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Uncertainty of coherent states of QHO

  1. Feb 4, 2012 #1
    1. The problem statement, all variables and given/known data
    What is the uncertainty in x and p of the coherent state
    [itex]|z> = e^{-|z|^2/2} \sum\frac{z^{n}}{\sqrt{n!}} |n>[/itex]


    2. Relevant equations

    ...

    3. The attempt at a solution
    This seems pretty straight-forward to me. You just find the expected value of (let's say) x:
    [itex]<z|x|z>[/itex], and use the relation between <x> and uncertainty. And you use the ladder operator definition of x:
    [itex] \hat{x}= \sqrt{ \frac{\hbar}{m \omega}}( \hat{a} + \hat{a}^{+})[/itex]

    When I work out the math using the fact that [itex] \hat{a}|n> = \sqrt{n}|n-1>[/itex] and [itex] \hat{a^{+}}|n> = \sqrt{n+1}|n+1>[/itex]
    I run in to:
    A*<n-1|n> + B*<n|n+1>
    (Where A and B are summations).

    So my question is: what are <n-1|n> and <n|n+1>? They are energy eigenkets for the QHO But are they orthogonal? And if they are...what do? (Since everything then goes to 0). And if they aren't, what is their inner product?

    Thanks.
     
    Last edited: Feb 4, 2012
  2. jcsd
  3. Feb 5, 2012 #2

    fzero

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    Indeed [itex]\langle n | n\pm 1\rangle =0[/itex]. However, you've oversimplified things. What you should really find is an expression like

    [tex] \sum_{n, n'} \left( A_{nn'} \langle n' | n -1 \rangle + B_{nn'} \langle n' | n +1 \rangle \right). [/tex]

    There are nonvanishing terms appearing in the sum which you can evaluate using [itex]\langle n | n' \rangle = \delta_{nn'}.[/itex]
     
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