Register to reply

Q.M. harmonic oscillator

by spdf13
Tags: harmonic, oscillator
Share this thread:
spdf13
#1
Feb1-04, 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=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 appriciate it.
Phys.Org News Partner Science news on Phys.org
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
Tom Mattson
#2
Feb3-04, 03:05 PM
Emeritus
Sci Advisor
PF Gold
Tom Mattson's Avatar
P: 5,532
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)
spdf13
#3
Feb4-04, 07:13 PM
P: 10
I think you're right. Thanks for the help.


Register to reply

Related Discussions
Harmonic Oscillator Introductory Physics Homework 12
Harmonic Oscillator Advanced Physics Homework 2
Harmonic oscillator Introductory Physics Homework 3
Harmonic Oscillator Introductory Physics Homework 4
Harmonic oscillator Quantum Physics 3