How to Find the Expectation Value of an Operator with a Constant Commutator?

[Domnu]

Problem
Consider an operator \hat{A} whose commutator with the Hamiltonian \hat{H} is the constant c... ie [\hat{H}, \hat{A}] = c. Find \langle A \rangle at t > 0, given that the system is in a normalized eigenstate of \hat{A} at t=0,corresponding to the eigenvalue a.<br /> <br /> <b>Attempt Solution</b><br /> We know that<br /> <br /> \frac{\partial \langle A \rangle}{dt} = \langle \frac{i}{\hbar} [\hat{H}, \hat{A}] + \frac{\partial \hat{A}}{\partial t} \rangle = \langle \frac{i c}{\hbar} + 0 \rangle = \frac{i c}{\hbar}.<br /> <br /> Is this correct? (I'm just confirming that d\hat{A}/dt = 0 since we're in an eigenstate of \hat{A}). But this means that the expected value of A is complex... clearly, \hat{A} is not Hermitian then, right?

 

[Dick]

None of this is very clear. What's the exact problem? ic/hbar may be real if c is complex. It's certainly true that the expectation values of a hermitian operator are real. That I'll give you, clearly.

 

[Domnu]

Hmm... I couldn't edit my previous post, so here's the new problem... (slight LaTeX error in previous post):

Problem
Consider an operator \hat{A} whose commutator with the Hamiltonian \hat{H} is the constant c... ie [\hat{H}, \hat{A}] = c. Find \langle A \rangle at t > 0, given that the system is in a normalized eigenstate of \hat{A} at t=0, corresponding to the eigenvalue a.

This is the correct problem. Notation-wise, we have that \langle A \rangle denotes the expected value of the operator \hat{A} operating upon some wavefunction \psi... here, we know that for our wavefunction, [\hat{H}, \hat{A}] = c.