This preview shows pages 1–3. Sign up to view the full content.
MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II
Instructor: Prof. Robert Field
5.74 RWF Lecture #4
4 – 1
The WignerEckart Theorem
It is always possible to evaluate the angular (universal) angular part of all matrix elements, leaving behind a
(usually) unevaluated radial integral. Unfortunately (as in the separation of the hydrogen atom
wavefunction into angular and radial factors), the radial factor depends on the values of angular momentum
magnitude quantum numbers, but not angular momentum projection quantum numbers. The WignerEckart
Theorem provides an automatic way of evaluating the angular parts of matrix elements of many important
types of operators. The radial factor is called a “reduced matrix element.” Often, explicit relationships
between many reduced matrix elements may be derived by multiple applications of the WignerEckart
Theorem or by evaluating the matrix element directly for one extreme set of quantum numbers (“stretched
state”) where one unique basis state in one basis is by definition identical to one unique basis state in a
different basis set.
The WignerEckart theorem applies to systems which have lower than spherical (atoms) or cylindrical
(linear molecules) symmetry. Any symmetry at all will suffice.
It is essential to express all operators in spherical tensor form. It will become clear that the same operator
may be expressed in several different spherical tensor forms. These are useful for evaluation of reduced
matrix elements.
The central idea is that operators are classified according to their transformation properties under rotation.
The same transformations (Wigner rotation matrices) describe the transformation properties of angular
momenta. The WignerEckart Theorem may be viewed as a generalization of the coupling of separate

ΑΜ
Α
⟩
and

ΒΜ
Β
⟩
angular momentum basis states to form coupled

ΑΒ
C
Μ
C
⟩
basis states.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
3
5.74 RWF Lecture #4
4 – 2
jm
j
j
k
j
′
−
k
k
njm
jq
()
TA
j
(
nj
TA
()
j
n
′′
m
′
j
n
′′
=−
1)
−
m
j
q
m
′
j
The 3j coefficient is what you would expect for coupling
j
1
=
j
′
,
m
j
1
=
m
′
j
j
2
=
k
m
j
=
q
,
to
to form
j
3
=
j
,
m
j
=
m
j
. The factor with the two pairs of vertical lines is called a reduced matrix element.
It
does not depend on m, m
′
j
, or q!
For examples (the matrix elements of the
q
= 0 operator component are always easiest to evaluate,
especially when one chooses an
m
j
= j
basis state):
1
Tj
0
=
j
z
0
2
Tj
j
0
(,
)
=
j
2
−
1
2
2
0
=
6
/
(
3
j
−
j
)
.
z
Using the WE theorem
jm
j
1
j
′
1
−
1
jm
j
0
m
(
j
j
′
,
j
j
−
m
j
0
m
′
j
but we also know
jm
jz
j
m
j
j
jj
′
=
δδ
m
j
m
′
j
m
j
.
Thus, using the analytical expression for the 3j coefficient
−
12
−
m
j
0
m
j
j
mj
(
j
+
j
+
1)
]
,
j
1
j
1
−
j
[
)
(
/
we evaluate the reduced matrix element:
1
)
12
/
j
j
′ = δ
jj
′
[
j
(
j
+
j
+
]
.
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '04
 RobertField
 Chemistry

Click to edit the document details