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PHYS 385 Lecture 22  Spherical harmonics  II
22  1
©2003 by David Boal, Simon Fraser University.
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Lecture 22  Spherical harmonics  II
What's important
:
•
associated Legendre functions
•
quantized angular momentum
Text
: Gasiorowicz, Chap. 11
In the previous lecture, we separated out the angular part of the Schrödinger equation
and solved for the azimuthal part
(
) of the wavefunction as well as the special case
m
= 0 of the polar function
(
) in the equation:
sin
d
d
sin
d
d
+
l
(
l
+
1)sin
2

m
2
[ ]
=
0
(1)
The solution set for
m
= 0 is the Legendre polynomials
(
) =
P
L
(cos
), and we showed
that
l
is required to have integer values.
Associated Legendre functions
When
m
≠
0, more work is required to solve the equation.
First, perform the substitution
x
= cos(
)
so that Eq. (1) becomes Eq. (8) in the previous lecture:
d
dx
1

x
2
( 29
dP
dx
+
l
(
l
+
1)

m
2
(1

x
2
)
P
=
0.
(2)
Expanding the first term:
d
2
P
l
m
(
x
)
dx
2

2
x
dP
l
m
(
x
)
dx

x
2
d
2
P
l
m
(
x
)
dx
2
+
l
(
l
+
1)

m
2
1

x
2
P
l
m
(
x
)
=
0
(3)
or
(1

x
2
)
d
2
P
l
m
(
x
)
dx
2

2
x
dP
l
m
(
x
)
dx
+
l
(
l
+

m
2
1

x
2
P
l
m
(
x
)
=
(4)
This equation looks like trouble because of the (1
x
2
) in the denominator of the square
brackets.
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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
 Angular Momentum, Momentum, Quantum Physics

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