This preview shows pages 1–2. Sign up to view the full content.
PHY4604
Problem Set 4
Department of Physics
Page 1 of 3
PHY 4604 Problem Set #4
Due Wednesday October 18, 2006 (in class)
(Total Points = 110, Late homework = 50%)
Reading:
Griffiths Chapter 3 and the Appendix.
Useful Integrals:
π
=
∫
+∞
∞
−
dx
x
x
2
2
)
(
sin
.
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.
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.
 Spring '07
 Field
 mechanics, Work

Click to edit the document details