PHYS 385 Lecture 6  Probability current
6  1
©2003 by David Boal, Simon Fraser University.
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Lecture 6  Probability current
What's important
:
•
continuity equation
•
probability current
Text
: Gasiorowicz, Chap. 3
We have now provided the motivation for the freeparticle Schrödinger equation in one
dimension
i
h
(
x
,
t
)
t
= 
h
2
2
m
2
(
x
,
t
)
x
2
.
(one dimension)
The interpretation of the wavefunction is that its complex square is equal to the
probability density at
x
and
t
:
[
probability density at
x, t] = 
(
x
,
t
)
2
Note that
(
x
,
t
) has units (!) which depend on the dimensionality of the equation:
in one dimension,
(
x
,
t
) has units of [
length
]
1/2
in three dimensions,
(
x
,
t
) has units of [
length
]
3/2
.
Now, we can't just say "gee, it looks like a probability"  there are some properties which
must be established.
First, we turn to fluids (no longer part of most physics curricula)
and derive the continuity equation.
Fluids
What is of interested to us here is how the local density of a fluid changes as it moves.
Let's consider a tube with cross sectional areas
A
L
and
A
R
at each end.
The fluid enters
at the left with speed
v
L
and leaves at the right with speed
v
R
.
The density need not be the same at each end, and we call
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 Spring '09
 DavidBoal
 Current, Quantum Physics, probability density function, charge density

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