The Schroedinger equation is a recipe for calculating |Ψ(t)>
when |Ψ(t0)> is known. The Schroedinger
equation is linear. The correspondence between |Ψ(t)>
and |Ψ(t0)> is therefore linear.
There
exists a linear operator that transforms |Ψ(t)> into |Ψ(t0)>.
|Ψ(t)> = U(t,t0) |Ψ(t0)>.
What are the properties of this evolution operator?
- (a) U(t0,t0) = I.
- (b)
(iħ∂/∂t)U(t,t0)|Ψ(t0)> = H(t)U(t,t0)|Ψ(t0)>, for any |Ψ(t0)>.
Note: |Ψ(t0)> is time-independent.
Therefore
(iħ∂/∂t)U(t,t0) = H(t)U(t,t0).
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(tn,t1) = U(tn,tn-1)U(tn-1,tn-2)...U(t3,t2)U(t2,t1).
|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,t0), which is a product of
infinitesimal evolution operators, is unitary.
U†(t,t0)U(t,t0) = U(t,t0)U†(t,t0)
= I.
- (e) If H does not explicitly depend on time, then we can solve
(iħ∂/∂t)U(t,t0) = HU(t,t0) for U(t,t0).
We find
U(t,t0) = exp(-iH(t - t0)/ħ).
Let |En> be an eigenstate of H and let |Ψ(t0)> = |En>.
Then
|Ψ(t)> = exp(-iH(t - t0)/ħ)|En> = exp(-iEn(t
- t0)/ħ)|En>
is also an
eigenstate. If H does
not explicitly depend on time and the system is in an eigenstate of H at t = t0,
then it will remain in this eigenstate. The system is in a stationary
state.