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Q.M. harmonic oscillator 
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#1
Feb104, 12:24 AM

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=Ns,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=Ns,n=N+s)*sum(m=Ns,m=N+s) <n I (a+a') I m> Exp[i(EmEn)t/h] (note a' is "a dagger") =(1/2s) (some constants)*sum(n=Ns,n=N+s)*sum(m=Ns,m=N+s) {sqrt(m) <n I m1> + sqrt(m+1) <n I m+1>} Exp[i(EnEm)t/h] (note <n I m1>=delta(n,m1) and <n I m+1>=delta(n,m+1). =(1/2s) (some constants)*sum(n=Ns,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. 


#2
Feb304, 03:05 PM

Emeritus
Sci Advisor
PF Gold
P: 5,532

That single term is going to be of the form: sqrt(N+1)exp(iωt)+sqrt(N)exp(iωt) 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(iωt)+exp(iωt)]=2sqrt(N)cos(ωt) 


#3
Feb404, 07:13 PM

P: 10

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



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