QMGreensite_merged

13.1. SPINWAVEFUNCTIONS 215

where

Ω=

eB

Mc

(13.62)

Ωisknownasthe”cyclotronfrequency.”TheeigenstatesofthisHamiltonianarethen
simplyχ±,becauseHisproportionaltoσz,andχ±aretheeigenstatesofσz:

Hχ+ = E+χ+ where E+=

1

2

̄hΩ

Hχ− = E−χ− where E−=−

1

2

̄hΩ (13.63)

EigenstatesoftheHamiltonianarestationarystates;anelectroninsuchastate
willremaininthateigenstateindefinitely.Butsupposeinsteadthattheelectronspin
isinitiallyinaneigenstateofSxorSy.Whatwenowshowisthattheelectronspin
willthentendtoprecessaroundthez-axis.
First,somenotation.Define”spin-up”statesand”spin-down”statesinthex,y,
andz-directionstobethe±^12 ̄heigenstatesofSx, Sy, Szrespectively:

Sxαx =

1

2

̄hαx Sxβx=−

1

2

̄hβx

Syαy =

1

2

̄hαy Syβy=−

1

2

̄hβy

Szαz =

1

2

̄hαz Szβz=−

1

2

̄hβz (13.64)

Weknowalreadythat

αz=χ+=

[

1

0

]

and βz=χ−=

[

0

1

]

(13.65)

Findingtheeigenstatesαy,βyrequiresfindingtheeigenstatesofthePaulimatrixσy.
Itisinstructivedothisindetail.
Thefirstthingtodoistofindtheeigenvaluesofσy. Theeigenvalueequationis

σy%u=λ%u (13.66)

andtheeigenvaluesaredeterminedbysolvingthesecularequation

det[σy−λI]= 0 (13.67)

whereIistheunitmatrix. ForthePaulimatrixσy,thisis

det

[

−λ i

−i −λ

]

=λ^2 − 1 = 0 (13.68)

sotheeigenvaluesare
λ 1 =+1 λ 2 =− 1 (13.69)