Main Question or Discussion Point
I want to know why the time-derivative acts as though it's Hermitian under conjugation. I have read elsewhere that the time-derivative isn't a true operator in the quantum mechanical sense but I don't understand why that's the case, and if that's the case I still don't understand why [itex] \partial_t^\dagger = \partial_t [/itex]. From what I do understand, in quantum mechanics a operator acts on a state vector which then gives us a new vector. This seems to be the case for the time-derivative, at least in the Schrodinger picture. For spatial derivatives we can use integration by parts to deal with the conjugate operators of the derivatives (i.e. in a scalar product) but since we're not integrating through time I can't find such a method to deal with the time derivative. So can anyone show me explicitly why the time-derivative is Hermitian?