What is time evolution operator? Could you explain to me (level 1 years undergrade) what is a time evolution operator? I read on Wikipedia, and it confuses me. 
 A: It's a map from the quantum state space onto itself that represents how the state changes with time. Simplest example: a spin half spin state is a $1\times 2$ column vector $\psi$ holding the two probability amplitudes for the system to be in spin up and spin down. Now leave the system isolated wait for a time $t$. In QM, the operator is linear, so that state change wrought by the time delay $t$ is described by $\psi\mapsto U(t)\,\psi$, where $U(t)$ is a unitary $2\times 2$ matrix. We call $U(t)$ the time evolution operator.
It's important to recall that quantum state evolution is deterministic. Utterly so. The issue of what happens when we take measurements is a wholly different issue and the time evolution operator has nothing to do with this nondeterministic part of QM.
The time evolution operator is the quantum state space analogue of the State Transition Operator in linear systems theory. In systems theory, the state generally isn't a vector of probability amplitudes, so the state transition operators are general invertible matrices rather than unitary ones. But the principles and mathematical ideas are exactly the same. If the system is not time varying, it can be shown that the time evolution operator / state transition operator is of the form $\exp(i\,H\,t)$, where $H$ is a constant $2\times 2$ matrix in our spin up/spin down example. $H$ is the system's Hamiltonian.
