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Change for position to energy basis

  1. Nov 29, 2011 #1

    lrf

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    1. The problem statement, all variables and given/known data

    Give expressions for computing the matrix elements Xmn of the matrix X representing the position operator X in the energy basis (using eigenvectors of the Harmiltonian operator)

    Also told to consider the example of the harmonic oscillator where energy eigenvalues are En=(1/2+n)hω

    2. Relevant equations

    Xmn=<em|X|en>

    H|en>=En|en>

    3. The attempt at a solution

    I'm thrown off a bit by how Xmn is defined here - if it is originally in the |x> basis, why is Xmn defined using |em> and |en>. Shouldn't these be inserted using the completeness relation to convert the matrix into the energy basis representation?

    Here goes...
    Xmn=<em|X|en>
    Xmn=ƩƩ<em|x><x|X|x'><x'|en>
    Xmn=ƩƩem(x)X(x,x')en(x')
     
  2. jcsd
  3. Nov 29, 2011 #2

    dextercioby

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    You can use the algebraic formulation of the harmonic oscilator and write X in terms of a and a^dagger. The action of a and a^dagger on the standard basis (eigenvectors of N and H) is already known, so...
     
  4. Nov 30, 2011 #3

    lrf

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    Sorry, still very confused!

    So use -h2/2m d2ψ/dx2+1/2mω2x2ψ=Eψ how?
     
  5. Nov 30, 2011 #4

    lrf

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    or use 1/2P2+1/2m2X2=H ?
     
  6. Nov 30, 2011 #5

    dextercioby

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    No, X and P need to be replaced by the raising and the lowering ladder operators, a and [itex] a^{\dagger} [/itex]. You should be familiar with them, I hope...
     
  7. Nov 30, 2011 #6

    lrf

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    yes, got it now, thank you for the push in the right direction!
     
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