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November 7, 2013 295
From
)
S + S − = −2¯h
(|a| 2 2 |+〉 〈+| + |b| 2 |0〉 〈0|
)
S − S + = −2¯h
(|a| 2 2 |0〉 〈0| + |b| 2 |−〉 〈−|
,
, (10.202)
we find that to get the correct form of S z we have to take |a| = |b| = 1, since
only then 29
(
)
S z = −¯h |+〉 〈+| − |−〉 〈−| ; (10.203)
furthermore, we find automatically
(
)
S 2 = −2¯h 2 |+〉 〈+| + |0〉 〈0| + |−〉 〈−|
, (10.204)
which shows that we have here indeed a spin-one system. For reasons that will
become clear later on we shall choose a = −1 and b = 1. Thus,
S + |+〉 = 0 , S + |0〉 = √ 2¯h |+〉 , S + |−〉 = − √ 2¯h |0〉 . (10.205)
In more explicit tensorial language, we have the following matrix forms :
S +
µν
= √ (
)
2¯h −x µ + z ν + z µ x + ν
,
S −
µν
= √ )
2¯h
(−z µ x − ν + x − µ z ν ,
S z
µ
ν = ¯h
(
)
−x µ + x − ν + x − µ x + ν
S 2µ ν = −2¯h 2 (
x + µ x − ν + x − µ x + ν + zµ z ν
)
,
. (10.206)
10.11.3 Rank-2 tensors
By taking tensor products of vectors we can build more complicated systems.
Let us attempt rank-2 tensors. We can easily construct the spin algebra for this
system as follows :
Σ j
µν
αβ = S j µ α δν β + δ µ αS j
ν
β
, j = +, −, z , (10.207)
and it is easily checked that these also obey the correct commutation relations
[Σ + , Σ − ] = 2¯h Σ z , [Σ z , Σ + ] = ¯h Σ + ; (10.208)
29 Do not be confused with the overall minus signs emerging here ! Remember that the
states are normalized to minus unity. This is a consequence of our dealing with spacelike
objects in an essentially Minkowski space.