into±-eigenspaces of the matrixσ|p·p|. In our discussion of the Bloch sphere in
section 7.5 we explicitly found that (see equation 7.6)
u+(p) =1
√
2(1 +p 3 )(
1 +p 3
p 1 +ip 2)
(34.7)
provides a normalized element of the + eigenspace ofσ|p·p| that satisfies
Ωu+(p) =u+(Rp)Similarly, we saw that
u−(p) =1
√
2(1 +p 3 )(
−(p 1 −ip 2 )
1 +p 3)
(34.8)
provides such an element for the−eigenspace.
Another way to construct such elements is to use projection operators. The
operators
P±(p) =1
2
( 1 ±
σ·p
|p|)
provide projection operators onto these two spaces, since one can easily check
that
P+^2 =P+, P−^2 =P−, P+P−=P−P+= 0, P++P−= 1
Solutions can now be written as
ψ ̃E,±(p) =αE,±(p)u±(p) (34.9)for arbitrary functionsαE,±(p) on the sphere|p|=
√
2 mE, where theu±(p) in
this context are called “spin polarization vectors”. There is however a subtlety
involved in representing solutions in this manner. Recall from section 7.5 that
u+(p) is discontinuous atp 3 =−1 (the same will be true foru−(p)) and any
unit-length eigenvector ofσ·pmust have such a discontinuity somewhere. If
αE,±(p) has a zero atp 3 =−1 the productψ ̃E,±(p) can be continuous. It
remains a basic topological fact that the combinationψ ̃E,±(p) must have a
zero, or it will have to be discontinuous. Our choice ofu±(p) works well if this
zero is atp 3 =−1, but if it is elsewhere one might want to make a different
choice. In the end one needs to check that computed physical quantities are
independent of such choices.
Keeping in mind the above subtlety, theu±(p) can be used to write an
arbitrary solution of the Pauli equation 34.3 of energyEas
(
ψ 1 (q,t)
ψ 2 (q,t))
=e−iEt(
ψ 1 (q)
ψ 2 (q)