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314 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

whereaandbareanyconstants. Thereisalsodefinedaninnerproduct<u|v>
betweenbraandketvectors
<u|v>=<v|u>∗ (21.61)

withthebi-linearityproperty

[a<q|+b<r|][c|s>+d|t>]=ac<q|s>+ad<q|t>+bc<r|s>+bd<r|t>
(21.62)
wherea,b,c,dareallconstants.
Inordertodistinguishquantitativelybetweenvectorsinthevectorspace, itis
importanttointroducewhatamountstoareferenceframe,calledabasis. Abasis
foraD-dimensional vector spaceisaset ofD orthonormalvectors{|en >, n=
1 , 2 ,...,D}
<en|em>=δmn (21.63)

suchthatanyelementofthevectorspacecanbewrittenasalinearcombination

|v>=

∑D

m=1

vm|em> (21.64)

Thisexpressioniscompletelyanalogoustotheexpansionofvectorsthatisoftenseen
inclassicalmechanics
F%=Fx%i+Fy%j+Fz%k (21.65)

where%i,%j, %kareunitvectorsinthex,y,andz-directionsrespectively.
Takingtheinnerproductofeq. (21.64)withthebravector<en|

<en|v>=<en|

∑D

m=1

vm|em> (21.66)

andusingthebi-linearityproperty(21.62)

<en|v>=

∑D

m=1

vm<en|em> (21.67)

andtheorthonormalityproperty(21.63)

<en|v>=

∑D

m=1

vmδnm (21.68)

wefind:

TheComponent(or”Wavefunction”)ofVector|v> inthebasis{|en>}

vn=<en|v> (21.69)