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Exponential of creation/annihilation operators

  1. Feb 1, 2010 #1

    I found on this webpage:


    page 1, on the bottom


    [tex] e^{\phi^* a } f(a^{\dagger} , a ) = f(a^{\dagger} + \phi^*, a) e^{\phi^* a }[/tex]

    I have tried to prove this, writing both as taylor series, but the problem is to understand the "hint" :(
    Last edited: Feb 1, 2010
  2. jcsd
  3. Feb 1, 2010 #2


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    Science Advisor

    For the benefit of other readers, this pdf performs a quick derivation of the
    coherent state path integral (in which a coherent state resolution of unity is used
    between time slices rather than the usual resolutions using momentum and position

    The "hint" is

    which refers to this trick:

    [a, g(a^\dagger)] ~=~ \frac{\partial g(a^\dagger)}{\partial a^\dagger}

    as may be shown by induction (perhaps modulo a sign and/or some
    factors of i and [tex]\hbar[/tex], depending on one's conventions).

    Start with [tex]g(a^\dagger) = a^\dagger[/tex], then progress to
    [tex]g(a^\dagger) = (a^\dagger)^2[/tex], to see the pattern,
    then prove it for [tex]g(a^\dagger) = (a^\dagger)^n[/tex] by induction.
    Then you can extend the result to any well-behaved analytic function.)

  4. Feb 1, 2010 #3
    ah yes now its clear! Thank you

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