QMGreensite_merged

13.1. SPINWAVEFUNCTIONS 217

Notethattheinnerproduct

αy·βy=

1

2

[1,−i]

[

1

−i

]

= 1 +(−i)^2 = 0 (13.78)

vanishes,soαyandβyareorthogonal,asexpected.
Bysimilarmeanswecanfindtheeigenstatesofσx,withtheresult:

αx =

1

√

2

[

1

1

]

βx=

1

√

2

[

1

− 1

]

αy =

1

√

2

[

1

i

]

βy=

1

√

2

[

1

−i

]

χ+=αz =

[

1

0

]

χ−=βz=

[

0

1

]

(13.79)

Exercise: Obtain the eigenstates αx andβx of Sx by the same meansused for
obtainingtheeigenstatesofSy.

Toanalyzethevariationwithtimeoftheelectronmagneticmomentinanexternal
magneticfield,weuse thetime-dependentSchrodingerequationwithHamiltonian
(13.61)

ih ̄∂tψ=

1

2

̄hΩσzψ (13.80)

or,inmatrixform,

[

∂tψ+

∂tψ−

]

=−i

1

2

Ω

[

1 0

0 − 1

][

ψ+

ψ−

]

=

[

−^12 iΩψ+

1

2 iΩψ−

]

(13.81)

Thesearetwoindependentfirst-orderdifferentialequations,oneforψ+andonefor
ψ−,andthegeneralsolutioniseasilyseentobe

ψ(t)=

[

ae−iΩt/^2

beiΩt/^2

]

(13.82)

whereaandbareconstants
Supposethatattimet= 0 theelectronisina”spin-up”stateinthex-direction.
Thismeansthattheconstantsaandbaredeterminedtobe

a=b=

1

√

2

(13.83)