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Volume of N dimensional phase space

  1. Nov 6, 2006 #1
    Hi guys,

    I have a volume integral in 3D phase space that looks like:

    [tex]\int \frac{4\pi p^2 dp}{h^3}

    [/tex]

    Now, I want to generalize to N dimensions. How does this look:


    [tex]\int \frac{\frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}p^N dp}{N!h^{3N}}
    [/tex]

    Essentially, I've changed the 4 pi (which I think is the volume of a 2 sphere) into a generalized volume for an N sphere, and made some changes in the powers.

    how does this look?
     
  2. jcsd
  3. Nov 7, 2006 #2

    dextercioby

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    Multiplied by p^{N-1} (with N the dimension of the momentum space) you need to have the area of the N-1 sphere.

    What is with the N factorial ?

    Daniel.
     
  4. Nov 10, 2006 #3
    I'm not sure I understand your answer Daniel.

    Do you mean I need to multiply my answer by p^{N-1}?
     
  5. Nov 13, 2006 #4

    dextercioby

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    It should be something like

    [tex] \int_{\Omega} dV =\int_{0}^{\infty} p^{n-1} dp\int_{\partial \Omega} dS_{\Omega} [/tex]

    The second integral is the integral giving the area of the "n-1 sphere embedded in R^{n}.

    Daniel.
     
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