's quite technical for general operators but basically it's just a diagonalization of the operator in the "basis" of "eigenvectors" (scare quotes because operators on infinite-dimensional spaces need not have eigenvalues or eigenvectors). –  Marek Feb 23 '11 at 16:57
May be I was not clear in my question.I should have I asked some justification of self adjoint operators as observable.You may say we want real eigen values etc.But why do we assume eigen values to be set of observed values...Is this framework necessary--I mean can we deduce that it is necessary to deal with linear operators,eigen values from some assumptions...And my problem regarding momentum was why is the fourier representation related to momentum which was some time derivative in Classical Physics. –  Ket Feb 23 '11 at 17:49
@Ket: In this case I have misunderstood your question. You can analyze an observable O through true-false questions of the form "Is the value of O in the set E?" for Borel sets E in $\mathbb R$. This leads to projection-valued measures on $\mathbb R$. Then the spectral theorem (again) gives the connections to self-adjoint operators. I find this natural, but that is subjective. To write this in detail would be more than a comment and as I said I understood your question differntly. –  MBN Feb 23 '11 at 18:01
Thanks for the reference.I will look at it. –  Ket Feb 23 '11 at 18:12