# Q.M. harmonic oscillator

by spdf13
Tags: harmonic, oscillator
 P: 10 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 . 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: =(1/2s) (some constants)*sum(n=N-s,n=N+s)*sum(m=N-s,m=N+s) 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) + sqrt(m+1) } Exp[i(En-Em)t/h] (note =delta(n,m-1) and =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 appriciate it.
Emeritus
PF Gold
P: 5,539
 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 appriciate 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)
 P: 10 I think you're right. Thanks for the help.

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