1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    maajdl

    User Avatar
    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:
    http://en.wikipedia.org/wiki/Particle_in_a_box
    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

    maajdl

    User Avatar
    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]

    and

    [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.

    Thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Matrix elements of position operator in infinite well basis
Loading...