3Expectation and uncertainty
IB Quantum Mechanics
3.2 Ehrenfest’s theorem
We will show that, in fact, quantum mechanics is like classical mechanics.
Again, consider a normalizable state Ψ(
x, t
) satisfying the time-dependent
Schr¨odinger equation, i.e.
i~
˙
Ψ = HΨ =
ˆp
2
2m
+ V (ˆx)
Ψ.
Classically, we are used to
x
and
p
changing in time. However, here
ˆx
and
ˆp
are
fixed in time, while the states change with time. However, what does change
with time is the expectations. The expectation values
hˆxi
Ψ
= (Ψ, ˆxΨ), hˆpi
Ψ
= (Ψ, ˆpΨ)
depend on t through Ψ. Ehrenfest’s theorem states the following:
Theorem (Ehrenfest’s theorem).
d
dt
hˆxi
Ψ
=
1
m
hˆpi
Ψ
d
dt
hˆpi
Ψ
= −hV
0
(ˆx)i
Ψ
.
These are the quantum counterparts to the classical equations of motion.
Proof. We have
d
dt
hˆxi
Ψ
= (
˙
Ψ, ˆxΨ) + (Ψ, ˆx
˙
Ψ)
=
1
i~
HΨ, ˆxΨ
+
Ψ, ˆx
1
i~
H
Ψ
Since H is Hermitian, we can move it around and get
= −
1
i~
(Ψ, H(ˆxΨ)) +
1
i~
(Ψ, ˆx(HΨ))
=
1
i~
(Ψ, (ˆxH − H ˆx)Ψ).
But we know
(ˆxH − H ˆx)Ψ = −
~
2
2m
(xΨ
00
− (xΨ)
00
) + (xV Ψ − V xΨ) = −
~
2
m
Ψ
0
=
i~
m
ˆpΨ.
So done.
The second part is similar. We have
d
dt
hˆpi
Ψ
= (
˙
Ψ, ˆpΨ) + (Ψ, ˆp
˙
Ψ)
=
1
i~
HΨ, ˆpΨ
+
Ψ, ˆp
1
i~
H
Ψ
Since H is Hermitian, we can move it around and get
= −
1
i~
(Ψ, H(ˆpΨ)) +
1
i~
(Ψ, ˆp(HΨ))
=
1
i~
(Ψ, (ˆpH − H ˆp)Ψ).
Again, we can compute
(ˆpH − H ˆp)Ψ = −i~
−~
2
2m
((Ψ
00
)
0
− (Ψ
0
)
00
) − i~((V (x)Ψ)
0
− V (x)Ψ
0
)
= −i~V
0
(x)Ψ.
So done.
Note that in general, quantum mechanics can be portrayed in different
“pictures”. In this course, we will be using the Schr¨odinger picture all the time,
in which the operators are time-independent, and the states evolve in time. An
alternative picture is the Heisenberg picture, in which states are fixed in time,
and all the time dependence lie in the operators. When written in this way,
quantum mechanics is even more like classical mechanics. This will be explored
more in depth in IID Principles of Quantum Mechanics.