(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For a given total energy E_{0}compute and compare a time average and a phase space average of x^{2}for the harmonic oscillator. The one-dimensional Hamiltonian is

[tex] H = \frac{p^2}{2m}+\frac{m\omega^2}{2}x^2 [/tex]

Reminder: the time average is defined as

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

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

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

2. Relevant equations

3. The attempt at a solutionFirst, for the time average, all I can think of is that for a harmonic oscillator [tex] x = a\cos (\sqrt{k/m}t+\phi ) [/tex]. 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 [tex] \phi [/tex] are given the information in the problem.

Any hints/suggestions would be greatly appreciated.

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# Homework Help: Time average vs. phase space average

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