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Quantum expectation values

by misterpickle
Tags: quantum, wave mechanics
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misterpickle
#1
Sep28-10, 04:56 PM
P: 12
I'm not sure why PhysicsForums.com isn't displaying my latex properly so I have attached a PDF of the question.

1. The problem statement, all variables and given/known data
Show that, for a 3D wavepacket,

[itex] \frac{d\langle x^2 \rangle}{dt} $=$ \frac{1}{m}(\langle xp_{x} \rangle+\langle p_{x}x \rangle) [/itex]

3. The attempt at a solution
Through several pages of algebra and taking divergences I have come to the following.

[itex] \frac{d\langle x^2\rangle}{dt} $=$ \frac{ih}{2m}\int\int\int{x^{2}(\psi\frac{\partial^{2}{\psi^{*}}}{\part ial{x^{2}}}-\psi^{*}\frac{\partial^{2}{\psi}}{\partial{x^{2}}})+4x(\psi\frac{\parti al{\psi^{*}}}{\partial{x}}}-\psi^{*}\frac{\partial{\psi}}{\partial{x}})\,dx\,dy\,dz [/itex]

I know the values of <x> and <p> and that my equation must, somehow (if correct), equate to the problem statement. I'm not quite sure if I just don't know how to finish this problem or if my solution thus far is incorrect.
Attached Files
File Type: pdf quantum5.pdf (91.5 KB, 6 views)
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