18.8 Spin
Earlier, we showed that both integer and half integer angular momentum could satisfy (See section
14.4.5) the commutation relations for angular momentum operatorsbut that there is no single
valued functional representation for the half integer type.
Some particles, like electrons, neutrinos, and quarks have half integerinternal angular momen-
tum,also calledspin. We will now develop aspinor representationfor spin^12. There are no
coordinatesθandφassociated with internal angular momentum so the only thing we haveis our
spinor representation.
Electrons, for example, have total spin one half. There are no spin3/2 electrons so there are only
two possible spin states for an electron. The usual basis states are the eigenstates ofSz. We know
from our study of angular momentum, that the eigenvalues ofSzare +^12 ̄hand−^12 ̄h. We will simply
represent the +^12 ̄heigenstate as the upper component of a 2-component vector. The−^12 ̄heigenstate
amplitude is in the lower component. So the pure eigenstates are.
χ+=(
1
0
)
χ−=(
0
1
)
An arbitrary spin one half state can be represented by a spinor.
χ=(
a
b)
with the normalization condition that|a|^2 +|b|^2 = 1.
It is easy toderive(see section 18.11.6)the matrix operators for spin.
Sx=̄h
2(
0 1
1 0
)
Sy=̄h
2(
0 −i
i 0)
Sz=̄h
2(
1 0
0 − 1
)
These satisfy the usual commutation relations from which we derived the properties of angular
momentum operators. For example lets calculate the basic commutator.
[Sx,Sy] =̄h^2
4[(
0 1
1 0
)(
0 −i
i 0)
−
(
0 −i
i 0)(
0 1
1 0
)]
=
̄h^2
4[(
i 0
0 −i)
−
(
−i 0
0 i)]
=
̄h^2
2(
i 0
0 −i)
=i ̄h̄h
2(
1 0
0 − 1
)
=i ̄hSzThe spin operators are an (axial)vector of matrices. To form the spin operator for an arbitrary
direction ˆu, we simply dot the unit vector into the vector of matrices.
Su= ˆu·S~ThePauli Spin Matrices,σi, are simply defined and have the following properties.
Si ≡̄h
2σi