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PHYS385 Lecture 23  Angular momentum
23  1
©2003 by David Boal, Simon Fraser University.
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Lecture 23  Angular momentum
What's important:
•
quantum numbers of spherical harmonics
•
ladder operators
Text:
Gasiorowicz Chap. 10
In deriving the solutions to the angular parts of the three dimensional Schrödinger
equation, we found that two quantum numbers,
and
m
arose naturally from the
polynomial solutions of the relevant differential equations.
We show now that the
quantum numbers have more than just mathematical significance.
Quantized angular momentum
In lecture 20, we established the form for the angular momentum operator in polar
coordinates as
L
2
= 
h
2
1
sin
sin
+
1
sin
2
2
2
(1)
From lecture 21, we know that the spherical harmonics satisfy the equation

1
sin
sin
+
1
sin
2
2
2
Y
=
l
(
l
+
1)
Y
(2)
Combining these two equations shows that the result of applying the
L
2
operator to
Y
(
,
) is:
L
2
Y
(
,
) =
l
(
l
+1)
h
2
Y
(
,
).
(3)
In other words, angular momentum is a conserved quantity, from the eigenvalue
equation, whose length is given by

L
 = 
L
2

1/2
= (
l
(
l
+1))
1/2
h
(4)
The most important feature of this relationship is that it is not the same as the Bohr
condition

L
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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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