QMGreensite_merged

218 CHAPTER13. ELECTRONSPIN

LetT = 2 π/ΩbetheperiodcorrespondingtoanangularfrequencyΩ. Then,from
thesolution above, weseethat thewavefunction fortheelectronspin”precesses”
aroundthez-axis:

ψ(0) = αx (assumption)

ψ(

T

4

) =

1

√

2

[

e−iπ/^4

eiπ/^4

]

=e−iπ/^4 αy

ψ(

T

2

) =

1

√

2

[

e−iπ/^2

eiπ/^2

]

=e−iπ/^2 βx

ψ(

3 T

4

) =

1

√

2

[

e−^3 iπ/^4

e^3 iπ/^4

]

=e−^3 iπ/^4 βy

ψ(T) =

1

√

2

[

e−iπ

eiπ

]

=e−iπαx (13.84)

Thusthespindirectionoftheelectronprecessesaroundthez-axiswithaprecession
frequencyf = Ω/ 2 π. The factthat aspin 1 / 2 particleprecesses inthe presence
ofanexternalmagneticfieldisofconsiderablepracticalimportancefortheNuclear
MagneticResonance Imagingtechnique inmedicine,whichtakesadvantage ofthe
precessionofatomicnucleiofspin 1 /2.

• Spin-OrbitCoupling

Evenwhen thereisno external magnetic field,the frequenciesof the spectral
lines of the Hydrogen atom arenot exactly as predicted by Bohr. Certain lines
ofthespectrum,when observedusingahigh-resolutionspectrometer,arefoundto
actuallybetwoclosely spacedspectrallines;thisphenomenonisknownasatomic
finestructure. Historically,itwasanattempttoexplainthisfinestructureofspectral
linesinthealkalielementsthatledGoudsmitandUhlenbeck,in1925,toproposethe
existenceofelectronspin.
Tounderstand theGoudsmit-Uhlenbeckreasoning, consider an elecronmoving
withvelocity%vacrossastatic electricfield, suchasthe Coulombfield due toan
atomicnucleus. Accordingto theoryof specialrelativity,theelectromagneticfield
asitappearsintherestframeoftheelectronisnolongerentirelyelectric,butalso
containsamagneticcomponent

B% = −√^1

1 −v

2

c^2

%v

c

×E%

≈ −

1

Mc

%p×E% (13.85)

Giventhattheelectron hasamagneticmoment%μ,thereshouldbe aninteraction
energyduetotheelectronmagneticmoment,intherestframeoftheelectron

H′=−%μ·B% (13.86)