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foranychoiceofi,jwithi+=j.
Sofar, allthis reasoninghasbeenveryhypothetical. Ifquantumstatescarry
noexperimentallyinaccessible”hiddeninformation,”thenthewavefunctionsmustbe
eithersymmetricorantisymmetricwithrespecttoaninterchangeofpositionsand
spins.Nowcomesanabsolutelyastoundingfact,knownas
TheSpin StatisticsTheorem
Systems of identical particles withinteger spins = 0 , 1 , 2 ...are describedby
wavefunctionsthataresymmetricundertheinterchangeofparticlecoordinatesand
spin.Systemsofidenticalparticleswithhalf-integerspins=^12 ,^32 ,^52 ...aredescribedby
wavefunctionsthatareantisymmetricundertheinterchangeofparticlecoordinates
andspin.
ParticleswithintegerspinareknownasBosons. Examplesarepi-mesons(spin
zero),photons(spin1),andgravitons(spin2). Particleswithhalf-integerspinare
knownasFermions. Examplesincludeelectrons,protons,andneutrinos(spin1/2),
theOmegahyperon(spin3/2),andthe(hypothetical)gravitino(spin3/2)predicted
byatheoryknownassupergravity.
TheSpin-Statistics Theoremiscalledatheorembecauseitcan beprovenfrom
ahandfulof axioms(causality, locality, Lorentzinvariance...) inthecontext of a
relativisticextensionofquantummechanicsknownasquantumfieldtheory.Wewill
touchbrieflyonquantumfieldtheoryattheveryendofthesemester,buttheproof
of theSpin-Statistics Theoremisnoteasy, andissimply beyondthescopeof the
course. Letusjustnoteherethatitsimplicationshavebeentested,experimentally,
inmanyways.Themostimportantconsequenceforatomicphysicsisknownas
ThePauli Exclusion Principle
Notwoelectronscanbeinexactlythesamequantumstate.Thisisthe fact whichmakes chemistry possible, and leads to theremarkable
repetitionof chemicalproperties amongtheelements whichissummarizedin the
PeriodicTable.ButinordertounderstandtheExclusionPrinciple,andhowitfollows
fromtheSpin-StatisticsTheorem,itsusefultofirstlookatsomesimpleexamplesof
2-electronwavefunctions.