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Creation/Annihilation operator help

  • Thread starter cahill8
  • Start date
  • #1
31
0

Homework Statement


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

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).
 

Answers and Replies

  • #2
911
18
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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