Webpage title: How to Calculate <x> for a One-Dimensional Harmonic Oscillator

[spdf13]

Here's the problem:

A one dimensional harmonic oscillator has mass m and frequency w. A time dependent state psi(t) is given at t=0 by:

psi(0)=1/sqrt(2s)*sum(n=N-s,n=N+s) In>
where In> are the number eigenstates and N>>s>>1.

Calculate <x>. Show it varies sinusoidally; find the frequency and amplitude. Compare the amlitude and frequency to the corresponding values of a classical harmonic oscillator.


Here's how I proceeded:

<x>=(1/2s) (some constants)*sum(n=N-s,n=N+s)*sum(m=N-s,m=N+s) <n I (a+a') I m> Exp[i(Em-En)t/h]

(note a' is "a dagger")

=(1/2s) (some constants)*sum(n=N-s,n=N+s)*sum(m=N-s,m=N+s) {sqrt(m) <n I m-1> + sqrt(m+1) <n I m+1>} Exp[i(En-Em)t/h]

(note <n I m-1>=delta(n,m-1) and <n I m+1>=delta(n,m+1).

=(1/2s) (some constants)*sum(n=N-s,n=N+s) {sqrt(m+1) Exp[-iwt] + sqrt(m) Exp[iwt]}

This is where I get stuck. I don't know if I'm supposed to make some approximation since N>>s>>1, and approximate the term in the {} as sqrt(m) cos (wt), or if I'm just completely wrong from the start. If someone can help, I'd really appreciate it.

 

[quantumdude]

Originally posted by spdf13
This is where I get stuck. I don't know if I'm supposed to make some approximation since N>>s>>1, and approximate the term in the {} as sqrt(m) cos (wt), or if I'm just completely wrong from the start. If someone can help, I'd really appreciate it.

If N>>s, then all the numbers in the range of the index of summation [N-s,N+s] are approximately equal to N. That is, it is (approximately) as though you only have a single term.

That single term is going to be of the form:

sqrt(N+1)exp(-i&omega;t)+sqrt(N)exp(i&omega;t)

The thing that is screwing this up from being a sinusoid is the fact that the two terms have different coefficients. Now is the time to invoke N>>1. Do that to approximate as follows:

sqrt(N)[exp(-i&omega;t)+exp(i&omega;t)]=2sqrt(N)cos(&omega;t)

 

[spdf13]

I think you're right. Thanks for the help.