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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:

  1. whether it can be and why it can.
  2. if it can't, why it can't, and how the actual proof should go.
Qmechanic
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Abe Jacob
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    Hello Abe Jacob. This StackExchange article discusses the Solution of the time dependent Schrödinger equation in a very general framework https://physics.stackexchange.com/q/103503/273635 maybe this could give you a bit of an overview. – jan0155 Oct 08 '21 at 14:17
  • U is a matrix, not a number. You can't write something like $\frac{U'}{U}$, It's ambiguous. It should be either $U^{-1}U'$ or $U'U^{-1}$ – KP99 Oct 08 '21 at 15:37
  • @KP99 I know, but I couldn't see another way to solve the ODE. The last line of writing above my 2 questions addresses this. – Abe Jacob Oct 14 '21 at 12:12

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