PHYS 385 Lecture 20  Schrödinger equation in 3D
20  1
©2003 by David Boal, Simon Fraser University.
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Lecture 20  Schrödinger equation in 3D
What's important
:
•
Schrödinger equation in 3D
•
angular momentum operators
Text
: Gasiorowicz, Chap. 10
3D Schrödinger equation
As discussed in Lec. 17, the timeindependent Schrödinger equation for a free particle
in three dimensions reads:

h
2
2
m
d
2
dx
2
+
d
2
dy
2
+
d
2
dz
2
u
E
(
x
,
y
,
z
)
=
E u
E
(
x
,
y
,
z
).
(1)
Adding a timeindependent potential can easily be accommodated through

h
2
2
m
d
2
dx
2
+
d
2
dy
2
+
d
2
dz
2
u
E
(
x
,
y
,
z
)
+
V
(
x
,
y
,
z
)
u
E
(
x
,
y
,
z
)
=
E u
E
(
x
,
y
,
z
)
(2)
Now, for many situations of interest, the potential:
•
is between two objects, both of whose motion must be considered
•
depends only on the radial separation
r
, socalled central potentials.
Twoparticle wavefunctions in onedimension were introduced in Lec. 16, where we
showed that if the potential energy depended only on the relative separation
x
rel
, the
wavefunction could be written as a product of independent wavefunction for the relative
motion and the cm motion.
The same is true for three dimensions for central potentials
V
(
r
).
To review the notation (
bold
indicates vectors):
R
cm
= (
m
1
r
1
+
m
2
r
2
) / (
m
1
+
m
2
)
and
(3)
r
=
r
1

r
2
.
With this replacement, the arguments of the plane wave
p
1
r
1
+
p
2
r
2
=
P
cm
R
cm
+
pr
,
(4)
where the total and relative wavevectors are
P
cm
=
p
1
+
p
2
and
p
= (
m
2
p
1

m
1
p
2
) / (
m
1
+
m
2
).
(5)
The total kinetic energy then becomes
E
=
P
cm
2
/ 2
M
total
+
p
2
/ 2
μ
.
(6)