Chapter 9
The Harmonic Oscillator
ThereareonlyaveryfewpotentialsforwhichtheSchrodingerequationcanbesolved
analytically. Themostimportantoftheseisthepotential
V(x)=1
2
k(x−x 0 )^2 (9.1)oftheharmonic oscillator,becauseso manyof thesystemsencounteredinnuclear
physics,condensedmatterphysics,andelementaryparticlephysicscanbetreated,
toafirstapproximation,asasetofcoupledharmonicoscillators.
Thereareseveralreasonswhytheharmonicoscillatorpotentialshowsupsoof-
ten. Firstofall,anyfinitepotentiallookslikeaharmonicoscillatorpotentialinthe
neighborhoodofitsminimum. Supposetheminimumofapotentialisatthepoint
x=x 0 .ExpandthepotentialinaTaylorseriesaroundx 0
V(x)=V(x 0 )+(
dV
dx)x=x 0(x−x 0 )+1
2
(
d^2 V
dx^2)x=x 0(x−x 0 )^2 +... (9.2)BecauseV(x)isaminimumatx 0 ,thefirstderivativeofthepotentialvanishesthere,
so
V(x)=V(x 0 )+1
2
(
d^2 V
dx^2)x=x 0(x−x 0 )^2 +... (9.3)Forsmalldisplacements,thisisjustaharmonicoscillator.
Secondly, any timetheclassical equations of motionarelinear, itmeansthat
wearedealingwithaharmonicoscillator,orasetofcoupledharmonicoscillators.
Newton’slawofmotionF=maisgenerallynon-linear,sinceF(x)isusuallyanon-
linearfunctionofx.However,ifFdependslinearlyonx,itfollowsthatthepotential
dependsquadraticallyonx,whichimpliesaharmonicoscillator. Nowitsohappens
thatanynon-dispersivewaveequation,e.g.
1
v^2∂^2 y
∂t^2=
∂^2 y
∂x^2