thanks all! It did turn out to be a terminology thing. I had thought that the TDSE meant that the Hamiltonian must be time dependent, but in the context that the confusion arose we are talking about the state vector evolving in time, with the phase factor mentioned.
Actually I'm not sure why...
Some notes I have been given have some kind of typo in them I think. There is a suggestion that time dependent schrodinger equations can be solved by expanding initial states in terms of stationary states, and that seemed like a problem. I wanted to make sure my intuition is correct, but I...