PHY3063 Spring 2007
Problem Set 7
Department of Physics
Page 1 of 3
PHY 3063 Problem Set #7
Due Thursday April 5 (in class)
(Total Points = 110, Late homework = 50%)
Reading:
Finish reading Tipler & Llewellyn
Chapter 5 and 6.
Problem 1 (25 points):
.
(a) (2 points)
Show that the sum of two hermitian operators is hermitian
(b) (2 points)
Suppose that H
op
is a hermitian operator, and
α
is a complex number.
Under what
condition (on
α
) is
α
H
op
hermitian?
(c) (2 points)
When is the product of two hermitian operators hermitian?
(d) (2 points)
If
dx
d
O
op
=
, what is
↑
op
O
?
(e) (2 points)
Show that
↑
↑
↑
=
op
op
op
op
A
B
B
A
)
(
.
(f) (2 points)
Prove that [AB,C] = A[B,C] + [A,C]B, where A, B, and C are operators.
(g) (2 points)
Show that
dx
df
i
x
f
p
op
x
h
−
=
)]
(
,
)
[(
, for any function f(x).
(h) (2 points)
Show that the antihermitian operator, I
op
, has at most one real eigenvalue (Note:
antihermitian means that
op
op
I
I
−
=
↑
).
(i) (2 points)
If A
op
is an hermitian operator, show that
0
2
>≥
<
op
A
.
(j) (2 points)
The parity operator, P
op
, is defined by P
op
Ψ
(x,t) =
Ψ
(x,t).
Prove that the parity
operator is hermitian and show that
1
2
=
op
P
, where 1 is the identity operator.
Compute the
eigenvalues of the parity operator.
(k) (2 points)
Idempotent operators have the property that
op
op
P
P
=
2
.
Determine the eigenvalues
of the idempotent operator P
op
and characterize its eigenvectors.
(l) (3 points)
Unitary operators have the property that
1
=
=
↑
↑
op
op
op
op
U
U
U
U
, where 1 is the
identity operator.
Show that the eigenvalues of a unitary operator have modulus 1.
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 Spring '07
 Field
 Physics, Work, Hilbert space, op, Department of Physics

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