Tensors for Physics

248 13 Spin Operators

The density operator can be expressed as a linear combination of the projection
operatorsP(m), introduced in Sect.13.2.2,viz.

ρ=

∑s

m=−s

cmP(m), cm=〈P(m)〉,

∑s

m=−s

cm= 1. (13.39)

Notice that tr{P(m)}=1. The coefficientscmdetermine the relative weight of the
magnetic sub-statem. For a system ofNparticles, whereN(m)particles are in the
sub-statem, one has

cm=

N(m)

N

, N=

∑s

m=−s

N(m). (13.40)

The (13.39) is analogous to the representation of a vector as a linear combination of
basis vectors. The projectorsP(m)play the role of the basis vectors, the coefficients
cmare the relevant components.
An ensemble of spins is said to beunpolarized, when all magnetic sub-states
occur equally, i.e. whencm= 1 /( 2 s+ 1 ), for allm. This meansρdoes not depend
ons, it is just proportional to the unit operator 1. Due to the normalization condition,
the density operatorρ 0 for an unpolarized state is

ρ 0 =

1

2 s+ 1

1. (13.41)

Averages in the unpolarized state are denoted by〈...〉 0. The trace formulas presented
above are such averages. Averages in a partially polarized state are discussed in the
next section.

13.5.2 Expansion of the Spin Density Operator

The deviation of the quantum mechanical spin density operatorρfrom its isotropic
or unpolarized state can be expanded with respect to irreducible spin tensors. This is
similar to the description of the deviation from isotropy of the orientational distrib-
ution functionf(u)of Sect.12.2. There, the expansion is with respect to irreducible
tensors of rank, constructed from the components of the classical vectoru.Inthe
classical case, in principle, tensors of all ranks, from=1upto=∞are needed
for a complete characterization, cf. (12.6). For spins, on the other hand, irreducible
tensors of ranks>max≡ 2 svanish. Thus in the spin case, the expansion runs
from=1upto= 2 s. Here the expansion is formulated as