New Version of the Rayleigh–Schrdinger Perturbation Theory(4)

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.

Theusualformulationofthe rst-orderdegen-erateperturbationtheorycanbederivedfromEq.(20)asfollows.SubstituteEqs.(18)and(19)intoEq.(20),andweobtainforapointx

E1

j j 10

¥jd H1 0 x 1a0 H0 E0 0j 01a000TABLEI______________________________________

PerturbationenergiesEnforthegroundstateoftwo

coupledharmonicoscillators(28);E0istheexactzero-orderenergy.

n

En

2

0.750000000 0.93750000 0.0234375000 0.0133056640 0.00031534830 0.0132794910.0240443106 0.07430306870.234920366 0.845542553.34568731 14.494654868.195859 346.325411888.18252 11000.399468201.932 448367.103115424.0 22813412.6

.(23)

Bymultiplyingthenumeratoranddenominatorby(H0 E0),weobtain

E1

and

E1

j j 0

¥jd 1a0H1 0 x 0j 1a00(24)

j 1

a

d0

j

0 x j 0

j 1

a

d0

j 0 j H1 0 x .(25)

Further,bymultiplyingthisequationfromtheleft

i)*

sidebythecomplexconjugatefunction (0(x),in-tegratingoverx,andassumingorthonormalityof

j)

thefunctions (0(x),astandardsecularproblemisobtained:

01234567891011121314151617181920

j 1

W E a

d0

ij

1ij

j 0 0,i 1,...,d0,(26)

2 2

222222

H x1 x2 x1x2 x1 x2 .

12

(28)

where

i j

Wij 0 H1 0 .

(27)

3.Examples

Fortheone-dimensionalproblems,thismethod

hasalreadybeenusedwithverygoodresults(see[3–8]).Forthisreason,severaltwo-dimensionalproblemswithincreasingcomplexityareinvesti-gatedhere.

3.1.COUPLEDANHARMONICOSCILLATORSAsthe rstexample,wediscusstheproblemhavingonlybounddiscretestates.Wecalculatedtheperturbationenergiesforthegroundstateoftwononlinearlycoupledharmonicoscillators[9–11]:

Tocompute nfromEq.(6)intheregionx1 [ 11,11],x2 [ 11,11],weusedagridofpoints158 158,160 160,162 162,164 164,as-sumedthatthefunctions nequalzeroattheborderofthisregion,andsolvedthecorrespondingsystemofdifferenceequationsindouble-precisionaccu-racyinFortran.Thesamegridofpointswasusedinallexamplesgivenbelow,onlytheintegrationre-gionwasdifferent.

Toeliminatetheeffectofanon-zerostepofthegridtheperturbationenergieswereextrapolatedtoanin nitelydensegridbymeansoftheRichardsonextrapolation.Thisextrapolationissubstantialforincreasingtheaccuracyoftheresults.Theground-stateperturbationenergiesareshowninTableI.Onlythedigitswhichagreeinthecalculationsforthepointsx [0,0]andx [1,1]areshown.Itisseenthattheenergiesdependonthechoiceofthepointxonlyverylittle.AlldigitsshowninTableIalsoagreewithanindependentcalculationmade

328VOL.99,NO.4