What are the Expectations for Momentum and Energy in Quantum Mechanics?

[Grand]

Homework Statement

We have a particle of mass m and potential energy V(x), wavefn $$\psi(x,t)$$
What are the expectations for the momentum $$p_x$$, $$p_x^2$$ and the energy.

 

[AlexChandler]

We have momentum and energy operators that we can use here. The momentum operator is $$\hat{p}$$ and the energy operator is $$\hat{E}$$ where
$$\hat{p} = \hbar / i \frac{\partial }{\partial x}$$
and
$$\hat{E} = i \hbar \frac{\partial }{\partial t}$$

For any function $$g(x)$$ we have the expectation value as:

$$\langle g(x) \rangle = \int^{\infty}_{-\infty} \psi^\star (x,t) g(x) \psi (x,t)\,dx$$

Then we have

$$\langle p \rangle = \hbar / i \int^\infty_{-\infty} \psi^\star (x,t) \frac{\partial }{\partial x} \psi (x,t)\,dx$$

and

$$\langle E \rangle = i \hbar \int^\infty_{-\infty} \psi^\star (x,t) \frac{\partial }{\partial t} \psi (x,t)\,dx$$

and also

$$\langle p^2 \rangle = \hbar^2 \int^\infty_{-\infty} \psi^\star (x,t) \frac{\partial^2 }{\partial x^2} \psi (x,t)\,dx$$