What is time evolution?

[Swapnil]

What is time evolution? Is it a term only applicabe to matter waves or does it apply to other waves as well?

[Fredrik]

In quantum mechanics, the state of any physical system is represented by a vector. Suppose that |\alpha\rangle is such a vector. Time evolution is the process

|\alpha\rangle\rightarrow e^{-iHt}|\alpha\rangle

where H is the Hamiltonian operator.

You can think of the state vector as a representation of all properties of the system, in the past, present, and future. The effect of the time evolution operator is then to transform our state vector to the state vector that another observer would use to describe the same system. This would be an observer whose clock shows zero t seconds after ours does.

That point of view is called the Heisenberg picture. (If we're using the Heisenberg picture, I prefer to call it time translation rather than time evolution).

Another point of view is the Schrödinger picture. Here we think of the state vector as a time-dependent quantity:

|\alpha;t\rangle=e^{-iHt}|\alpha\rangle

We think of this as the state of the system at time t. It's easy to verify that this time dependent state vector satisfies the Schrödinger equation (because the time evolution operator does):

i\frac{\partial}{\partial t}|\alpha;t\rangle=H|\alpha;t\rangle