Time evolution operator in QM

Question

I am reading a introduction to quantum mechanics right now. There is a part about the time evolution operator:

\begin{align*} i\hbar \partial_t \,\psi(\vec r, t) = \hat H(t)\, \psi(\vec r,t) \end{align*} is the time dependent Schrödinger-equation. If we assume that for each $\psi(\vec r, t_0)$ there is a unique solution $\psi(\vec r, t)$, then we can define an operator

$$U(t,t_0): \mathcal H \to \mathcal H,\, \psi(\vec r, t_0) \mapsto \psi(\vec r, t)$$

This operator is linear, since the Schrödinger equation is linear and it is unitary, since $\partial_t \langle\psi(\vec r, t)| \psi(\vec r, t)\rangle = 0$. I am totally happy with that. I can also accept, that $U(t,t_0) = e^{-i(t-t_0)\hat H/ \hbar}$, if $\hat H$ is time independent, where $e^{-i(t-t_0)\hat H/ \hbar}$ is defined over how it acts on the eigenvectors of $\hat H$.

But I have no idea, what the next sentence in my book means, and there is no good explanation. Is says there:

The differential equation, together with the initial condition ($U(t_0,t_0) = Id$) is equivalent to the integral equation: \begin{align*} U(t,t_0) = 1 - \frac{i}{\hbar } \int_{t_0}^t ds\, \hat H(s) U(s,t_0) \end{align*}

So my problem is basically, I don't understand this at all :/. How can I integrate operators, what does that even mean? Are there any good examples, where this integral makes sense? This is probably a really stupid question, but I am happy if someone could spare two minutes to help me.

Answer 1

This is a very good question, but a mathematical one. The expression you quoted from the book is the part-"summation" of the Dyson expansion of the unitary evolution operator. To quote from Reed and Simon, theorem X.69 (Vol. II, p. 282)

Let $t\mapsto H(t)$ a strongly continuous map of $\mathbb{R}$ into the bounded self-adjoint operators on a Hilbert space $\mathcal{H}$. Then there is a unitary propagator on $\mathcal{H}$ so that, for all $\psi\in\mathcal{H}$, $$\phi_s (t) = U(t,s) \psi $$ satisfies $$ \frac{d}{dt} \phi_s (t) = -i H(t) \phi_s (t) \ ; \ \phi_s (s) = \psi$$

The proof starts by explicitely exhibiting the unitary propagator as

$$ U(t,s) \phi = 1 +\sum_{n=1}^{\infty} (-i)^n \int_{s}^{t} \int_{s}^{t_1} ... \int_{s}^{t_n} H(t_1)... H(t_n) \phi \ dt_n ... \ dt_1 $$

What the book did is just ~resum~ the infinite expression to the right of $ H(t_1) $ into another U (the minus vs. plus sign after the unit vector comes from the different convention for evolution). Now we have no longer an integral of a product of operators, but of Hilbert space-valued functions. This is just an iteration of a Bochner-type integral.