Expressing expectation values of a particle moving in a periodic potential

[RJLiberator]

Homework Statement

A particle moving in a periodic potential has one-dimensional dynamics according to a Hamiltonian $\hat H = \hat p_x^2/2m+V_0(1-cos(\hat x))$

a) Express $\frac{d <\hat x>}{dt}$ in terms of $<\hat p_x>$.
b) Express $\frac{d <\hat p_x>}{dt}$ in terms of $<sin(\hat x)>$.
c) Write a time-dependent Schrödinger equation for this problem in real space.

Homework Equations

The Attempt at a Solution

Let's start with a. I am highly confused here, but there seems to be various routes I can go.

Usually I would calculate the expectation value <x> from a wave function. Can I still do this here with the Hamiltonian? Just straight up integrate H*xH over all space and then take that derivative and find a way to express it in terms of <px> (thus I'd have to take the expectation value for the momentum of the Hamiltonian?

I've been trying some things but running into a wall with this method.
Any tips on how to start off this problem would be great, I can then work on it and get back to this thread.

 

[RedDelicious]

No you definitely do not want to try to integrate the operators.

Ehrenfest's theorem gives you a very useful set of relations that you can use to solve this problem.