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PHYS 507
Answers to Homework III (Fall ’05)
1.
(a)
[
X,Y
]
†
= (
XY

Y X
)
†
=
Y
†
X
†

X
†
Y
†
= [
Y
†
,X
†
] = [
Y,X
] =

[
X,Y
].
(b)
(
A
†
A
)
†
=
A
†
(
A
†
)
†
=
A
†
A
. Let

φ
i
be an eigenket of
A
†
A
with eigenvalue
λ
. Let

φ
i
be normalized. In that case
λ
=
h
φ

A
†
A

φ
i
=
h
φ

A
†
Aφ
i
=
h
Aφ

Aφ
i
=

Aφ

2
,
i.e., norm square of
A

φ
i
. Since normsquare cannot be negative (either positive
or zero) we have
λ
≥
0.
2.
ψ
(
x
0
) =
h
x
0

ψ
i
=
N
exp
±

x
0
2
4
σ
2
+
ikx
0
¶
.
(a)
N
= 1
/
√
σ
(2
π
)
1
/
4
.
(b)
˜
ψ
(
p
0
) =
r
2
σ
¯
h
1
(2
π
)
1
/
4
exp
±

σ
2
¯
h
2
(
p
0

¯
hk
)
2
¶
.
(c)
h
p
i
= ¯
hk ,
›
p
2
ﬁ
= ¯
h
2
k
2
+
¯
h
2
4
σ
2
,
Δ
p
=
¯
h
2
σ
.
3.
(a)
Obvious from Fourier transform expression.
(b)
For any odd function
f
(
p
0
) (i.e.,
f
(

p
0
) =

f
(
p
0
)) we have
h
f
(
p
)
i
=
R
f
(
p
0
)
˜
P
(
p
0
)
dp
0
=
0 since the integral of any odd function is 0. For this reason
h
p
i
=
h
p
17
i
= 0.
4.
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