New Version of the Rayleigh–Schrdinger Perturbation Theory(8)

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.

FIGURE2.First-orderperturbationfunction 1forthe

groundstateoftheHe´non–HeilesHamiltonian(31,

32).

FIGURE4.Third-orderperturbationfunction 3forthe

groundstateoftheHe´non–HeilesHamiltonian(31,32).

E1,E3,...equalzero.Innumericalcalculations,theseenergiesdifferfromzerowithanaccuracyof7–9digits.Theeven-perturbationenergiesE2,E4,...,arenegative,andtheirabsolutevaluein-creasesrapidlywiththeorderoftheperturbationtheory.

ResultsinTablesIV–VIIindicatethattheabso-lutevaluesoftheperturbationenergiesEnincreasewiththeenergydifferenceoftheexcitedstateandthegroundstate.

Concludingthissection,ourmethodofsolvingtheperturbationproblemgivesgoodresultsintheallinvestigatedcasesincludingthemostdif cultHe´-non–HeilesHamiltonian.TheabsolutevaluesoftheperturbationenergiesEnincreaserapidlyforlargen.

4.Conclusions

Insummary,themethoddescribedinthispaperissimpleandef cientalternativetotheusualfor-mulationoftheperturbationtheory.Itcanbeusedforone-dimensionalaswellasmultidimensionalproblemsandfornondegenerateaswellasdegen-erateeigenvalues.Itsmainadvantagesareeasycal-culationofthelarge-orderperturbationsandthepossibilityto ndtheperturbationcorrections,evenincaseswhenonlyafewzero-orderboundstatesexist.Theanalyticandnumericalresultsfortheinvestigatedexamplesshowthatourversionoftheperturbationtheoryyieldsresultswithgoodaccu-racyandcanbeusedatlargeorders.

Thenumericalresultsshowinallinvestigatedexamplesthattheabsolutevaluesoftheperturba-tionenergiesEnincreaserapidlyforlargen.Theyalsoindicatethattheperturbationseries(4)aredivergent,asymptoticseriesinthesecases.Theyareobviouslyrelatedtodifferentasymptoticbehaviorofthewavefunctionscorrespondingtothezero-orderHamiltonianH H0andthefullHamiltonianH H0 H1forx3 .Therefore,oneshouldbeverycarefulwhentruncatingsuchperturbationse-riesatlow

orders.

FIGURE3.Second-orderperturbationfunction 2for

thegroundstateoftheHe´non–HeilesHamiltonian(31,

32).

332VOL.99,NO.4