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Matrix mechanics

  1. Dec 11, 2007 #1
    1. The problem statement, all variables and given/known data
    My book says that we can express

    [tex] p_{nm} = -i \sqrt{M \omega \hbar} \left( \delta_{n,m-1}\sqrt{m} - \delta_{n,m+1} \sqrt{m+1}\right)[/tex]

    and

    [tex] x_{nm} = \sqrt{\hbar/2M\omega} \left( \delta_{n,m-1} \sqrt{m} +\delta_{n,m+1}\sqrt{m+1}\right)[/tex]

    for the simple harmonic oscillator potential.

    I want to calculate [p,x] = i h-bar.
    2. Relevant equations



    3. The attempt at a solution
    What I am saying is that when I calculate

    [tex] \sum_{k}x_{nk}p_{km} - \sum_{k}p_{nk}x_{km} [/tex]

    I get 0. Can someone check that? If I need to I can post more work.
     
    Last edited: Dec 11, 2007
  2. jcsd
  3. Dec 11, 2007 #2
    Shouldn't there be some creation and annihilation operators in here? Usually they're written as "a" with or without a little dagger.
     
  4. Dec 11, 2007 #3
    No, I think you are thinking of something else. x_nm is defined as <x_n|x|x_m> for example. |x_n> is nth eigenstate of the SHO Hamiltonian.
     
  5. Dec 11, 2007 #4
    Ah, right. I remember them now.

    These are not the x and p operators, they are expectation values. [p,x] = i h-bar is for the operators. The expectation values are just numbers, and the commutator of two numbers is always zero.
     
  6. Dec 11, 2007 #5
    I am saying that [tex] \sum_{k}x_{nk}p_{km} - \sum_{k}p_{nk}x_{km} [/tex] is equal to 0 for each n and each m. So the entire matrix is equal to zero.

    The problem in my book says, "show that [x,p] = ihbar holds as a matrix equation."
     
  7. Dec 12, 2007 #6

    robphy

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    What do you get for
    [tex] \sum_{k}x_{nk}p_{km}[/tex]?
     
  8. Dec 12, 2007 #7
    [tex] (1/2)i\hbar\delta_{n,m} [/tex]

    and I get the same for the

    [tex] \sum_{k}p_{nk}x_{km} [/tex]

    All would be well if I got [tex] \sum_{k}p_{nk}x_{km} [/tex] equal to minus that, but I checked my algebra several times and I just don't know what is going on.
     
  9. Dec 12, 2007 #8

    malawi_glenn

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    [p,x] = i h-bar

    Just insert what p and x are in the linear combination of annihilation and creation operators, and use their commuting algeras.
     
  10. Dec 12, 2007 #9
    OK, I'm over my denseness now. You're calculating [tex]<x_n|[p,x]|x_m>[/tex], right?

    So, for [tex] \sum_{k}p_{nk}x_{km}[/tex] you should get that the [tex]\delta_{n,m}[/tex] terms actually cancel and the remaining terms have [tex]\delta_{n+2,m}[/tex] and [tex]\delta_{n-2,m}[/tex], and that for [tex] \sum_{k}x_{nk}p_{km}[/tex] you get those same terms, minus the final (correct) answer, proportional to [tex]\delta_{n,m}[/tex] of course. Can you show some work?
     
    Last edited: Dec 12, 2007
  11. Dec 12, 2007 #10

    malawi_glenn

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    ehrenfest : can you please tell us exacly what you want to calculate?
     
  12. Dec 12, 2007 #11

    nrqed

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    Malawi, I think it's pretty clear what he is calculating. He has the matrix elements of p and x and wants to calculate the commutator using the matrix representation of those oeprators.
     
  13. Dec 12, 2007 #12

    malawi_glenn

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    Ok, 2Tesla also asked. If this is a exercise in dealing with the matrix representation, or if he just wants [p,x]

    Using the template is good. Saying exactly what the problem are. Now one can inteprent his problem to be:

    I want to calculate [p,x] = i h-bar for the simple harmonic oscillator potential.
     
  14. Dec 12, 2007 #13
    I see where I messed up. I just totally botched the replacement of n and m's with k's using the Kronecker deltas. Thanks.
     
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