Tensors for Physics

242 13 Spin Operators

These projectors have the properties

P(m)P(m

′)

=δmm′P(m),

∑s

m=−s

P(m)= 1. (13.9)

Thus they are orthogonal, idempotent as any projector, and they form a complete set.
Theprojectorscanbeexpressedintermsofpowersh·s,analogoustotheHamilton-
Cayley relation. In particular, one has

P(m)=

∏

m′=m

(h·s−m′)

m−m′

. (13.10)

Clearly, in the product, the magnetic quantum numbers run over all allowed values,
exceptm. The highest power ofh·soccurring in (13.10)is2s.
For spins=^12 , the projection operators are

P(^1 /^2 )=

1

2

+h·s, P(−^1 /^2 )=

1

2

−h·s. (13.11)

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.
Fors=1, the projection operators are

P(±^1 )=

1

2

h·s( 1 ±h·s), P(^0 )=( 1 −h·s)( 1 +h·s). (13.12)

13.3 Irreducible Spin Tensors

13.3.1 Defintions and Examples

The-rank irreducible tensor constructed from the components of the spin operator
sis the symmetric traceless tensor

sμ 1 sμ 2 ···sμ =Δ()μ 1 μ 2 ···μ,ν 1 ν 2 ···νsν 1 sν 2 ···sν. (13.13)

Here the symmetrization matters. This is in contradistinction to tensors constructed
from vectors whose components commute.
The second rank irreducible tensor is explicitly given by

sμsν =

1

2

(sμsν+sμsν)−

1

3

s(s+ 1 )δμν. (13.14)