4.1. ANEWREPRESENTATIONOFMOTION 47
=
a b 0... 0
(4.4)
whichseemstosuggestthattheparticlecouldbeinintervals 1 and 2 atthesame
time? Inaclassical mechanics, aparticle isdefinitelyin asingleinterval atany
giventime,soonlytheunitvectors%enarephysicallymeaningful. Theverynatural
mathematicaloperationofadditionofvectorswouldthereforehavetoberegarded,
inthisrepresentationofclassicalmotion,asphysicallymeaningless.
Ontheotherhand,wehavealreadyseenthatthedeBrogliewavefunctionψ(x,t)
givesprobabilisticinformationaboutthelocationoftheelectron. Therefore,letusgive
thefollowing”BornInterpretation”totheN-dimensionalvector,whosecomponents
ψnareallowedtobecomplexnumbers:
ψ% = ψ 1 %e^1 +ψ 2 %e^2 +...+ψN%eN=
ψ 1
ψ 2
ψ 3.
.
ψN
(4.5)
TheprobabilityPnthatanelectron,inthestaterepresentedbythevectorψ%,will
befounduponmeasurementtobeinthen-thintervalofthetube,isequaltosquared
modulus
Pn=ψ∗nψn (4.6)Digression
Wemustpauseforamomenttodiscusstheinnerproductofvectorswithcomplex
components. Thenormofavectorisdefinedassquarerootoftheinnerproductof
avectorwithitself
|v|=√
%v·%v (4.7)wheretheinnerproductisrepresentedastheproductofarowvectortimesacolumn