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PHY4604 Fall 2007
Problem Set 4
Department of Physics
Page 1 of 3
PHY 4604 Problem Set #4
Due Wednesday October 17, 2007 (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.
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 Spring '07
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