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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 $(t_0,t)$, the ket vector $\left|\psi(t)\right>$ representing the dynamical state of the system at the later time $t$ is exactly determined by specifying $\left|\psi(t_0)\right>$ the ket vector at the previous initial time $t_0$.

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

Consequently the correspondence between $\left|\psi(t_0)\right>$ and $\left|\psi(t)\right>$ is linear and defines a certain linear operator $U(t_0,t)$ called the evolution operator.

$\displaystyle\left|\psi(t)\right>=U(t_0,t)\left|\psi(t_0)\right>$.

For a conservative system the Hamiltonian does not depend explicitly upon the time. So $U(t_0,t)$ can be deduced by the Einstein’s postulate: $E=\hbar \omega$. We write the Schrödinger equation:

$\displaystyle H\left| u_E(t_0)\right>=E\left| u_E(t_0)\right>$

We postulate that the eigenvector $\left| u_E(t)\right>$ evolves in time according to the law:

$\displaystyle \left| u_E(t)\right>=e^{-i\hbar \omega(t-t_0)}\left| u_E(t_0)\right>$

or

$\displaystyle \left| u_E(t)\right>=e^{-iH(t-t_0)/\hbar}\left| u_E(t_0)\right>$

Consequently:

→  $\displaystyle U(t_0,t)=e^{-iH(t-t_0)/\hbar}$

Differentiating we obtain:

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

satisfying the initial condition: $U(t_0,t_0)=1$

For a non conservative system the Hamiltonian depends explicitly upon time so the Einstein’s postulate losses all meaning. We postulate that $U(t_0,t_0)$ is solution of the previous differential equation even when the system is not conservative, but $U(t,t_0)$ is defined by the integral equation (Fredholm eq.):

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

In order that the norm of $\left|\psi (t)\right>$ remain constant in time, it is necessary and sufficient that $H$ be Hermitian. If $H$ is Hermitian, $U(t_0,t_0)$ is an unitary operator:

$UU^{\dagger}=U^{\dagger}U=1$

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