PHY 4604 Fall 2010 – Homework 6
Due at 5:00 p.m. on Friday, November 12. Please turn in your homework in
class before the deadline, or else bring it to NPB 2162. (You may push it under
the door if NPB 2162 is unoccupied.) No credit will be available for homework
submitted after the start of class on Monday, November 15.
Answer all four questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
1.
Compatible observables.
Two observables Ω and Λ are said to be “compatible” if the
corresponding operators commute, i.e., [
ˆ
Ω
,
ˆ
Λ] = 0. Compatible observables can be
chosen to have a complete set of simultaneous eigenstates
{
ω, λ
i}
such that
ˆ
Ω

i
=
ω

i
and
ˆ
Λ

i
=
λ

i
.
Example (based on Problem 123 in
Modern Quantum Mechanics
by J. J. Sakurai):
Consider a threedimensional Hilbert space. In a certain basis, operators
ˆ
Ωand
ˆ
Λare
represented by
ˆ
Ω=
a
00
0

a
0

a
,
ˆ
Λ=
b

ib
0
ib
0
,
where
a
and
b
are both real.
(a) Show that
ˆ
ˆ
Λcommute
.
(b) Find the spectrum of
ˆ
Ω and the spectrum of
ˆ
Λ.
(c) Find a complete orthonormal set of simultaneous eigenstates of
ˆ
ˆ
Λ. Label
each eigenstate according to its eigenvalues of
ˆ
ˆ
Λ.
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 Fall '07
 Field
 mechanics, Work

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