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How to get the prefactor for the Planck Distribution

  1. Jun 30, 2008 #1
    So I have a question on the Planckian (from one of my PhD qualifier practice problems). I'm trying to derive the Planck distribution from the Bose-Einstein distribution, which is,

    [tex]f(\epsilon) = \dfrac{1}{e^{\frac{\epsilon - \mu}{k_B T}} - 1}[/tex]

    Now I know that I can set the chemical potential equal to zero, and let [tex]\epsilon = \hbar \omega = h \nu [/tex] to get,

    [tex]f(\epsilon) = \dfrac{1}{e^{\frac{h \nu}{k_B T}} - 1}[/tex]

    However, this is only the distribution function. There's some sort of prefactor that I need in order to turn it into an energy density. My question is: how do I derive this prefactor? I know it must have something to do with deriving the density of states for a bose gas, since the total occupation is,

    [tex] \int_0 ^\infty g(\epsilon) f(\epsilon) d\epsilon[/tex]

    I'm also fairly certain that I need to somehow express energy in terms of the wave vector and convert to spherical coordinates. But how exactly does this work? I'm still not sure how the idea of a density of states applies to a Bose gas, since the Pauli principle doesn't apply. I know I could just look up the Planckian, but I'd like to understand this so I could reproduce it on a qualifying exam (should the need arise). Help would be appreciated, thanks.
     
  2. jcsd
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