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Creation/Annihilation operator help

  1. Oct 20, 2010 #1
    1. The problem statement, all variables and given/known data
    I'm trying to go through something shown in a lecture and the part I'm stuck on is shown here.
    [tex]\frac{\partial}{\partial t}\int^t_0 \mbox{du }e^{i \omega(t-u)}\hat{b}(u)=\hat{b}(t)[/tex]

    [tex]\hat{b}(u)[/tex] is an annihilation operator

    3. The attempt at a solution
    Can someone explain how this step is made? Obviously a delta function has to come in somewhere but I don't know how to do that with a derivative (only another integral).
     
  2. jcsd
  3. Oct 20, 2010 #2
    this follows from a theorem in calculus

    Theorem: Suppose that a function [itex]g:[c,d]\rightarrow [a,b][/itex] (not necessarily a
    onto function) is differentiable, and a function [itex]f:[a,b]\rightarrow \mathcal{R}[/itex]
    is continuous. If

    [tex] H(t)=\int_a^{g(t)} f(u)\, du [/tex]

    with [itex]t\in [c,d][/itex], then H is differentiable and

    [tex] \large H'=(f(g(t))g'(t) [/tex]

    in your example

    [tex]f(u)=e^{i \omega(t-u)}\hat{b}(u)[/tex]

    and [itex]g(t)=t[/itex]

    so

    [tex]H'(t)=(f(g(t))g'(t)[/tex]

    [tex]g'(t)=1[/tex]

    so

    [tex]H'(t)=f(t)[/tex]

    which gives what you are seeking
     
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