Tensors for Physics

250 13 Spin Operators

operators given in (13.11). Thus the relative difference of the occupation numbers is
a measure for the degree of polarization of spin 1/2 particles. It is determined by

c 1 / 2 −c− 1 / 2 =

N(^1 /^2 )−N(−^1 /^2 )

N(^1 /^2 )+N(−^1 /^2 )

=〈 2 hνsν〉=hμPμ. (13.47)

To obtain the last equality, the spin density (13.46) and the trace formula (13.29)are
used. By definition, the coefficientscmare positive and bounded by 1. Thus one has
− 1 ≤h·P≤1. A state withh·P=±1, associated withc 1 / 2 = 1 ,c− 1 / 2 = 0
andc 1 / 2 = 0 ,c− 1 / 2 =1, respectively, is completely polarized, with respect to the
preferential directionh. The cases in between the complete polarizations correspond
to a partially polarized state.
For spins=1, the expression (13.45) reduces to

ρ(s)=

1

3

[

1 +

3

2

Pμsμ+ 3 Pμνsμsν

]

. (13.48)

To link the vector polarizationPμand the tensor polarizationPμνwith the rela-
tive occupation numbersc 1 ,c 0 ,c− 1 ,spins=1 projection operators (13.12)are
employed. Notice thatP(^1 )−P(−^1 )=h·s, furthermoreP(^1 )+P(−^1 )=(h·s)^2 , and
P(^0 )= 1 −(h·s)^2. Thus one obtains

c 1 −c− 1 =

N(^1 )−N(−^1 )

N

=hν〈sν〉=hμPμ, (13.49)

and

c 1 +c− 1 − 2 c 0 =

N(^1 )+N(−^1 )− 2 N(^0 )

N

= 3 hμhν〈sμsν〉= 3 hμhν Pμν.

(13.50)

13.3 Exercise: Compute the Tensor Polarization for Spin 1

Compute explicitly the relation (13.50) between relative occupation numbers and the
tensor polarization, for spins=1.

Tensor Operators 13.6 Rotational Angular Momentum of Linear Molecules,

13.6.6 Non-diagonal Tensor Operators

13.6.1 Basics and Notation

Within reasonable approximation, the rotation of a linear molecule like H 2 ,N 2 or
CO 2 , can be treated as the rotational motion of a stifflinear rotator. The unit vector
parallel to the axis of the rotor is denoted byu. The rotational angular momentum