Time average vs. phase space average

[iomtt6076]

Homework Statement

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

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

Homework Equations

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.