1
Physics 316
Solution for homework 6
Spring 2005
Cornell University
I.
EXERCISE 1
Compute the position uncertainty
Δ
x
in the ground state
Φ
0
(
x
) =
1
p
√
πa
e

ξ
2
2
,
ξ
=
x
a
,
a
=
r
¯
h
mω
0
.
(1.1)
for the harmonic oscillator potential. Use
Δ
x
=
p
h
(
x
 h
x
i
)
2
i
=
q
h
x
2
i  h
x
i
2
and the expectation values
h
x
i
=
Z
∞
∞
x

Φ
0
(
x
)

2
dx ,
›
x
2
fi
=
Z
∞
∞
x
2

Φ
0
(
x
)

2
dx .
(1.2)
(prepared by Steve Drasco)
First we will compute the average position
h
x
i
h
x
i
=
Z
∞
∞
dx x

Φ
0

2
= 0
,
(1.3)
since
x

Φ
0

2
is an odd function, and the integration boundaries are symmetric about the origin. Here we have used a
general rule which applies to any odd function
f
(

x
) =

f
(
x
)
Z
A

A
dx f
(
x
) =
Z
0

A
dx f
(
x
) +
Z
A
0
dx f
(
x
)
,
=

Z
0
A
dx f
(

x
) +
Z
A
0
dx f
(
x
)
,
=

Z
A
0
dx f
(
x
) +
Z
A
0
dx f
(
x
)
,
= 0
.
(1.4)
We now compute
›
x
2
fi
›
x
2
fi
=
Z
∞
∞
dx x
2

Φ
0

2
,
=
1
a
√
π
Z
∞
∞
dx x
2
e

Ax
2
,
(1.5)
where we have defined
A
=
a

2
. Now note that
x
2
e

Ax
2
=

∂
∂A
e

Ax
2
,
(1.6)
so that we have
›
x
2
fi
=

1
a
√
π
∂
∂A
Z
∞
∞
dx e

Ax
2
,
=

1
a
∂
∂A
A

1
/
2
,
=
a
2
2
,
(1.7)
2
where we have used the known Gaussian integral
Z
∞
∞
dx e

Ax
2
=
r
π
A
.
(1.8)
To prove this relation let
I
=
R
∞
∞
e

Ax
2
. Then we have
I
2
=
Z
∞
∞
dx
Z
∞
∞
dy e

A
(
x
2
+
y
2
)
,
=
Z
∞
0
dr
Z
2
π
0
rdφ e

Ar
2
,
= 2
π
Z
∞
0
dr re

Ar
2
,
=
π/A,
(1.9)
where in the second line we have converted from cartesian coordinates with area element
dxdy
, to polar coordinates
with area element
rdφdr
. These results, Eqs. (1.3) and (1.7), give
Δ
x
=
q
h
x
2
i  h
x
i
2
=
a
√
2
≈
0
.
7
a.
(1.10)
So on average, measurements should find the particle at the origin, and these measurements should fluctuate on a
scale of about 70% of the ground state’s classical maximum extension
a
.
II.
EXERCISE 2
What is the expectation value
h
V
i
and the uncertainty
Δ
V
=
p
h
V
2
i  h
V
i
2
of the harmonic oscillator potential
V
=
1
2
mω
2
0
x
2
for the ground state wave function? This corresponds to the fact that the energy of a classical harmonic
oscillator is on average split in equal parts between kinetic an potential energy.
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 Spring '05
 HOFFSTAETTER
 Physics, Work, NZ, ground state, Enx

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