Chapter 4
The Quantum State
Wetakeforgrantedtheideathateveryobject,ateverymomentintime,iscenteredat
adefinitepointinthree-dimensionalspace,andthatoveraspanoftime,thelocation
oftheobjecttracesatrajectorythroughthatspace. Wearesureofthisbecauseof
ourlongexperience,beginningatbirth,withthebehaviorofmacroscopicobjects,i.e.
objectsonthescaleofamicronorlarger,objectswhichwecansee,objectswhichwe
havetheillusionofobservingcontinuously.^1
Itisnow timetoabandonthe viewthatobjectsmove intrajectoriesinthree-
dimensionalspace,atleastinthecaseofmicroscopicobjectssuchaselectrons,pho-
tons,andatoms.Thisisfortworeasons. First,ifaparticlemovesinatrajectory,then
itmusthaveadefinitepositionandmomentumateverymomentintime.However,
as wehavediscussedinthe caseoftheHeisenbergmicroscope, thereisno exper-
imentwhichcanverifythis. Second,andmoreimportantly, themereassumption
thatelectronsandphotonsfollowdefinitetrajectories%x(t)leadstoconclusionswhich
disagreewithexperiment,suchastheDavisson-Germerexperimentdiscussedinthe
lastlecture,andmanyotherexamplestofollowlaterinthecourse.
How,then,doesanelectrongetfromonepointtoanother,withouttracingapath
throughspace?Theansweristhattheelectrondoesfollowapath;however,itisnot
apathintheordinarythree-dimensionalspaceweareusedto.
4.1 A New Representation of Motion
Consideraparticlemovingbackandforth,accordingtothelawsofclassicalphysics,in
athinclosedpipeoflengthL.AplotofitstrajectorymightlooklikeFig.[4.1].This
isthestandardwayofrepresentingone-dimensionalmotioninclassicalmechanics.
Toeveryinstantoftimetthereisacorrespondingnumberx(t),whichinthiscaseis
thedistancefromoneendofthepipe. Isthereanyotherconceivablerepresentation
ofmotioninonedimension?
(^1) Noobservationistrulycontinuous,becauseofthediscrete,photoncompositionoflight.