# Evolution Operator

On the scale of quantum phenomena the evolution of a quantum system **ceases*** to be strictly causal* (There exist no net separation between the system itself and the observing instrument).

*Postulate (1): an isolated quantum system evolves in an exactly predictable manner.*

If the system was not subjected to any observation during the time interval , the ket vector representing the dynamical state of the system at the later time is exactly determined by specifying the ket vector at the previous initial time .

*Postulate (2): the linear superposition of states is preserved in the course of time.*

Consequently the correspondence between and is linear and defines a certain linear operator called the *evolution operator*.

.

For a *conservative *system the Hamiltonian does not depend explicitly upon the time. So can be deduced by the *Einstein’s postulate*: . We write the Schrödinger equation:

We postulate that the eigenvector evolves in time according to the law:

or

Consequently:

→

Differentiating we obtain:

satisfying the initial condition:

For a *non conservative *system the Hamiltonian depends explicitly upon time so the Einstein’s postulate losses all meaning. We postulate that is solution of the previous differential equation even when the system is not conservative, but is defined by the integral equation (Fredholm eq.):

→

In order that the norm of remain constant in time, it is necessary and sufficient that be Hermitian. If is Hermitian, is an *unitary operator*: