zje
Mar7-11, 07:13 PM
1. The problem statement, all variables and given/known data
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]
2. Relevant equations
d<x>/dt = <p/m>
3. 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.
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]
2. Relevant equations
d<x>/dt = <p/m>
3. 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.