The Schrödinger equation is given by:
$$ i \hbar \partial_t |\Psi\rangle = \hat{H}|\Psi\rangle $$
The right hand side is just an operator acting on a state vector, so we are free to consider its expectation value
$$ \langle \Psi|\hat{H}|\Psi\rangle $$
which would just be the average of the energies we would measure if we measured identically prepared systems many times. But from the equality this should be equal to:
$$ i\hbar \langle \Psi|\partial_t|\Psi\rangle $$
But what does this even mean? I know that the time derivative is somewhat unique in not being an operator in quantum mechanics, so I could see this term not even making sense. But I haven't done anything questionable here: I just hit both sides of the Schrödinger equation with the bra $\langle \Psi |$, and got something reasonable on the right hand side.
Is there a reasonable interpretation of the term on the left?
