328 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
WenextconsiderabasisforHilbertSpaceconsistingoftheeigenstates|φn>of
theHarmonicoscillatorHamiltonianHho.TheHilbertSpacewehavediscussedsofar
consistsofthephysicalstatesofaparticlemovinginonedimension. TheHarmonic
Oscillatorrepresentationisjustabasisinthatspace, andcanbeusedwhetheror
nottheparticleisactuallymovinginaharmonicoscillatorpotential(althoughthat
wouldbethemostusefulapplication).
FromtheorthonormalityofeigenstatesofaHermitianoperator,thewavefunction
oftheeigenstate|φn>is
φn(m)=<φm|φn>=δmn m= 0 , 1 , 2 ,.... (21.156)Inotherwords, thewavefunctionof eigenstates of theharmonic oscillator, inthe
harmonicoscillatorrepresentation,canberepresentedasinfinite-dimensionalcolumn
vectors:
φ 0 =
1 0 0...
φ 1 =
0 1 0...
φ 2 =
0 0 1...
..... (21.157)
Infact,becausen= 0 , 1 , 2 ,...isadiscreteindex,all”eigenfunctions”
ψ(n)=< φn|ψ > can be thoughtof as thecomponents of acolumn vector. In
particular,aneigenfunctionofposition|x>,intheHO-representation,is
ψx(n) = <φn|x>
= <x|φn>∗
= φn(x) (21.158)Inotherwords,aneigenstateofpositionhastheformofacolumnvector
ψx=
φ 0 (x)
φ 1 (x)
φ 2 (x)
.
.
.
(21.159)
Thentheorthogonalityofpositioneigenstatesimplies
δ(x−y) = <x|y>