QMGreensite_merged

68 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE

or,moreexplicitly [

−

̄h^2

2 m

d^2

dx^2

+V(x)

]

φ(x)=Eφ(x) (5.35)

isknownastheTime-IndependentSchrodingerEquation.
Equation(5.34)isan exampleof an Eigenvalue Equation. This isalinear
differentialequationinwhichonehastosolvesimultaneouslyforasetofconstants,
knownastheeigenvalues,andasetofcorrespondingfunctionsknownastheeigen-
functions. InthecaseoftheSchrodingerequation,theconstantEiscalledthethe
”energyeigenvalue,”andthefunctionφ(x)iscalledthe”energy eigenfunction”or
”energyeigenstate.”
ToeachenergyeigenvalueEthereisatleastone(andsometimesmorethanone)
energyeigenstate,andtoeacheigenstatetherecorrespondsasolution

ψ(x,t)=φ(x)e−iEt/ ̄h (5.36)

ofthetime-dependentSchrodingerequation. Suchsolutionsarealsoknownassta-
tionarystatesbecausethetime-dependenceiscontainedentirelyinanoverallphase.
Thismeansthattheprobabilitytofindaparticleintheneighborhoodofpointx,i.e

P!(x)=ψ∗(x,t)ψ(x,t)!=φ∗(x)φ(x)! (5.37)

isconstantintime.
Let{φα(x),Eα}denoteacompletesetofeigenstatesandcorrespondingeigenval-
ues,inthesensethatanyothereigenstatewhichisnotinthesetcanbeexpressedasa
linearcombinationofthosethatare.Thenthegeneralsolutiontothetime-dependent
Schrodingerequationis
ψ(x,t)=

∑

α

cαφα(x)e−iEαt/ ̄h (5.38)

iftheenergyeigenvaluesareadiscreteset,or

ψ(x,t)=

∫

dαcαφα(x)e−iEαt/ ̄h (5.39)

iftheenergyeigenvaluesspanacontinuousrange,oracombination

ψ(x,t)=

∑

α

cαφα(x)e−iEαt/ ̄h+

∫

dαcαφα(x)e−iEαt/ ̄h (5.40)

ofsumandintegral,ifsomeeigenvaluesarediscreteandothersarefoundinacon-
tinuousrange.
Mostoftheeffortinquantummechanicsgoesintosolvingthetime-independent
Schrodingerequation(5.35). Onceacomplete setof solutionstothat equationis
found,thegeneralsolution(eq. (5.40))tothetime-dependentproblemfollowsim-
mediately.