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Express for Operator of coordinate in momentum representation

  1. Feb 9, 2007 #1
    1. The problem statement, all variables and given/known data

    1. Obtain an expression for operator of coordinate in momentum representation. To this end
    begin with definition of the average coordinate
    x = ∫ψ*(x)xψ(x)dx
    express the wave functions as wave packets in terms of plane waves, and rewrite the
    expression for average coordinate in such a way that it would have a form of an operator
    acting on the wave functions in the momentum representation.

    2. Relevant equations

    Not sure



    3. The attempt at a solution


    Where do I start? I'm not sure what the question is even asking.
     
  2. jcsd
  3. Feb 9, 2007 #2

    Dick

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    What kind of an operator O(p) might you apply to a plane wave exp(i*p*x) to create an expectation value integral like you have written?
     
  4. Feb 9, 2007 #3
    I would apply an x operator. I don't see the point though.
     
  5. Feb 9, 2007 #4

    Dick

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    Exactly. But you want to express the x operator in terms of p. Here's big hint. How do you express the p operator in terms of x? The point is that the same problem can be expressed in different equivalent variables. And problems that can be hard in one variable are easy in another. This is practice. That's the point.
     
    Last edited: Feb 9, 2007
  6. Feb 10, 2007 #5
    So I just use the fact that <p>=d<x>/dt and that is the whole problem?
     
  7. Feb 10, 2007 #6
    How can you compute <x> and <p> as functions of time?
     
  8. Feb 10, 2007 #7

    Dick

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    You didn't answer my 'hint' about what the operator p looks like in the x-representation. If you want to show x and p don't commute, how do you represent p?
     
  9. Feb 10, 2007 #8
    well we would represent p by (h bar/i)*d/dx to show x and p do not commute
     
  10. Feb 10, 2007 #9

    Dick

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    Great! That's p in x-representation. Now take a plane wave and apply that and see how it 'pulls down' p. Then think of a way to 'pull down' x using an operator written in terms of p.
     
  11. Feb 10, 2007 #10
    x = ∫(p/h bar*i) dp ?

    I am not sure how to represent x in terms of p
     
  12. Feb 10, 2007 #11

    Dick

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    What is the form of a plane wave solution?
     
  13. Feb 10, 2007 #12
    Isn't the plane wave of the form exp(i(kx- wt))?
     
  14. Feb 10, 2007 #13

    Dick

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    Yes. Now apply the p operator you sent me (the d/dx one).
     
  15. Feb 10, 2007 #14
    so i get hk*exp(i(kx-wt)) and hk = p by deBroglie relations,correct?
     
  16. Feb 10, 2007 #15

    Dick

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    Right again! Now in the same way define an operator using p that will make x appear in front of the plane wave.
     
  17. Feb 10, 2007 #16
    The thing is that p is not in the exponent expression. I could pull out x using the operator -i*d/dk and d/dk=h*d/dp. But then I would have to express k in the exponent as k=p/h.
     
  18. Feb 11, 2007 #17

    Dick

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    I don't see why that should pose a problem?
     
  19. Feb 11, 2007 #18
    I think I got it now. Thanks. Muchly appreciated.
     
  20. Dec 21, 2010 #19
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