The equation below is the time-dependent Schroedinger equation for the time evolution operator $U$.
$$i\hbar \frac{\partial}{\partial t}U(t,t_0) = HU(t,t_0).$$
According to my prof. this can be solved as a differential equation:
(my attempt) Assume both U and H are dependent on $t$. Let $ \frac{\partial U}{\partial t} = U'$ ($t_0$ is not a variable). $$ U' = -\frac{i}{\hbar} HU $$ $$\frac{U'}{U} = -\frac{i}{\hbar} H $$ $$ \int \frac{U'}{U} dt = -\frac{i}{\hbar} \int H dt$$ Use standard integral $$\ln |U| = -\frac{i}{\hbar} \int H dt $$
This works provided the unitary operator can be treated as a function. Can someone explain:
- whether it can be and why it can.
- if it can't, why it can't, and how the actual proof should go.
