# SHO in phase space

1. Sep 20, 2006

### quasar987

Consider a simple harmonic oscillation in 1 dimension: x(t)=Acos(wt+k). If the enegy of this oscillator is btw E and E+$\delta E$, show that the probability the the position of the oscillator is btw x and x+dx is given by

$$P(x)dx=\frac{1}{\pi}\frac{dx}{\sqrt{A^2-x^2}}$$

Hint: calculate the volume in phase space when the energie is btw E and E+$\delta E$ and when the position is btw x and x+dx, and compare this volume with the total volume when the oscillator is anywhere but in the same energy interval.

For a given energy E, it's easy to see that the path of the oscillator in phase space is an ellipse of semi axes A and mwA.

I could write the semi axes of the ellipse representing the energy E and E+$\delta E$ by A+$\delta A$ and (m+$\delta m$)(w+$\delta w$)(A+$\delta A$) but I fear that would not be very practical... :/

I could then find an expression for the difference in area of the 2 ellipses as a function of x, differentiate that, multiply by dx and finally divide by the total difference in area of the 2 ellipses and I would be done.

Actually I already tried that with the case where only A was "allowed" to vary and not m or w, and it did not work. So I'm very much open to any suggestion!

Last edited: Sep 20, 2006