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Homework Help: Matrix elements of position operator in infinite well basis

  1. Jul 10, 2014 #1
    1. The problem statement, all variables and given/known data
    Find the eigenfunctions of a particle in a infinite well and express the position operator in the basis of said functions.

    2. Relevant equations

    3. The attempt at a solution

    Tell me if I'm right so far (the |E> are the eigenkets)
    [itex]X_{ij}= \langle E_i \vert \hat{X} \vert E_j \rangle = \int dx \int dx' \langle E_i \vert x \rangle \langle x \vert \hat{X} \vert x'\rangle \langle x'\vert E_j \rangle [/itex]
    [itex] \int dx \int dx' \Psi_i^*(x) x\delta_{x, x'} \Psi_j(x') = \int dx \Psi_i^*(x) x \Psi_j(x) [/itex]
    Last edited: Jul 10, 2014
  2. jcsd
  3. Jul 10, 2014 #2


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    Gold Member

    Modify your notation for the eigenfunctions.
    Using E to denote an eigenfunction is very confusing.
    You can read on wikipedia about the eigenfunctions for an infinite well:
    If I guessed your notations correctly, you are starting in the right direction.
  4. Jul 10, 2014 #3
    The eigenvalues Ei are only used to denote their respective eigenkets, the wavefunctions are denoted by the index of the eigenvalue. I think I've seen this in many books.

    The last integral is correct then? I was not sure how to handle the delta.
  5. Jul 10, 2014 #4


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    Gold Member

    Yes it is correct.
    By definition, I would say.
    It looks more like a change of notation rather than any derivation of something!

    [tex]X_{ij}= \langle \Psi_i \vert \hat{X} \vert \Psi_j \rangle[/tex]

    Which is indeed the integral you have written!

    The state is independent of its representation.
    Therefore I see no need to distinguish between

    [tex]\langle \Psi_i \vert \hat{X} \vert \Psi_j \rangle[/tex]


    [tex]\langle E_i \vert \hat{X} \vert E_j \rangle[/tex]
  6. Jul 10, 2014 #5
    Yes, you are right, but I first needed to state it in terms of the wavefunctions, and I wasn't quite sure if I had done it right.

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