Chapter 1
The Classical State
Inthefirstquarterofthiscentury,itwasdiscoveredthatthelawsofmotionformulated
byGalileo,Newton,Lagrange,Hamilton,Maxwell,andmanyothers,wereinadequate
toexplainawiderangeofphenomenainvolvingelectrons,atoms,andlight.Aftera
greatdealofeffort,anewtheory(togetherwithanewlawofmotion)emergedin1924.
Thattheoryisknownasquantummechanics,anditisnowthebasicframeworkfor
understandingatomic,nuclear,andsubnuclearphysics,aswellascondensed-matter
(or”solid-state”)physics. Thelawsofmotion(duetoGalileo, Newton,...) which
precededquantumtheoryarereferredtoasclassicalmechanics.
Althoughclassicalmechanicsisnowregardedasonlyanapproximationtoquan-
tummechanics,itisstilltruethatmuchofthestructureofthequantumtheoryis
inherited fromtheclassicaltheorythat itreplaced. Sowe beginwithalightning
reviewof classicalmechanics, whose formulationbegins (butdoesnotend!) with
Newton’slawF=ma.
1.1 Baseball, F = ma, and the Principle of Least
Action
Takeabaseballandthrowitstraightupintheair. Afterafractionofasecond,or
perhapsafewseconds, thebaseballwillreturntoyourhand. Denotetheheightof
thebaseball,asafunctionoftime,asx(t);thisisthetrajectoryofthebaseball. If
wemakeaplotofxasafunctionoft,thenanytrajectoryhastheformofaparabola
(inauniformgravitationalfield,neglectingairresistance),andthereareaninfinite
numberofpossibletrajectories.Whichoneofthesetrajectoriesthebaseballactually
followsisdeterminedbythemomentumofthebaseballatthemomentitleavesyour
hand.
However,ifwerequirethatthebaseballreturnstoyourhandexactly∆tseconds
afterleavingyourhand,thenthereisonlyonetrajectorythattheballcanfollow.For
abaseballmovinginauniformgravitationalfielditisasimpleexercisetodetermine