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Expectation values of the quantum harmonic oscillator

  1. Nov 30, 2016 #1
    1. The problem statement, all variables and given/known data

    Show the mean position and momentum of a particle in a QHO in the state ψγ to be:

    <x> = sqrt(2ħ/mω) Re(γ)
    <p> = sqrt (2ħmω) Im(γ)

    2. Relevant equations

    ##\psi_{\gamma} (x) = Dexp((-\frac{mw(x-<x>)^2}{2\hbar})+\frac{i<p>(x-<x>)}{ħ})##


    3. The attempt at a solution

    I put ψγ into the equation

    <p> = ∫[ψ(x,t)]* (-iħψ'(x)) dx

    Which gave me

    (-(mω(x-<x>)/ħ)+((<p>)(i/ħ))*ψγ which I can't help but feel is taking me further away from what I'm looking for. Are there easier alternative route to doing this?

    Thank you in advance
     
    Last edited: Nov 30, 2016
  2. jcsd
  3. Nov 30, 2016 #2

    PeroK

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    How is ##\psi_{\gamma}## defined? That expression has no ##\gamma## in it, but it does have the expected values of ##x## and ##p##.

    It's hard to read your latex. I'll post some you can copy in a minute.
     
  4. Nov 30, 2016 #3

    PeroK

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    Some latex:

    ##\psi_0 = (\frac{m \omega}{\pi \hbar})^{\frac14}exp(-\frac{mwx^2}{2\hbar})##

    ##\psi_{\gamma} = ?##

    If you reply to this post, it will give you something to cut and paste.
     
  5. Nov 30, 2016 #4
    I've edited it accordingly
     
  6. Nov 30, 2016 #5

    PeroK

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    What's ##\gamma##?
     
  7. Nov 30, 2016 #6
    i haven't got that information, ψγ represents coherent states.
     
  8. Nov 30, 2016 #7

    PeroK

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    Given that the required answer (for the expectation values) depends on ##\gamma##, it must be a parameter in the state. You need to check the question.
     
  9. Nov 30, 2016 #8
    Just realised I have missed a little bit out,

    γ is a complex parameter and
    a_ψγ(x) = γψγ(x)

    That's 100% all the information I have now
     
  10. Nov 30, 2016 #9

    PeroK

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    That's entirely different. That means that ##\psi_{\gamma}## is an eigenstate of the lowering operator, corresponding to eigenvalue ##\gamma##.

    The expression you quoted for ##\psi_{\gamma}## before makes no sense.

    Hint: can you express the position and momentum operators in terms of the raising and lowering operators?

    HInt #2: I suspect you can do this using Linear Algebra, without resorting to integration.
     
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