46 CHAPTER4. THEQUANTUMSTATE
LetusimaginesubdividingthelengthofthepipeintoNequalintervalsoflength
!=L/N, asshowninFig. [4.2]. Iftheparticleisinthe firstinterval,we choose
torepresentitsstatenotbyanumber,suchasx 1 ,butratherbyanN-dimensional
vector
%e^1 =
1 0 0... 0
(4.1)
Iftheparticleisinthesecondinterval,thiswillberepresentedby
%e^2 =
0 1 0... 0
(4.2)
andsoon,downtotheN-thintervalwhichisdenotedby
%eN=
0 0 0... 1
(4.3)
Thepositionoftheparticleisthereforeapproximated,atanygiventime,byoneof
the%enunitvectors,andastheparticlemovesfromoneintervaltoanother,theunit
vector”jumps”discontinuouslyfromsome%ektoeither%ek+1or%ek−^1 ,dependingon
whichwaytheparticleismoving. IfthenumberofintervalsNislargeenough,the
particlepositioncanberepresented,asafunctionoftime,toarbitraryprecision.
Areasonableobjectiontothisrepresentationofmotionisthat,inintroducingaset
ofNorthogonalvectors{%en},wehavealsointroducedanN-dimensionalspacewhich
containslinearcombinations ofthosevectors. What, forexample, isthe physical
meaningofavectorsuchas
%v = a%e^1 +b%e^2