142 CHAPTER9. THEHARMONICOSCILLATOR
isalinearequationofmotion. Thisequationmightdescribeatransverse waveon
astring,orasoundwaveinasolid,oranelectromagneticplanewave, propagating
inthex-direction. Any systemthatsatisfiesawaveequationofthisformcan be
thoughtofasaninfinitesetofharmonicoscillators.Toseethis,wedefinetheFourier
transformY(q,t)ofthewavefunctiony(x,t)atanytimetas
Y(q,t)=
∫∞
−∞
dxy(x,t)eiqx (9.5)
withtheinversetransform
y(x,t)=
∫∞
−∞
dq
2 π
Y(q,t)e−iqx (9.6)
Thefunctionsy(x,t)andY(q,t)containthesameamountofinformationaboutthe
wave attimet,sinceonefunctioncan beobtainedfromtheother. Now multiply
bothsidesof(9.4)byeiqx,integrateoverx,andthenintegratebyparts:
∫
dx
1
v^2
∂^2 y
∂t^2
eiqx =
∫
dx
∂^2 y
∂x^2
eiqx
∂^2
∂t^2
∫
dx
1
v^2
y(x,t)eiqx =
∫
dxy(x,t)
∂^2
∂x^2
eiqx
1
v^2
∂^2
∂t^2
∫
dxy(x,t)eiqx = −q^2
∫
dxy(x,t)eiqx
1
v^2
∂^2
∂t^2
Y(q,t) = −q^2 Y(q,t) (9.7)
Butthisissimplytheequationofmotionofaharmonicoscillator
accelleration
(
∂^2 Y
∂t^2
)
∝−displacement (Y) (9.8)
Weconcludethatthewaveequation(9.4)issecretlyaninfinitenumberofharmonic
oscillatorequations;oneequationforeachwavenumberq.
Wave equations show up everywherein classicalphysics. Waves propagatein
solids,liquids,gases,andplasmas;theypropagateintheelectromagneticfield,and
even(accordingtoEinstein’stheoryofgeneralrelativity)inthegravitationalfield.
Allofthesesystems canthereforeberegarded,at somelevelof approximation,as
setsofharmonicoscillators.Thisiswhytheharmonicoscillatorpotentialisthemost
importantproblemtosolveinquantumphysics. Fortunately,itisaproblemwitha
simpleandelegantsolution.
9.1 Raising and Lowering Operators
Letaandbbeanytworealnumbers,anddefine
c=a+ib (9.9)