Where does the factor (2π)³ come from in QM Fourier transform convention?

[Bapelsin]

Homework Statement

I'm trying to follow the solution to a homework problem in QM, and I don't fully understand this step. Where does the factor $$(2\pi)^3$$ come from?

$$\int d^3re^{-i\vec{p}\cdot\vec{r}}\int{\frac{d^3p'}{(2\pi)^32E_{p'}}\left(a(\vec{p}')e^{-i(E_{p'}t-\vec{p}'\cdot\vec{r})}+a^{\dagger}(\vec{p}')e^{+i(E_{p'}t-\vec{p}'\cdot\vec{r})}\right) =$$
$$=\int{\frac{d^3p'}{(2\pi)^32E_{p'}}\left(a(\vec{p}')e^{-iE_{p'}t}(2\pi)^3\delta(\vec{p}-\vec{p}')+a^{\dagger}(\vec{p}')e^{+iE_{p'}t}(2\pi)^3\delta(\vec{p}+\vec{p}')\right)$$

Homework Equations

See above.

Any help appreciated. Thanks!

 

[phyzguy]

When you do a Fourier transform, which is what the position to momentum space transform is, you have to put a $$2\pi$$ somewhere because this is the period of the complex exponential. There are a variety of conventions for where to put the $$2\pi$$ . Read this: (http://en.wikipedia.org/wiki/Fourier_transform), especially the part about "Other conventions". The convention in QM is not to put it in the exponential, which means it has to go outside as a normalizing factor. Hope this helps.