Time average vs. phase space average

[iomtt6076]

1. The problem statement, all variables and given/known data
For a given total energy E0 compute and compare a time average and a phase space average of x2 for the harmonic oscillator. The one-dimensional Hamiltonian is

H = \frac{p^2}{2m}+\frac{m\omega^2}{2}x^2

Reminder: the time average is defined as

\langle x^2\rangle =\frac{1}{t}\int_0^t x^2\tau\,d\tau

we will be mostly interested in the long time limit. The phase space average is

\overline{x}^2=\frac{\int\delta (E_0-H)x^2\,dx\,dp}{\int\delta (E_0-H)\,dx\,dp}


2. Relevant equations



3. The attempt at a solution First, for the time average, all I can think of is that for a harmonic oscillator x = a\cos (\sqrt{k/m}t+\phi ) . I can then substitute this in the given integral for time average, which I can then evaluate. The problem is that I don't know what a and \phi are given the information in the problem.

Any hints/suggestions would be greatly appreciated.

[Maxim Zh]

The amplitude a can be calclated since the energy E0 is given.
The initial phase \phi doesn't affect the average x2.

By the way, the definition of the time average has wrong dimension. Is it a typing error?

[iomtt6076]

Thank you for the reply.

The amplitude a can be calclated since the energy E0 is given.

Yes, I've now found out that E = (1/2)m\omega^2 A^2.

By the way, the definition of the time average has wrong dimension. Is it a typing error?

Well, I've copied it exactly the way it shows up on the problem set, so maybe the professor made a typo?

[Maxim Zh]

The common time average definition is:


<f(t)> = \frac{1}{t} \int_0^t f(\tau)\,d\tau.


May be the brackets around the \tau are missed in your definition?

[iomtt6076]

Okay, I see what you're saying; I agree it should be

\frac{1}{t}\int_0^t x^2(\tau)\,d\tau

Thanks, I got it now.