Physics 3220 – Quantum Mechanics 1 – Fall 2008
Problem Set #9
Due Wednesday, Oct 29 at 2pm
9. 1
Commutators
[15 pts]
Prove the following commutator identities:
A)
[A+B,C] = [A,C] + [B,C]
B)
[AB, C] = A[B,C] + [A,C]B
C)
df
[f (x),p]
i
dx
=
h
, where x and p are the position and momentum operators (in 1D)
Next, given our usual definition of the Hamiltonian operator,
H
=
p
2
2
m
+
V
(
x
)
,
D) show that
[
H
,
p
] =
i
h
V
/
x
E) What is [H, x]?
(Quick check: does this result make sense when inserted into the right side
Griffiths’ eqn 3.71?)
F) In 2 dimensions,
p
x
= -
i
h
/
x
,
p
y
= -
i
h
/
y
.
What is
[
p
x
,
p
y
]
?
(Note: Why do we care about commutators? They tell us whether observables are “compatible”,
they are the basis for generalized uncertainty principles, and commutators with the Hamiltonian H
teach us about time dependence of expectation values. So these rather formal looking relations turn
out to have lots of practical use in quantum mechanics!)
9. 2
Quantum measurements [
20 pts]
An operator
ˆ
A
(representing observable A) has two normalized eigenstates
ψ
1
and
ψ
2
, with
eigenvalues a
1
and a
2
. Operator
ﾵ
B
(representing observable B) has two normalized eigenstates
φ
1
and
φ
2
, with eigenvalues b
1
and b
2
.
Suppose these eigenstates are related by the following:
ψ
1
=
2
3
f
1
+
5
3
f
2
,
y
2
= -
5
3
f
1
+
2
3
f
2
,
A) Show us that (assuming
φ
1
and
φ
2
are properly normalized),
ψ
1
and
ψ
2
are normalized too.
B) Let’s start in some unspecified random state, and then observable A is measured. Further, assume
that you DO in fact measure the particular value a
1
- what is the state of the system immediately
after this measurement?
C) Immediately after the measurement of A (which, recall, happened to yield a