Physics 6210/Spring 2007/Lecture 10
Lecture 10
Relevant sections in text:
§
1.7
Gaussian state
Here we consider the important example of a Gaussian state for a particle moving in
1d. Our treatment is virtually identical to that in the text, but this example is suﬃciently
instructive to give it again here.
We deﬁne this state by giving its components in the position basis,
i.e.,
its wave
function:
ψ
(
x
) =
h
x

ψ
i
=
±
1
π
1
/
4
√
d
²
exp
±
ikx

x
2
2
d
2
²
.
You can check as a good excercise that this wave function is normalized:
1 =
h
ψ

ψ
i
=
Z
∞
∞
dx
h
ψ

x
ih
x

ψ
i
=
Z
∞
∞
dx

ψ
(
x
)

2
.
Roughly speaking, the wave function is oscillatory with wavelength
2
π
k
but sitting in a
Gaussian envelope centered at the origin. The probability density for position is a Gaussian
centered at the origin with width determined by
d
. Thus this state represents a particle
“localized” near the origin within a statistical uncertainty speciﬁed by
d
.
Let us make this more precise and exhibit some properties of this state. As an exercise
you can check the following results.
h
X
i
=
h
ψ

X

ψ
i
=
Z
∞
∞
dx x

ψ
(
x
)

2
= 0
.
so that the mean location of the particle is at the origin in this state. Next we have
h
X
2
i
=
h
ψ

X
2

ψ
i
=
Z
∞
∞
dx x
2

ψ
(
x
)

2
=
d
2
2
.
Thus the dispersion in position is
h
Δ
X
2
i
=
d
2
2
,
i.e.,
d
√
2
is the standard deviation of the probability distribution for position. Next we have
h
P
i
=
h
ψ

P

ψ
i
Z
∞
∞
dx ψ
*
(
x
)
¯
h
i
d
dx
ψ
(
x
) = ¯
hk,
telling us that, on the average, this state has the particle moving with momentum ¯
hk
, and
h
P
2
i
=
h
ψ

P
2

ψ
i
Z
∞
∞
dx ψ
*
(
x
)(

¯
h
2
)
d
2
dx
2
ψ
(
x
) =
¯
h
2
2
d
2
+ ¯
h
2
k
2
,
1
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so that the momentum uncertainty is
h
Δ
P
2
i
=
¯
h
2
2
d
2
.
Thus the momentum uncertainty varies reciprocally with
d
relative to the position uncer
tainty. The product of position and momentum uncertainties is as small as allowed by the
uncertainty relation. One sometimes calls

ψ
i
a
minimum uncertainty state
.
Because the Fourier transform of a Gaussian function is another Gaussian, it happens
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 Spring '07
 M
 Physics, mechanics, Vector Space, Particle, 1D, product vectors

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