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QM: need help with intermediate step

  1. Mar 14, 2010 #1
    1. The problem statement, all variables and given/known data

    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 [tex](2\pi)^3[/tex] come from?

    [tex]\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) = [/tex]
    [tex]=\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)[/tex]

    2. Relevant equations

    See above.

    Any help appreciated. Thanks!
     
  2. jcsd
  3. Mar 14, 2010 #2

    phyzguy

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    Science Advisor

    When you do a Fourier transform, which is what the position to momentum space transform is, you have to put a [tex]2\pi[/tex] somewhere because this is the period of the complex exponential. There are a variety of conventions for where to put the [tex]2\pi[/tex] . 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.
     
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