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)