Solve Griffith's Problem 1.12: Ehrenfest's Theorem

[stunner5000pt]

Homework Statement

Griffith's problem 1.12
Calculate $d\left<p\right>/dt.$

Answer $$\frac{d\left<p\right>}{dt} = \left<\frac{dV}{dx}\right>$$

2. The attempt at a solution

so we know that

$$\left<p\right> = -i\hbar \int \left(\Psi^* \frac{d\Psi}{dx}\right) dx$$

so then

$$\frac{d\left<p\right>}{dt} = -i\hbar \int \left( \frac{\partial\Psi^*}{\partial t} \frac{\partial\Psi}{\partial x} + \Psi^* \frac{\partial^2 \Psi}{\partial t \partial x} \right) dx$$

im not quite sure if one can simplify this further ... i mean we can't integrate wrt x because all the terms in the integrand have x dependance... don't they?? Should i intergate by parts to proceed??

I think a couple of extra terms would be required, no?

Thanks for the help!

 

[da_willem]

Now there is this famous equation for $$\frac{\partial \Psi}{\partial t}$$, what's it called again...

 

[stunner5000pt]

Now there is this famous equation for $$\frac{\partial \Psi}{\partial t}$$, what's it called again...

shhhhhhhhh you

i got the required answer anyway

thansk for your help