Harmonic Oscillator Expectation Values

AI Thread Summary
The discussion centers on calculating the time-dependent expectation values <x> and <p> for a particle in a harmonic oscillator potential using a given wave packet. Participants express uncertainty about how to begin the calculations, particularly regarding the use of Ehrenfest's theorem and the raising/lowering operators. There is a consensus that calculating <x> first may simplify finding <p>, and some suggest using integration techniques, including substitution, to evaluate the integral for <x>. The complexity of the integral leads to questions about the best approach, with participants exploring different methods to simplify the calculations. Overall, the thread highlights the challenges of applying quantum mechanics concepts to this problem.
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Homework Statement


A particle of mass m that is confined to a harmonic oscillator potential V(x) = \frac{1}{2} m \omega^2 x^2 is described by a wave packet having the probability density,

|\Psi (x,t) |^2 = \left(\frac{m\omega}{\pi\hbar} \right )^{1/2}\textrm{exp}\left[-\frac{mw}{\hbar}(x - a\textrm{cos}\omega t)^2 \right ]

where \omega is a constant angular frequency and a is a positive real constant.

Calculate the time-dependent expectation values <x> and <p>. [Hint: Use Ehrenfest's theorem]

Homework Equations


d<x>/dt = <p/m>

The Attempt at a Solution


I'm not quite sure where to begin attacking this problem. I feel that if I can calculate <x>, then <p> should be easy given the equation above. I was thinking of trying the raising/lowering operators. Can I assume the particle is in the ground state since the only Hermite polynomial in \Psi is H_0 = 1? Is there an easier approach to this problem? I tried just calculating <x> using
\int \limits_{-\infty}^{\infty} \Psi(x,t)x\Psi^*(x,t)\textrm{d}x
but that was getting out of control fairly quickly.
 
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That integral is pretty straightforward to do with a simple substitution. What did you try?
 
I tried expanding the exponent term and couldn't find a way to integrate by parts that made it simpler. Looking at it again, would the simple substitution be u=(x - a*cos[wt]), du=dx?

Thanks for your reply!
 
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