New Version of the Rayleigh–Schrdinger Perturbation Theory(6)

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.

TABLEIII_____________________________________________________________________________________________

PerturbationenergiesEnforthegroundstateoftheBarbanisHamiltonian(30);E0istheexactzero-orderenergy.*n024681012141618202224262830

numEn

anEn

2

0.1041666665 0.0322627314 0.02298880368 0.024159131 0.0326818070 0.05343610684 0.1019221649 0.2217101989 0.541404172 1.46666250 4.366576807 14.17768870 49.8757770 189.0381799 768.15366

2

5

0.104166666667...48223 0.0322627314815...114407

0.0229888036908...

4976640346266143

0.0241591313924...

2360833242959

0.0326818070580...

7223692492800012969801730008377

0.0534361068462...

124680261346275858491

0.101922164944...

122328898150072320000010935414749213048671720261 0.221710199048...493230117341091594240000004441356782637499756905980351899

0.541404172625...

667033238517271928733626515967166703

1.46666250754...

454796679596048442320958259200000000

numan*TheenergiescalculatednumericallyfromEq.(12)aredenotedasEn.TheanalyticenergiesdenotedasEnwerecalculatedfor

n 0,2,...,20.Odd-orderperturbationenergiesequalzero.

cillator.Forexample,theground-statezero-orderwavefunctionequals

03 x,y 133/451/4y 15

150

13y2 e x 13/20 y 0 x 3 y ,

2

2

00 x,y

15

13

1/43/4

5e

x2 13/20 y2

(35)

0 x 0 y .(33)

and

Analogously,orthonormalexcitedstatewavefunc-tionscorrespondingtolowquantumnumberscanbewrittenintheform

21 x,y

3/41/4

135 1 4x2 ye x 13/20 y

10

2

2

01 x,y

15 2 x 1 y .

133/451/4ye x 13/20 y

2

2

(36)

0 x 1 y ,(34)

Nowwediscusstheground-stateperturbationproblemwiththezero-orderwavefunction

330VOL.99,NO.4