Tensors for Physics

244 13 Spin Operators

In theHeisenberg picture, the time dependence of an operatorOis governed by the
commutator with the relevant Hamilton operatorH:

dO

dt

=

i



[H,O]−. (13.18)

For particles with a magnetic momentμsλ, in the presence of a magnetic fieldBλ=
Bhλ, wherehis unit vector, one hasH=−μBhλsλ. Then the commutation relation
(13.16)forthe-th rank spin tensor implies

d

dt

sμ 1 sμ 2 ···sμ =−ωBhλ()μ 1 μ 2 ···μ,λ,ν 1 ν 2 ···νsν 1 sν 2 ···sν, (13.19)

with the precession frequencyωB=μB/.
The commutation relation for two second rank spin tensors is

[sμsν,sκsλ]−=i

{

μν,λ,αβ(sκsαsβ +sαsβsκ)

+μν,κ,αβ(sλsαsβ +sαsβsλ)

}

. (13.20)

Commutators of this type occur in applications, when the Hamilton operator involves
the second rank spin tensor, as in the case of a nucleus with an electric quadrupole
moment, e.g. the deuteron, in the presence of an electric field.

13.3.3 Scalar Products.

The scalar product or total contraction of two irreducible spin tensors of rankis
given by

sμ 1 sμ 2 ···sμ sμ 1 sμ 2 ···sμ =

!

( 2 − 1 )!!

S^20 S^21 ···S^2 l− 1. (13.21)

The factorN=( 2 −! 1 )!!occurs in the corresponding expression for the contraction
of irreducible tensors constructed from vectors with commuting components, cf.
(9.10) and (9.11). The factors

Sk^2 =s(s+ 1 )−

k

2

(

k

2

+ 1

)

, (13.22)

reflect quantum mechanical features of the spin. Notice, one hasSk^2 =0fork=
2 s, and the norm of the irreducible spin tensor of rank 2s+1 is zero, in accord
with (13.15). On the other hand, fors1 ands, the productS 02 S 12 ···Sl^2 − 1

approaches its ‘classical’ values^2 .