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Expectation values with annihilation/creation operators

  1. Oct 15, 2013 #1
    1. The problem statement, all variables and given/known data

    Calculate [itex]<i(\hat{a} - \hat{a^{t}})>[/itex]


    2. Relevant equations

    [itex]|\psi > = e^{-\alpha ^{2}/2} \sum \frac{(\alpha e^{i\phi })^n}{\sqrt{n!}} |n>[/itex]

    [itex] \hat{a}|n> = \sqrt{n}|n-1>[/itex]

    I derived:
    [itex] \hat{a}|\psi> = (\alpha e^{i\phi})^{-1}|\psi>[/itex]
    3. The attempt at a solution

    [itex]<i(\hat{a} - \hat{a^{t}})> = <\psi|i\hat{a}-i\hat{a^{t}}|\psi>
    [/itex]

    [itex]
    <\psi|i\hat{a}-i\hat{a^{t}}|\psi> = <\psi|i\hat{a}|\psi> - <\psi|i\hat{a^{t}}|\psi>
    [/itex]

    [itex]
    <\psi|i\hat{a}|\psi> - <\psi|i\hat{a^{t}}|\psi> = <\psi|i\hat{a}|\psi> - <-i\hat{a}\psi|\psi>
    [/itex]

    [itex]
    i(\alpha e^{i\phi})^{-1}<\psi|\psi> + i(\alpha e^{i\phi})^{-1}<\psi|\psi>
    [/itex]

    assuming [itex] \psi [/itex] is normalized,

    [itex]
    <\psi|\psi> = 1
    [/itex]

    [itex]
    <i(\hat{a} - \hat{a^{t}})> = 2i(\alpha e^{i\phi})^{-1}
    [/itex]

    Now, I think I did this correctly.. What I don't understand is the significance of
    [itex]
    <i(\hat{a} - \hat{a^{t}})>
    [/itex]

    Normally with expectation values, you can usually tell if your result is at least reasonable.. I don't understand what this expectation value is telling me, so I can't tell if my result is reasonable. =/

    Any help would be much appreciated!!
     
  2. jcsd
  3. Oct 15, 2013 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It would probably help to write ##\hat{a}, \hat{a}^\dagger## in terms of the position and momentum operators.
     
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