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Homework Help: Calculate expectation value of H'

  1. Aug 4, 2010 #1
    1. The problem statement, all variables and given/known data

    Calculate the expectation value of [tex]\hat{H}'[/tex] in the state [tex]\psi(x,t=0)[/tex].

    [tex]\hat{H}'=k(\hat{x}\hat{p}+\hat{p}\hat{x})[/tex]

    [tex]\psi(x,t=0)=A(\sqrt{3}i\varphi_{1}(x)+\varphi_{3}(x))[/tex],
    where [itex]A=\frac{1}{2}[/itex]

    2. Relevant equations



    3. The attempt at a solution

    I know it's found by:
    [tex]\left\langle\psi,\hat{H}'\psi\right\rangle[/tex],
    but it's been so long since I calculated this, so I'm not quite sure how to tackle/calculate it to be honest.

    So I hope you might be able to give me some pointers.


    Regards.
     
  2. jcsd
  3. Aug 4, 2010 #2

    diazona

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    Homework Helper

    To start with, you can see that the state ψ is given in terms of two basis states φ1 and φ3. What are the properties of those basis states? Specifically, do you know the expectation values of certain operators in those basis states?
     
  4. Aug 4, 2010 #3
    Ehhh, should I know any ? If so, I'm kinda blank. As I said, it's been a while since I did QM calculations, so I'm not really in the game atm.
     
  5. Aug 4, 2010 #4

    diazona

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    Homework Helper

    You have to be given some information about the states φ1 and φ3. It's impossible to do the problem if you don't know what those states are.

    Typically, φn is defined to be an energy eigenstate (i.e. an eigenstate of the Hamiltonian), and the number n is assigned so that either φ1 or φ0 is the eigenstate with the lowest energy. Maybe you're supposed to assume that. I guess if you don't have any other information, try making that assumption.

    Anyway, here's something you can do, even without knowing about the basis states: you know (correctly) that the expectation value is computed as
    [tex]\langle\psi\vert H'\psi\rangle[/tex]
    Plug in the definition you're given for ψ and use the distributive property. You should be able to reduce it to a sum of terms of the form
    [tex]\langle\varphi_m\vert H'\varphi_n\rangle[/tex]
    where m and n are integers, either 1 or 3.
     
  6. Aug 5, 2010 #5
    You're right about them being energy eigenstates. Just forgot to mention.
    And I think I got it now :)

    Thank you very much.
     
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