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Simple Problems in 1D
•
To Describe the motion of a particle in 1D, we
need the following four QM elements:
•
Putting them together yields the Schrödinger
wave equation:
)
(
)
(
t
H
t
dt
d
i
!
=
h
)
(
2
2
X
V
m
P
H
+
=
)
,
(
)
(
t
x
t
x
=
)
,
(
)
(
t
x
x
i
t
P
x
"
"
#
=
h
)
,
(
)
(
)
,
(
2
)
,
(
2
2
2
t
x
x
V
t
x
x
m
t
x
dt
d
i
+
"
"
#
=
h
h
Coordinate Representation
•
In coordinate representation, we can make the
following substitutions:
–
Example:
–
This is just a shortcut, whose validity is readily
verified via Dirac notation
)
,
(
)
(
t
x
t
"
x
X
!
x
i
P
!
!
"
h
!
"
"
#
=
$
x
x
x
dx
i
P
)
(
)
(
%
h
)
(
)
,
(
)
(
)
,
(
x
i
x
f
x
dx
P
X
f
x
"
#
#
$
%
&
h
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View Full DocumentWavevector basis
•
Instead of momentum, it is often convenient to
use wavevector states:
h
P
K
=
k
k
k
K
=
)
(
k
k
k
k
!
"
=
!
#
!
2
x
k
i
e
k
x
=
k
p
h
=
1
=
!
+
"
"
#
k
k
dk
[
]
i
K
X
=
,
[
]
0
,
=
K
P
k
k
k
k
k
k
p
p
p
p
!
=
!
"
=
!
"
=
!
"
=
!
h
h
h
h
1
)
(
1
)
(
)
(
"
k
=
h
p
p
=
h
k
Bound States and/or Scattering States
•
Problems dealing with motion in 1D fall into one
of two categories
1.
Boundstate problems:
–
V
(
x
) < E
over finite region only
–
Energy levels are discrete
–
Typical problem:
•
Find Energy eigenvalues:
{
E
n
};
n
=1,2,3,…
•
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 Fall '08
 MichaelMoore
 mechanics

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