Quantum expectation values

misterpickle

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,

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

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

[latex] \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 [/latex]

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

quantum5.pdf (91.5 KB, 6 views)