1.4. THECLASSICALSTATE 19
possible,bysolvingtheequationsofmotionexactly).Notethatthecoordinates{qa}
alonearenotenoughtospecifythephysicalstate,becausetheyarenotsufficientto
predictthefuture.Informationaboutthemomenta{pa}isalsorequired.
Thespaceofallpossible{qa,pa}isknownasphasespace. Forasingleparticle
movinginthreedimensions,therearethreecomponentsofpositionandthreecompo-
nentsofmomentum,sothe”physicalstate”isspecifiedby 6 numbers(x,y,z,px,py,pz),
whichcanbeviewedasapointin6-dimensionalphasespace. Likewise,thephysical
stateof asystemofN particlesconsistsof 3 coordinatesforeachparticle(3N co-
ordinatesinall),and 3 componentsofmomentumforeachparticle(3Nmomentum
componentsinall),sothestateisgivenbyasetof 6 Nnumbers,whichcanbeviewed
asasinglepointin 6 N-dimensionalspace.
Aswewillseeinthenextlectures,classicalmechanicsfailstopredictcorrectly
thebehaviorofbothlightandmatterattheatomiclevel,andisreplacedbyquantum
mechanics. Butclassicalandquantummechanicshavealotincommon: theyboth
assignphysicalstatestoobjects,andthesephysicalstatesevolveaccordingto1st-
orderdifferentialequations. Thedifferencelies mainlyin thecontrast between a
physical stateas understoodbyclassical mechanics, the ”classicalstate”, andits
quantumcounterpart,the”quantumstate”. Thisdifferencewillbeexploredinthe
nextfewlectures.