214 CHAPTER13. ELECTRONSPIN
where”spinup”and”spindown”refertothetwopossibilitiesoffindingsz=^12 and
sz=−^12 .Comparingthelasttwolines
Prob(spinup) =
∫
dxdydz|ψ+|^2
Prob(spindown) =
∫
dxdydz|ψ−|^2 (13.55)
Althoughψ+andψ−are,ingeneral,independentfunctions,therearemanyim-
portantcaseswherethespatialandspinpartsofthewavefunctionfactorize,i.e.
ψ=ψ(x)
[
a
b
]
(13.56)
AnexamplewehavejustseenaretheeigenstatesofmomentaandSzshownineq.
(13.50)and(13.51). Anotherexampleistheset ofeigenstatesofH,L^2 ,Lz,S^2 ,Sz,
whereHistheHydrogenatomHamiltonian.The”spin-up”eigenstatesare
φ(r,θ,φ)χ+=Rnl(r)Ylm(θ,φ)
[
1
0
]
(13.57)
whilethe”spin-down”eigenstatesare
φ(r,θ,φ)χ−=Rnl(r)Ylm(θ,φ)
[
0
1
]
(13.58)
Inketnotation,thestatesarelabeledbytheirquantumnumbers:
{|nlmssz>} where s=
1
2
, sz=±
1
2
(13.59)
Wecan getsome practiceinthe use ofspinor notationandspinmatrices, by
studyingtheprecessionofelectronspininanexternalmagneticfield.
- ElectronPrecession
Supposeanelectronisinastatewherep≈0,sotheelectronspinor,whileitcan
dependontime,doesnotdependonspace,i.e.
ψ(t)=
[
ψ+(t)
ψ−(t)
]
(13.60)
Inaddition,supposethatthereisanexternal,constantmagneticfield,ofmagnitude
B,inthez-direction. TheHamiltonianisthen
H =
p^2
2 M
−%μ·B%
≈
e ̄hBg
4 Mc
σz
≈
1
2
̄hΩσz (13.61)