Calculating Probability of Simple Harmonic Oscillator in Phase Space

[quasar987]

Consider a simple harmonic oscillation in 1 dimension: x(t)=Acos(wt+k). If the energy 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!

 

[Jhero]

Thank you for your post and for considering the simple harmonic oscillator in 1 dimension. The probability of the oscillator being between x and x+dx can be calculated by looking at the phase space volume for the given energy interval.

To start, let's consider the phase space volume for the oscillator when it is at energy E. As you mentioned, this volume is an ellipse with semi-axes A and mwA. The total area of this ellipse is given by πA√(A^2-x^2). Now, when the energy is varied by a small amount δE, the semi-axes of the ellipse will change by δA and δ(mwA). Therefore, the new area of the ellipse will be given by π(A+δA)√[(A+δA)^2-x^2]. Similarly, the phase space volume for the oscillator at energy E+δE will be given by π(A+δA)√[(A+δA)^2-x^2].

Now, to find the probability of the oscillator being between x and x+dx, we need to calculate the difference in area between the two ellipses at energies E and E+δE. This difference in area will be given by:

π(A+δA)√[(A+δA)^2-x^2] - πA√(A^2-x^2)

Expanding the terms and keeping only the first-order terms in δA, we get:

π(√(A^2-x^2) + δA/2) - πA√(A^2-x^2)

Simplifying, we get:

πA√(A^2-x^2)δA/2

Now, to find the probability of the oscillator being between x and x+dx, we need to divide this difference in area by the total volume of the phase space, which is given by the area of an ellipse with semi-axes (A+δA) and (mwA+δ(mwA)). The total area of this ellipse is given by:

π(A+δA)√[(A+δA)^2-(mwA+δ(mwA))^2]

Again, expanding the terms and keeping only the first-order terms in δA, we get:

π(√(A^2-x^2) + δA/2)