This preview shows pages 1–2. Sign up to view the full content.
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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 STEVEPOLLOCK
 mechanics

Click to edit the document details