Physics 424: Quantum Mechanics, Fall 2015
Reading/Homework Assignment 3
Due in Class Friday, 10/2/15
1
Reading
•
Finish reading Townsend Chapter 2. This is a long and fairly difficult
chapter, but it covers many important principles.
•
Start reading Townsend Chapter 3. You don’t need this for the prob
lems below, but we should start covering this material in class by the
end of the week.
Problems to work out and turn in (55 pts total)
1. (10 pts) Townsend problem 2.1  Show lim
N
→∞
(1 +
z/N
)
N
=
e
z
.
2. (5 pts) Consider the projection operator
ˆ
P
α
=

α
ih
α

where

α
i
is any
normalized quantum state.
This problem should be done with bras,
kets, and operators (no spinors or matrices written out), and so it is
valid for Hilbert spaces with any dimension
D
(so far we have used
D
= 2 in class).
(a) Show that
ˆ
P
2
α
=
ˆ
P
α
.
(b) Use result 2a to show that the only possible eigenvalues of
ˆ
P
α
are
zero and one.
(c) Show that

α
i
is an eigenvector of
ˆ
P
α
with eigenvalue one, and that
any state

β
i
that is orthogonal to

α
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 Fall '15
 Linear Algebra, mechanics, Work, Hilbert space, Quantum state