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zje
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Homework Statement
A particle of mass m that is confined to a harmonic oscillator potential [tex]V(x) = \frac{1}{2} m \omega^2 x^2[/tex] is described by a wave packet having the probability density,
[tex]|\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 ][/tex]
where [tex]\omega[/tex] 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 [tex]\Psi[/tex] is [tex]H_0[/tex] = 1? Is there an easier approach to this problem? I tried just calculating <x> using
[tex]\int \limits_{-\infty}^{\infty} \Psi(x,t)x\Psi^*(x,t)\textrm{d}x[/tex]
but that was getting out of control fairly quickly.
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