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Evolution Operator

November 25, 2008

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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