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 [tex]t=0,[tex] corresponding to the eigenvalue $$a$$.

Attempt Solution
We know that

$$\frac{\partial \langle A \rangle}{dt} = \langle \frac{i}{\hbar} [\hat{H}, \hat{A}] + \frac{\partial \hat{A}}{\partial t} \rangle = \langle \frac{i c}{\hbar} + 0 \rangle = \frac{i c}{\hbar}$$.

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, $$[\hat{H}, \hat{A}] = c$$.