###### From A. Messiah’s Quantum Mechanics p.722, Dover (1999)

Knowing that our quantum system is in a certain dynamical state at time $t_0$ we wish to determine its state at a later time $t$. The problem, therefore, is to determine as exactly as possible the operator $U(t,t_0)$ describing the evolution in time of the dynamical states of the system in the Schrödinger “representation” (picture). This operator $U(t,t_0)$ is completely determined once the Hamiltonian $H(t)$ of the system is given.

In the Schrödinger representation, the operator $U(t,t_0)$ is defined by the integral equation (or Fredholm equation):

$\displaystyle U(t,t_0)=1-i \hbar^{-1} \int_{t_0}^t H(t') U(t',t_0) dt'$

an equivalent expression of this law is the Schrödinger equation:

$\displaystyle i\hbar\frac{\mathrm{d}}{\mathrm{d} t}U(t,t_0)=H(t)U(t,t_0)$

with the initial condition:

$\displaystyle U(t_0,t_0)=1$

The solution of these equations is the central problem of the theory, because they express the fundamental law of evolution of the quantum system.

Since $H(t)$ is hermitean, $U$is unitary:

$\displaystyle U^{\dagger}(t,t')U(t,t')=U(t,t')U^{\dagger}(t,t')=1$

whence

$\displaystyle U^{\dagger}(t,t')=U(t',t)$

so, it is possible to write:

$\displaystyle U(t,t_0)=1-i \hbar^{-1} \int_{t_0}^t U(t,t') H(t') dt'$