254 13 Spin Operators
Notice that
tr{Pj}= 2 j+ 1.The summation overjhas to be taken with the allowed values, which may be all
integers, all even or all odd integers, depending on the symmetry and the nuclear
spin of the molecules.
By analogy to (12.65) and (13.42), the density operator is written as
ρ(J)=ρeq( 1 +Φ), (13.60)where the quantityΦ=Φ(t,J))specifies the deviation from the equilibrium. Anal-
ogous to the expansion of the orientational and the velocity distribution functions, as
well as the spin density operator, it is possible to expandΦwith respect to orthog-
onal basis tensors which here depend onJ. The first term of the expansion, and the
most relevant one in some applications, involves the second ranktensor polarization
proportional to〈JμJν〉. More specifically, the tensor operator
φTμν=√
15
2
c 0 −^1 c(J^2 )[
J^2
(
J^2 −
3
4
)]− 1 / 2
JμJν, (13.61)is introduced, where the superscriptTstands for Tensor. It is normalized according to
〈φμνφμ′ν′〉eq=Δμν,μ′ν′.Here
〈...〉eq=Tr{...ρeq} (13.62)indicates the average evaluated with the isotropic equilibrium density operatorρeq.
The weight functionc(J^2 )can be chosen appropriately. The normalization requires
thatc 02 =〈c^2 〉eq. The average
aTμν=〈φμνT〉 (13.63)is referred to astensor polarization. The choicec=1in(13.61) means that the
basis tensor is essentially constructed from a unit vector parallel toJ. The case
c(J^2 )=[J^2 (J^2 −^34 )]^1 /^2 implies an expansion tensor operator which is similar
to the expansion functionVμVν used for the velocity distribution, just with the
classical velocityVreplaced by the angular momentum operatorJ.