- #1

pleasehelpmeno

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## Homework Statement

Why does <0|[itex]\frac{1}{(2\pi)^3}[/itex]∫ [itex]\hat{a}^{\dagger}(t,r)[/itex] [itex]\hat{a}(t,r)[/itex] d[itex]^{3}[/itex] [itex]\textbf{k}[/itex] |0> = [itex]\frac{1}{\pi^2}∫[/itex]|β|^2 k^2 dk.

Where [itex]\hat{a} [/itex] and [itex]\hat{a}^{\dagger}[/itex] and its conjugate are bogulobov transformations given by:

[itex]\hat{a}[/itex](t,k) = [itex]\alpha[/itex](t)a(k) + β(t)[itex]b^{\dagger}[/itex](-k).

In the ground state a|0> =0 etc.

I am fairly certain it is some sort of table integral but i am not sure and want to prove it, any help or suggestions would be appreciated. I have taken the conjugate of the aforementioned a and multiplied it though but I don't understand why d[itex]^{3}[/itex] [itex]\textbf{k}[/itex] becomes k^2 dk. and how the pi factor changes, i.e. i get

[itex]\frac{1}{(2\pi)^3}[/itex]∫ [itex]b^{\dagger}[/itex](-k)b(-k)|β|^2 d[itex]^{3}[/itex] [itex]\textbf{k}[/itex]