QMGreensite_merged

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)