240 13 Spin Operators
Here,[.., ..]−indicates the commutator. Much as (7.87) implies the relation (7.88),
the spin commutation relation is equivalent to
ελμνsμsν=isλ, (13.2)or
s×s=is. (13.3)Clearly, the components of the quantum mechanical angular momentum do not com-
mute, in contradistinction to the components of the classical angular momentum for
which the corresponding cross product vanishes.
The scalar product of two spinsvector operators is
s·s=sμsμ=s(s+ 1 ) 1. (13.4)Here 1 stands for the unit operator in the spin space. In matrix representation, this
is just the( 2 s+ 1 )×( 2 s+ 1 )unit matrix. This unit operator is omitted frequently,
when no danger of confusion exists.
13.1.2 Spin 1=2 and Spin 1 Matrices
The spin matrices fors= 1 /2, the spin matricessx,sy,szare
1
2(
01
10
)
,
i
2(
0 − 1
10
)
,
1
2
(
10
0 − 1
)
. (13.5)
Apart from the factor 1/2, these are the Pauli matricesσx,σy,σz.
Fors=1, the spin matricessx,sy,szare
1
√
2
⎛
⎝
010
101
010
⎞
⎠, √i
2⎛
⎝
0 − 10
10 − 1
01 0
⎞
⎠,
⎛
⎝
10 0
00 0
00 − 1
⎞
⎠. (13.6)
13.1 Exercise: Verify the Normalization for the Spin 1 Matrices
Compute explicitlysx^2 +sy^2 +s^2 zfor the spin matrices (13.6) in order to check the
normalization relation (13.4).