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Harmonic oscillator phase space integral

  1. Jul 2, 2010 #1
    Hi all,

    I am having trouble with a certain integral, which I got from Quantum Physics by Le Bellac:
    [tex]\int dxdp\;\delta\left( E - \frac{p^2}{2m} - \frac{1}{2}m\omega^2x^2 \right) f(E) [/tex]
    The answer to this integral should be [tex] 2\pi / \omega\; f(E) [/tex].

    My attempts so far:
    This integral is basically a line integral around the perimeter of the (in general) ellipse traced out in phase space. Therefore I tried parameterizing the ellipse using x = acos(t) and p = bsin(t). From here, I've tried two different things:
    1) Even though the original integral is 2D, I rewrite both dx and dy to dt, and then finally arrive at (since now the argument of the dirac delta is always equal to 0, along the curve):
    [tex] -ab \int_0^{2\pi} sin(t)cos(t)dt = 0 [/tex]
    2) Rewrite using the arc length segment
    [tex] ds = \sqrt{ x'^2 + p'^2 } dt [/tex]
    this results in a complicated integral, which I am unable to solve. Also, by trying to simply use approximations for the perimeter of an ellipse turns out in other answers :(.
    Last edited: Jul 2, 2010
  2. jcsd
  3. Jul 2, 2010 #2


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    There's a property of the Dirac Delta distribution that is worth remembering, and I think will help you greatly with this integral:

    [tex]\delta(f(x))=\sum_i \frac{\delta(x-x_i)}{|f'(x_i)|}[/tex]

    Where [itex]f(x)[/itex] is any continuously differentiable function, with roots (assumed to be simple) [itex]x_i[/itex].
  4. Jul 3, 2010 #3
    Thanks a lot gabbagabbahey! I have solved the problem :)!
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