QMGreensite_merged

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)