PHYS 507
Homework II (Fall ’06)
Assigned:
November 17, 2006, Friday.
Due:
November 26, 2006, Sunday.
1.
Consider a state of a particle in 1D having the following positionspace wavefunction
ψ
(
x
0
) =
N
exp


x
0

a
¶
,
where
a
is a distance and
N
is an appropriate normalization factor.
(a)
Find
N
and then compute the momentumspace wavefunction
˜
ψ
(
p
0
).
(b)
Compute the probability density for momentum distribution
˜
P
(
p
0
) and sketch a
plot of it. Are opposite momenta values (
p
0
and

p
0
) equally probable?
(c)
Compute Δ
x
and Δ
p
and show that the uncertainty relation is satisfied.
Hints: (1) You can compute expectation values by using either
ψ
or
˜
ψ
. Use whichever
is convenient.
(2) Note that
h
p
2
i
=
h
ψ

p
2

ψ
i
=
h
pψ

pψ
i
=

pψ

2
. As a result,
h
p
2
i
can be computed
by using positionspace wavefunction by taking only a single derivative.
This is an
extremely useful trick.
(3) The Γfunction (factorial) integral might be useful in here
Z
∞
0
u
n
e

u
du
=
n
!
.
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 Spring '11
 starg
 Fundamental physics concepts, expectation values, Compute p2

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