Tensors for Physics

252 13 Spin Operators

ThediagonalpartofO(u)canbeexpressedasafunctionoftheangularmomentum
operator, viz.

∑

j

PjO(u)Pj

′

=O(J). (13.54)

Examples are presented next, the casej=j′is treated in Sect.13.6.6.
The trace operation Tr involves the summation over the magnetic quantum num-
bers, just like the trace operation tr for a spin, and an additional summation over the
rotational quantum numbers of the diagonal elements of an operatorO,

Tr{O}=

∑

j

∑j

m=−j

〈jm|O|jm〉=

∑

j

tr{Ojj}. (13.55)

An unbiased orientational average of an operator corresponds to( 2 j+ 1 )−^1 tr{..}.The
formulas for the traces tr of spinssalso apply for the rotational angular momentumJ.

13.6.3 Diagonal Operators

ObservablesO(u)which are even functions ofupossess a partO(u)which is diag-
onal inj. As an instructive example, the second rank irreducible tensor uμuν is
considered. By symmetry, it should be proportional to the symmetric traceless ten-
sor constructed from the components ofJ. Thus the ansatz

(uμuν)jj=cPjJμJν

is made. To determine the proportionality coefficientc, multiply this equation by
JμJνand useJ·u=0. The left hand side of the equation yields

JμJνPjuμuνPj=−

1

3

j(j+ 1 )Pj.

The right hand side is found to be

JμJνJμJνPj=

2

3

j 02 j 12 Pj,

by analogy to (13.23). One obtainsc=−^12 j 1 −^2. The abbreviations

j 02 =j(j+ 1 ), j 12 =j 02 −

3

4