PHYS 507
Homework II
Assigned:
October 15, 2003, Wednesday.
Due:
October 24, 2003, Friday, at 5:00 pm.
1.
Let
T
(
a
) be the translation operator corresponding to displacement
a
.
(a)
Calculate
T
(
a
)

p
0
i
where

p
0
i
is a momentum eigenket.
(b)
Let

φ
i
be the state obtained by translating the state

ψ
i
by a distance
a
, i.e.,

φ
i
=
T
(
a
)

ψ
i
. What is the relationship between the momentumspace wavefunctions
of these kets. (Relate
˜
φ
(
p
0
) to
˜
ψ
(
p
0
)).
2.
Let
A
= (
xp
+
px
)
/
2. Define the operator
S
(
μ
) by
S
(
μ
) = exp

i
¯
h
μA
.
(a)
Reexpress
S
(
μ
1
)
S
(
μ
2
) and
S
(
μ
)
†
in a suitable form. Is
S
(
μ
) unitary?
(b)
Calculate
i
¯
h
[
A, x
] and
i
¯
h
[
A, p
].
(c)
Calculate
S
(
μ
)
†
xS
(
μ
)
and
S
(
μ
)
†
pS
(
μ
)
.
(d)
Let

φ
i
=
S
(
μ
)

ψ
i
. Express the expectation values of the position and momentum
in the state

φ
i
in terms of expectation values in

ψ
i
. (i.e., what is
h
x
i
φ
=
h
φ

x

φ
i
in terms of
h
x
i
ψ
and what is
h
p
i
φ
in terms of
h
p
i
ψ
?)
3.
Let the positionspace wavefunction for a ket

ψ
i
be
ψ
(
x
0
) =
h
x
0

ψ
i
=
Ne
ikx
0
for
0
< x < a ,
0
otherwise
.
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 Spring '11
 starg
 Complex number, Schwarz, schwarz inequality, uncertainty relation, positionspace wavefunction, uncertainty relation be1

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