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Mircocanonical Damped Harmonic Oscillator

  1. Oct 18, 2010 #1
    I am supposed to find the number of mircostates for the following Hamiltonian

    [tex]\
    \begin{equation}
    \Sigma {(q_n+mwp_n)^2}<2mE
    \end{equation}
    [/tex]

    So I am attempting to take the integral as follows

    [tex]\
    \int e^{(q_n+mwp_n)^2} d^{3n}q d^{3n} p

    [tex\]

    I found a solution that tells me

    [tex]\
    \begin{equation}
    \int \exp{(ax^2+bxy+cx^2)^2} d^{3n}x d^{3n}y
    \end{equation}

    [/tex]

    which equals

    [tex]\
    \begin{equation}
    \pi^{m/2}/{det[A]}
    \end{equation}
    [/tex]

    where A is the 2-D matrix
    A=[a b
    b c]

    However, the determinant is zero so this doesn't work. I found this solution at http://srikant.org/thesis/node13.html .
    There is a bit more work shown on the website. My professor assured me that the solution is closed form.
    PS.
    Sorry for the bad latex...
    This is due tomorrow. So if you could help me fast that would be great!
     
    Last edited: Oct 18, 2010
  2. jcsd
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