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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 :(.

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# Harmonic oscillator phase space integral

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