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Consider a simple harmonic oscillation in 1 dimension: x(t)=Acos(wt+k). If the energy of this oscillator is btw E and E+[itex]\delta E[/itex], show that the probability the the position of the oscillator is btw x and x+dx is given by
[tex]P(x)dx=\frac{1}{\pi}\frac{dx}{\sqrt{A^2-x^2}}[/tex]
Hint: calculate the volume in phase space when the energie is btw E and E+[itex]\delta E[/itex] 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+[itex]\delta E[/itex] by A+[itex]\delta A[/itex] and (m+[itex]\delta m[/itex])(w+[itex]\delta w[/itex])(A+[itex]\delta A[/itex]) 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!
[tex]P(x)dx=\frac{1}{\pi}\frac{dx}{\sqrt{A^2-x^2}}[/tex]
Hint: calculate the volume in phase space when the energie is btw E and E+[itex]\delta E[/itex] 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+[itex]\delta E[/itex] by A+[itex]\delta A[/itex] and (m+[itex]\delta m[/itex])(w+[itex]\delta w[/itex])(A+[itex]\delta A[/itex]) 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!
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