Harmonic Oscillator Expectation Values

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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!
 
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
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