5Axioms for quantum mechanics
IB Quantum Mechanics
5.3 Evolution in time
– The state of a quantum system Ψ(t) obeys the Schr¨odinger equation
i~
˙
Ψ = HΨ,
where
H
is a Hermitian operator, the Hamiltonian; this holds at all times
except at the instant a measurement is made.
In principle, this is all there is in quantum mechanics. However, for practical
purposes, it is helpful to note the following:
Stationary states
Consider the energy eigenstates with
Hψ
n
= E
n
ψ
n
, (ψ
m
, ψ
n
) = δ
mn
.
Then we have certain simple solutions of the Schr¨odinger equation of the form
Ψ
n
= ψ
n
e
−iE
n
t/~
.
In general, given an initial state
Ψ(0) =
X
n
α
n
ψ
n
,
since the Schr¨odinger equation is linear, we can get the following solution for all
time:
Ψ(t) =
X
n
α
n
e
−iE
n
t/~
ψ
n
.
Example. Consider again the harmonic oscillator with initial state
Ψ(0) =
1
√
6
(ψ
0
+ 2ψ
1
− iψ
4
).
Then Ψ(t) is given by
Ψ(t) =
1
√
6
(ψ
0
e
−iωt/2
+ 2ψ
1
e
−3iωt/2
− iψ
4
e
−9iωt/2
).
Ehrenfest’s theorem (general form)
Theorem
(Ehrenfest’s theorem)
.
If
Q
is any operator with no explicit time
dependence, then
i~
d
dt
hQi
Ψ
= h[Q, H]i
Ψ
,
where
[Q, H] = QH − HQ
is the commutator.
Proof. If Q does not have time dependence, then
i~
d
dt
(Ψ, QΨ) = (−i~
˙
Ψ, QΨ) + (Ψ, Qi~
˙
Ψ)
= (−HΨ, QΨ) + (Ψ, QHΨ)
= (Ψ, (QH − HQ)Ψ)
= (Ψ, [Q, H]Ψ).
If
Q
has explicit time dependence, then we have an extra term on the right,
and have
i~
d
dt
hQi
Ψ
= h[Q, H]i
Ψ
+ i~h
˙
Qi
Ψ
.
These general versions correspond to classical equations of motion in Hamiltonian
formalism you will (hopefully) study in IIC Classical Dynamics.