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