IX. The WignerEckart theorem
We will now touch on a result that relates very deeply to the rotational
symmetry of space. However, its practical consequences are
somewhat limited, and so we will only go over it in faint detail. The
basic thing that we observe is that simply looking at how quantum
states behave under rotation gave us a terrific amount of insight into
the quantum theory of angular momentum. One can ask the
corresponding question about operators: how do quantum operators
behave when we rotate the system and what do we learn about
operators in general (and most importantly, their matrix
representations!) by studying their rotational character? We will find
many connections to our results for the addition of angular momenta.
a. Spherical Tensors
Key to the statement of the WignerEckart theorem is the definition of
spherical tensor operators.
This is rendered quite difficult by the
fact that most chemists and physicists do not know what a spherical
tensor is (never mind the operator part). This is just a geometrical
concept, and once again we will find that the transition to quantum
mechanics is trivial; all the “weirdness” is classical.
Under rotation, a classical observable
T
changes as
R
T
(
p
r
⎯
⎯ →
T
(
R
(
θ
⋅
R
r
(
θ
⋅
p
,
,
n
n
R
where “
⎯
⎯→
” indicates the act of rotation and
R
(
θ
is a matrix
n
rotates vectors by an angle
θ
about a unit vector
n,
as in the section
on angular momentum. Now, the interesting fact is that there are
certain
groups
of observables that are related to one another by
rotation. Further, one finds that these groups always have an
odd
number of members! Any such group of observables with
2
k
1
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 Spring '04
 RobertField
 Angular Momentum, tensor of rank, a. Spherical Tensors

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