Tensors for Physics

13.5 Density Operator 249

ρ(s)=ρ 0 ( 1 +Φ), ρ 0 =( 2 s+ 1 )−^1 ,Φ=

∑2s

= 1

bμ 1 μ 2 ···μsμ 1 sμ 2 ···sμ.

(13.42)

Clearly,Φis the deviation ofρfrom the isotropic density operatorρ 0. The expansion
coefficientsb...are the moments of the density. Computation of the average of the
-th rank spin tensor withρgiven by (13.42) and use of the trace formula (13.27)
yields

〈sμ 1 sμ 2 ···sμ〉≡tr{sμ 1 sμ 2 ···sμρ(s)}=

!

( 2 + 1 )

S^20 S 12 ···S^2 − 1 bμ 1 μ 2 ···μ.

(13.43)

Thetensor polarization Pμ 1 μ 2 ···μof rankis defined by

Pμ 1 μ 2 ···μ≡s−〈sμ 1 sμ 2 ···sμ〉,≥ 1. (13.44)

In terms of these tensor polarizations, the expansion (13.42) is equivalent to

ρ(s)=ρ 0

[

1 +

∑2s

= 1

s( 2 + 1 )!!

!S 02 S 12 ···S^2 − 1

Pμ 1 μ 2 ···μsμ 1 sμ 2 ···sμ

]

. (13.45)

For=1, the vectorPμoccurring here is calledvector polarization. It is the only
type of polarization possible for particles with spins= 1 /2. Of course, particles
with a larger spin may also have a vector polarization. Frequently, the term tensor
polarization is used forPμν, the case corresponding to=2. Particles with spin
s=1, or with a higher spin, can have this type of tensor polarization. The special
casess= 1 /2 ands=1 are discussed next.

13.5.3 Density Operator for Spin 1=2 and Spin

Electrons, protons and neutrons have spin 1/2, For them, the spin density operator
reads

ρ(s)=

1

2

[ 1 + 2 Pμsμ]. (13.46)

It is understood, that additive numbers, like the^12 here, have to be multiplied by the
appropriate unit matrix, when the spin operators are represented by matrices.
LetN(^1 /^2 )andN(−^1 /^2 )be the number of particles in the magnetic substatesm=
± 1 /2,N=N(^1 /^2 )+N(−^1 /^2 )is the total number of particles. The relative numbers
c± 1 / 2 =N(±^1 /^2 )/Nare determined by the averages〈P(±^1 /^2 )〉of the projection