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MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II
3–1
Instructor: Prof. Robert Field
5.74 RWF LECTURE #3
ROTATIONAL TRANSFORMATIONS AND SPHERICAL TENSOR OPERATORS
Last time;
3j for coupled
↔
uncoupled
transformation of operators as well as basis states
6j, 9j for replacement of one intermediate angular momentum magnitude by another
patterns — limiting cases — simple dynamics
minimum number of control parameters needed to fit or predict the
I
(
ω
) or
I
(
t
)
Today:
Rotation as a way of classifying wavefunctions and operators
Classification is very powerful — it allows us to exploit universal symmetry properties to
reduce complex phenomena to the unique, system specific characteristics
Recall finite group theory
*
Construct a reducible representation — a set of matrix transformations that reproduce the
symmetry element multiplication table
*
represent the matrices by their traces: characters
*
reduce the representation to a sum of irreducible representations
*
each irreducible representation corresponds to a symmetry species (quantum number)
*
selection rules, projection operators, integration over symmetry coordinates
*
the full rotation group is an
∞
dimension example, because the dimension is
∞
, there are an
infinite number of irreducible representations: the
J
quantum number can go from 0 to
∞
*
because the symmetry is so high, most of the irreducible representations are degenerate: the
M
J
quantum number corresponds to the 2
J
+1 degenerate components.
All of the tricks you probably learned in a simple point group theory are applicable to angular momentum
and the full rotation group.
Rotation of Coordinates
two coordinate systems:
XYZ fixed in space
xyz attached to atom or molecule±
2 angles needed to specify orientation of
z wrt Z
one more angle needed to specify orientation of xy by rotation about z±
EULER angles  difficult to visualize — several ways to define — you will need to invest some effort if you
want a deep understanding
Superficial path
θφ
l
m
θ φ
=
Y
l
m
(,
)
completeness
θφχ
R
(
,,)
l
=
∑
m
m
′
m
′
R
(
φθχ
l
l
,
,)
l
m
4
m
′
1
2
44
3
444
l
)
D
()
′
,
(,,
mm
* Rotation does not change
l
*
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View Full Document l
l
5.74 RWF LECTURE #3
3–2
[Nonlecture proof:
(i) [
l
2
,
l
] = 0,
j
(ii) rotation operators have the form
e
i
l
j
α
(rotate by
α
angle about the
ˆ
j
axis)
therefore: [
l
2
,
R
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.
 Spring '04
 RobertField
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