Copyright by Kirsten Viering 2006 - Raizen Lab - The University of ...

The product of the two rotation matrices Dk ⊗ Dj′ in eq. A.1 can be expressed
as a product of Clebsch-Gordan coefficients and a rotation matrix D K . The linear com-
bination of the products T(k, q ′ )|α ′ j ′ m ′′ 〉, that transforms under rotation as a particular
state |βKQ〉 of the basis functions of the rotation matrix D k is
|βKQ〉 = 〈kq ′ , j ′ m ′′ |KQ〉T(k, q ′ )|α ′ j ′ m ′′ 〉, (A.3)
q,m ′′
where 〈kq ′ , j ′ m ′′ |KQ〉 is a Clebsch-Gordan coefficient. Taking the inner product with
〈αjm|
〈kq ′ , j ′ m ′′ |KQ〉〈αjm|T(k, q ′ )|α ′ j ′ m ′′ 〉 = 〈αjm|βKQ〉 (A.4)
q,m ′′
can further be manipulated by multiplying both sides with 〈kq, j ′ m ′ |KQ〉 and doing
the summation over K and Q. The inner product 〈αjm|βjm〉 equals the reduced matrix
element 〈αj||T k||α ′ j ′ 〉, and the scalar product 〈αjm|βKQ〉 vanishes unless j = K and
m = Q.
〈αjm|T(k, q ′ )|α ′ j ′ m ′ 〉 =
〈αjm|βKQ〉〈kq, j ′ m ′ |KQ〉
KQ
= 〈kq, j ′ m ′ |jm〉〈αj||Tk||α ′ j ′ 〉. (A.5)
Hence we have successfully separated the dynamics from the geometrical properties
of the system.
By transforming the Clebsch-Gordan coefficients of eq. A.5 into Wigner-3j-
symbols we derive the Wigner-Eckart Theorem in the form used in this thesis,
〈αjm|T(k, q ′ )|α ′ j ′ m ′ 〉 = (−1) j−m
j k k ′
−m q m ′
〈αj||T k ||α ′ j ′ 〉. (A.6)
One should be careful with the conventions of the reduced matrix elements. We
have adopt the convention of Zare [12] and Edmonds [32], while others, e.g. Brink [33],
define the reduced matrix element by a factor of 1
√2j+1 smaller.
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