New Version of the Rayleigh–Schrdinger Perturbation Theory(2)

ABSTRACT: It has been shown in our preceding papers that the linear dependence of the perturbation wave functions on the perturbation energies makes possible to calculate the exact perturbation energies from the values of the perturbation wave functions co

KALHOUSETAL.

Wenotethat 0denotestheunperturbedwavefunctionoftheHamiltonianH0.Dependingontheprobleminquestion,itcanbetheground-stateaswellasexcited-statewavefunction.

Despitethewell-knownformulationsthatcanbefoundinanytextbookonquantummechanics,thereisonepropertyoftheperturbationtheorythathasbeennoticed[1,2]andused[3–9]onlyrecently.Ithasbeenshownthatthevalueoftheperturbationwavefunction n(x)atanarbitrarilychosenpointxdependslinearlyontheperturbationenergyEn.Thislineardependencemakesitpossibletodeter-minetheexactperturbationenergiesfromtheval-uesof n(x)fortwoarbitrarilychosentrialpertur-bationenergiesEnbysimplecalculation.Inthisway,thefunctions nwhicharenotquadraticallyintegrablecanbeusedtocalculatetheexactpertur-bationenergiesEnand,inthenextstep,thecorre-spondingexactperturbationfunctions n.

Thismethodhasafewadvantages.Incontrasttotheusualformulationoftheperturbationtheory,thismethodbasedonthecomputationof nfromEq.(6)foragivenenergyEncaneasilybepro-grammedforarbitrarylargeordersofthepertur-bationtheory.Forexample,200perturbationener-giesEnnecessaryfor ndingtheirlarge-orderbehaviourwerecalculatedin[7].Further,bysolv-ingEq.(6)numerically,boththediscreteandthecontinuouspartsoftheenergyspectrumistakenintoaccount,andtheperturbationenergiesEncanbecalculatedevenincaseswhenonlyafewboundstatesexist.Thelineardependenceof n(x)ontheenergyEnmakesitpossibletoavoidtheusualshootingmethodandreducethecomputationaltimesubstantially.Finally,wenotethatonlythewavefunctionsareneededinthismethodandnointegralshavetobecalculated.

Theaimofthispaperistoapplythismethodtoafewproblemsandtestitsnumericalproperties.

F x H0 E0 1 0 x

and

fn 1 x H0 E0 1H1 n 1 x

(8)

n 1

i 1

E

i

n i

x .(9)

ThegeneralsolutionofEq.(6)cancontainalsoatermcn 0(x)attheright-handsideofEq.(7),wherecnisanarbitraryconstant.Forthesakeofsimplic-ity,weassumecn 0here.AsitisseenfromEq.(7),theperturbationfunction n(En,x)dependsontheenergyEnwhichisnotyetknownandthepointx [x1,...,xN]inN-dimensionalspace.

Equations(7)–(9)showthatthestructureoftheperturbationfunctionsisverysimple.ItfollowsfromEq.(7)thatthefunction n(En,x)isalinearfunctionoftheenergyEn.Further,itisseenthatF(x)isafunctionindependentofn.Wenotealsothat,exceptforthecasethatEnistheexactperturbationenergy, n(En,x)isnotquadraticallyintegrableandhasnophysicalmeaning.

ThefunctionsF(x)andfn 1(x)arecalculatedfromEqs.(8)and(9)numericallywiththecondi-tionsF(xb) 0andfn 1(xb) 0,wherexbarepointsattheboundaryregionsuf cientlydistantfromthepotentialminimum.Thesameboundaryconditionsareusedforthefunction 0(x).

WenotethatthefunctionF(x)divergesintheexactcalculation,however,ithaslargebut nitevaluesinnumericalcalculations.Thefunctions n(En,x)fortheexactperturbationenergyEnarequadraticallyintegrable.Therefore,wecanassumetheyobeythecondition

n En,x F x .

(10)

2.SummaryoftheMethod

Firstwediscussanondegeneratemultidimen-sionalcase.Weassumethattheperturbationfunc-tions iandperturbationenergiesEiarealreadycomputedfori 0,...,n 1.SolutionofEq.(6)canbewrittenas

ItfollowsfromEqs.(7)and(10)thatthefunctions n(En,x)alsosatisfythecondition

n En,x fn 1 x .

(11)

Therefore,wecanneglect n(En,x)inEq.(7).TheformulafortheenergyEnthenreads

fn 1 x En .

F x

(12)

n En,x EnF x fn 1 x ,n 1,2,...,where

(7)

Thisequationcanbeusedatanarbitrarilychosenpointxinsidetheintegrationregionexceptforthe

326VOL.99,NO.4