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PHYS 385 Lecture 9
 Timeindependent Schrödinger equation
9  1
©2003 by David Boal, Simon Fraser University.
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Lecture 9  Timeindependent Schrödinger equation
What's important
:
•
timeindependent Schrödinger equation
•
eigenfunctions and eigenvalues
•
particle in a 1D box
Text
: Gasiorowicz, Chap. 4
Timeindependent Schrödinger equation
Let's apply the ideas of eigenfunctions to the
timedependent
Schrödinger equation to
extract the
timeindependent
Schrödinger equation.
In this, and the next several
lectures, we continue to work in one dimension.
Recall the explicit representation of the
Schrödinger equation:
i
h
(
x
,
t
)
t
= 
h
2
2
m
2
(
x
,
t
)
x
2
+
V
(
x
)
(
x
,
t
),
(1)
where we have made the potential energy a function only of position.
If an equation can
be segregated into parts which depend on only one variable, then a fruitful approach to
solving it is the socalled separation of variables approach.
As implemented here, that
means we should try to write the wavefunction
(
x
,
t
) as a product of two functions, one
containing all the position dependence and one containing all the time dependence:
(
x
,
t
) =
T
(
t
)•
u
(
x
).
(2)
Aside: not everyone uses this notation; frequently one sees
(
x
,
t
) for the complete
wavefunction, and
(
x
) for the position dependent part.
Substituting (2) into (1), and shuffling some functions to show what derivatives need
attention
i
h
u
(
x
)
dT
(
t
)
dt
= 
h
2
2
m
T
(
t
)
d
2
u
(
x
)
dx
2
+
T
(
t
)
V
(
x
)
u
(
x
).
Note how
∂
has become
d
.
Dividing by
T
(
t
)•
u
(
x
) yields
i
h
dT
(
t
)/
dt
T
(
t
)
=

(
h
2
/2
m
)(
d
2
u
(
x
)/
dx
2
)
+
V
(
x
)
u
(
x
)
u
(
x
)
(3)
Now, the time dependence is completely isolated on the left hand side, and the position
dependence is on the right hand side.
Since the LHS cannot depend on
x
, and the RHS
cannot depend on
t
, then NEITHER side can depend on
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 Spring '09
 DavidBoal
 Quantum Physics

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