The Schroedinger equation is a recipe for calculating |Ψ(t)>
when |Ψ(t_{0})> is known. The Schroedinger
equation is linear. The correspondence between |Ψ(t)>
and |Ψ(t_{0})> is therefore linear.

There
exists a linear operator that transforms |Ψ(t)> into |Ψ(t_{0})>.

|Ψ(t)> = U(t,t_{0}) |Ψ(t_{0})>.

**What are the properties of this** **evolution operator**?

- (a) U(t,t
_{0}) = I. - (b)
(iħ∂/∂t)U(t,t
_{0})|Ψ(t_{0})> = H(t)U(t,t_{0})|Ψ(t_{0})>, for any |Ψ(t_{0})>.

Therefore (iħ∂/∂t)U(t,t_{0}) = H(t)U(t,t_{0}).

Properties (a) and (b) completely define the evolution operator. - (c) |Ψ(t)> = U(t,t') Ψ(t')>, |Ψ(t')> = U(t',t'')|Ψ(t'')>.

Therefore |Ψ(t)> = U(t,t')U(t',t'')|Ψ(t'')> = U(t,t'')|Ψ(t'')>.

We can generalize:

U(t_{n},t_{1}) = U(t_{n},t_{n-1})U(t_{n-1},t_{n-2})...U(t_{3},t_{2})U(t_{2},t_{1}).

|Ψ(t)> = U(t,t')U(t',t'')|Ψ(t'')>. Let t’’ = t.

Then |Ψ(t)> = U(t,t')U(t',t)|Ψ(t)>, U(t,t')U(t',t) = I,

and interchanging the role of t’ and t, U(t',t)U(t,t') = I.

Therefore U(t,t')^{-1}= U(t',t). - (d) d|Ψ(t)> = |Ψ(t + dt)> - |Ψ(t)> = -(i/ħ)H(t)|Ψ(t)>dt

from the Schroedinger equation.

|Ψ(t + dt)> = [I - (i/ħ)H(t)dt]|Ψ(t)>.

Therefore

U(t + dt,t) = I - (i/ħ)H(t)dt

is the infinitesimal evolution operator.

U^{†}(t + dt,t) = I + (i/ħ)H(t)dt, since I and H are Hermitian.

U^{†}(t + dt,t)U(t + dt,t) = U(t + dt,t)U^{†}(t + dt,t) = I,

since for an infinitesimal operator we neglect terms higher than first order in dt. The infinitesimal evolution operator is a unitary operator. Therefore U(t,t_{0}), which is a product of infinitesimal evolution operators, is unitary.

U^{†}(t,t_{0})U(t,t_{0}) = U(t,t_{0})U^{†}(t,t_{0}) = I. - (e) If H does not explicitly depend on time, then we can solve

(iħ∂/∂t)U(t,t_{0}) = HU(t,t_{0}) for U(t,t_{0}). We find

U(t,t_{0}) = exp(-iH(t - t_{0})/ħ).

Let |E_{n}> be an eigenstate of H and let |Ψ(t_{0})> = |E_{n}>.
Then

|Ψ(t)> = exp(-iH(t - t_{0})/ħ)|E_{n}> = exp(-iE_{n}(t
- t_{0})/ħ)|E_{n}>

is also an
eigenstate. If H does
not explicitly depend on time and the system is in an eigenstate of H at t = t_{0},
then it will remain in this eigenstate. The system is in a **stationary
state**.