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7  1
©2003 by David Boal, Simon Fraser University.
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Lecture 7  Momentum operator
What's important
:
•
momentum operator
•
commutation relations
•
Hermitian operators
Text
: Gasiorowicz, Chap. 3.
Momentum operator
Once we have solved the Schrödinger equation
i
h
(
x
,
t
)
t
= 
h
2
2
m
2
(
x
,
t
)
x
2
(1)
for some particular condition (we'll introduce the form of the SE which includes
interactions in a moment) then we can calculate the mean position and its dispersion
from quantities like
<
x
> =
∫
*
x
dx
and
<
x
2
> =
∫
*
x
2
dx
.
(2)
Given the form of the wavepackets described earlier, it's obvious how to do the
mathematics of Eq. (1)  one just integrates over some functions which depend on
x
and
t
.
But what about a quantity like the momentum
p
?
What is the meaning of the
expression
<
p
> =
∫
*
p
dx
?
An appealing starting point is to return to the classical definition
p
=
mv
=
m
dx
/
dt
and PROPOSE
<
p
> =
m
d
<
x
>/
dt
.
(3)
We'll provide the motivation for this in a later lecture on the classical limit of
.
Thus,
<
p
> =
m
(
d
/
dt
)
∫
*
x
dx
.
(4)
The wavefunction is timedependent, so taking the time derivative inside the integral will
yield terms like
d
/
dt
.
However, there is no timedependence to
x
: it's just an
integration variable, not a function of
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This note was uploaded on 09/07/2009 for the course PHYS 385 taught by Professor Davidboal during the Spring '09 term at Simon Fraser.
 Spring '09
 DavidBoal
 Momentum, Quantum Physics

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