PHYS 385 Lecture 4  Wavepackets
4  1
©2003 by David Boal, Simon Fraser University.
All rights reserved; further copying or resale is strictly prohibited.
Lecture 4  Wavepackets
What's important
:
•
wavepackets in position and momentum
•
propagation of wavepackets
Text
: Gasiorowicz, Chap. 2
In the previous lecture, we derived the Fourier transform of a function
f
(
x
):
f
(
x
) = (2
π
)
1/2
∫

∞
+
∞
A
(
k
) exp(
ikx
)
dk
.
(1)
where
A
(
k
) = (2
π
)
1/2
∫

∞
+
∞
f
(
x
) exp(
ikx
)
dx
.
(2)
Here,
A
(
k
) is a function of the continuous variable
k
, interpreted as a wavevector
corresponding to the physical length
x
.
Let's do a Fourier decomposition of a
wavepacket, and then describe how the wavepacket propagates if
k
is treated as a
momentum via
p
=
h
k
.
Wavepackets
In Lec. 2, we describe how the uncertainty principle leads to the idea that motion in
quantum mechanics is probabilistic  the position and momentum of a particle are
described by probability densities
P
(
x
) and
P
(
p
) that have a distribution around some
mean value:
These functions look sort of Gaussian, as in exp(
k
2
), where
reflects the width of the
distribution.
Let's
suppose
that the
WAVE AMPLITUDE
of a particle in
k
space is
actually Gaussian, with a form
g
(
k
) = exp(
[
k

k
o
]
2
),
(3)
which is centered around
k
=
k
o
.
Compared to Eq. (2), we will drop the factors of (2
π
)
1/2
,
as is done by Gasiorowicz; what interests us is the width of the distribution, not the
normalization  in fact even with the missing (2
π
)
1/2
, Eq. (3) is not normalized to unity.
How does the amplitude in position behave if the amplitude in momentum is Gaussian?
P
(
x
vt
x
P
(
p
p
o
p
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View Full DocumentPHYS 385 Lecture 4  Wavepackets
4  2
©2003 by David Boal, Simon Fraser University.
All rights reserved; further copying or resale is strictly prohibited.
We start by evaluating
f
(
x
) =
∫

∞
+
∞
exp(
[
k

k
o
]
2
) exp(
ikx
)
dk
.
(4)
The first step in solving this integral is to combine the arguments of the exponentials

[
k

k
o
]
2
+
ikx
then rewrite
K
≡
k

k
o

K
2
+
i
(
K
+
k
o
)
x
.
Completing the square gives
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 Spring '09
 DavidBoal
 Momentum, Quantum Physics, Uncertainty Principle, David Boal, Wavepackets

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