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Homework Help: Help with Power Spectral Density Function derivation

  1. Mar 29, 2007 #1
    1. The problem statement, all variables and given/known data
    Given a two-level atom with transition frequency [tex]{ \omega }_{ ji } \equiv { \omega }_{a}[/tex] and spontaneous decay rate [tex]\gamma[/tex], we are asked to find an expression for the "power spectral density function" [tex]S(\omega)[/tex], in terms of [tex]\omega, {\omega}_{a}[/tex], and [tex]\gamma[/tex].

    2. The attempt at a solution

    Of course, it should be normalized to 1,
    [tex] \int_{- \infty }^{ \infty } d \omega S(\omega) = 1[/tex]

    I believe that the PSD function is just a Fourier Transform of the decay rate, right? If so,
    [tex]S(\omega)= \frac{ 1}{ \sqrt[ ]{ 2 \pi } } \int_{ - \infty }^{ \infty } \gamma (t) { e}^{ - \imath \omega t} dt[/tex]

    And then I would just need to double-check that it's normalized.

    Thoughts on this?
     
  2. jcsd
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