Solution of Quantum Anharmonic Oscillator with Quartic Perturbation

Transcription

1 ISS -79X (Paper) ISS (Olie) Vol.7, 0 Abstract Solutio of Quatum Aharmoic Oscillator with Quartic Perturbatio Adelaku A.O. Departmet of Physics, Wesley Uiversity of Sciece ad Techology, Odo, Odo State, igeria. Abajigi David Dele Departmet of Physics ad Electroics Adekule Ajasi Uiversity, Akugba- Akoko, Odo State, igeria. This study was desiged to obtai the eergy eigevalues for a Quatum Aharmoic Oscillator with Quartic Perturbatio Potetial. Two idepedet methods, the Dirac operator techique ad the umerov approach i solvig Schrodiger equatio, were used to solve the secod order differetial equatio obtaied from this system. A iterative procedure was carried out usig the fourth order Ruge-Kutta method o the trasformed secod order differetial equatio i lie with the umerov equatio. The results showed that the ormalized eigevalues obtaied from the Dirac operator techique, whe compared with eigevalues obtaied from the use of the Fourth order Ruge-Kutta method withi the umerov approach agreed closely whe the covergece i the perturbig potetial is weak, but the set of results diverges oly at high excitatio states. For the results from the two approaches to be closely compatible at high excitatio states, the choice of Zeta axis was made to satisfy the boudary coditios - < ζ< +. Keywords: QQAHO, Hamiltoia, Perturbatio, Dirac Operatio Techique, umerov Approach, Zeta (ζ), Ruge - Kutta Method ad Eigevalues.. Itroductio. Oe of the sources of progress of the scieces depeds o the study of the same problem from differet poit of view based o differet mathematical formalism. This why this study is focused o obtaiig a set of eergy eigevalues from two idepedet mathematical methods, the Dirac operator techique ad Fourth order Ruge-Kutta method withi the umerov approach i solvig Schrodiger equatio for purpose of establishig a simple but robust method for solvig Quatum Aharmoic Oscillator with Quartic Perturbatio Potetial. Determiatio of eergy eigevalues of the Schrodiger equatio via asymptotic iteratio method (AIM) has bee widely applied to establish eergy eigevalues of the Schrodiger type equatios [], [], arisig from the developmet of fast computers simulatios. Although the AIM formalism is very efficiet to obtai eigevalues of the Schrodiger equatio, it requires tedious calculatios i order to determie wave fuctios of systems which are ot exactly solvable ad thus the calculatio of wave fuctio ivolvig a large umber of terms will lose its simplicity ad accuracy [3]. Much of the problems ecoutered i givig solutios to quatum aharmoic oscillator with quartic perturbatio potetial were first oticed with the Rayleigh-Schrodiger perturbatio series for the simple system of the quartic aharmoic oscillator whose eigevalues diverged eve for small values of the couplig costats [], [5]. The quatum solutio for aharmoic oscillator with quartic perturbatio is very useful i φ field (the scalar field which iteracts with itself through the iteractio λφ) theory ad i the studies of quatum statistical properties of radiatio field iteractig with a cubic oliearity leadig to a quartic iteractio [7], [8]. A sigle mode of the radiatio field 38

2 ISS -79X (Paper) ISS (Olie) Vol.7, 0 iteractig with a optical fiber of cubic oliearity gives rise to the model of a quartic oscillator. Beyod the preset system which form the focus of this study, the quatum aharmoic oscillator with sextic, octic, ad the geeral oe perturbatio term, λx m, has bee studied more recetly, each with a diverse associated shortcomigs [9], [0], []. I this study, the two methods used to obtai the eergy eigevalues for the system uder cosideratio are classified uder two mathematical formalisms. The Dirac operator techique is termed as the eigevalues problem which takes care of the time developmet of wave fuctios i the Schrödiger cocept, ad Fourth order Ruge-Kutta method withi the umerov approach is a directly used to get the time developmet of the operators withi the Heiseberg frame-work. This paper is orgaized as follows after the brief itroductio i sectio. We solve the time idepedet Schrodiger equatio i sectio usig the Dirac operatio techique. I sectio 3 a detailed process of solutio for quatum aharmoic oscillator with quartic perturbatio potetial usig umerov approach is preseted. I sectio, the computatioal procedures ad results are preseted while sectio 5 cotais the discussio of the results ad coclusio. The Quatum Aharmoic Oscillator with Quartic Perturbatio Potetial is respectively captioed i the Hamiltoia ad Schrödiger equatio for the Dirac techique as p / H kx k x. () m d ( / kx ( k x ( E(. () m dx x By settig, a w d E ad d a d, equatio () ca be writte as d d / ( ) m ( ) k ( ) ( ).. (3) m a d This trasformatio is ecessary so that the Fourth order Ruge-Kutta method withi the umerov approach i solvig Schrodiger equatio ca be applied. 3. Solutio of Time Idepedet Schrodiger Equatio usig Dirac Operator Techique Two mai operators, âis the lowerig or aihilatio operator ad â + is the raisig or creatio operator, ecessary to specify the relatio betwee the positio ad the mometum are defied as follows. aˆ / [ x p / m] () aˆ / [ x ip / m] (5) where x = positio operator, m =mass of the particle, p =liear mometum operator =agular frequecy. The relatioship coied from the parameters show above is β = (6) where h /. The umber operator is defied as aˆ aˆ.. (7) The Hamiltoia of the harmoic oscillator i relatio to the operator defied i equatio (5) are give as H aˆ aˆ ).. (8) ad H aˆ aˆ ). (9) ( The wave fuctio of the harmoic oscillator, writte as that the product of aˆ aˆ satisfies ( for the state,, is properly defied such 39

3 ISS -79X (Paper) ISS (Olie) Vol.7, 0 a ˆ.. (0), a ˆ.() The Hamiltoia as expressed i equatio () ca be expressed as Hˆ Hˆ Hˆ.. () 0 where Ĥ 0 =Uperturbed Hamiltoia, represeted by ˆ p Ho kx, Ĥ = Hamiltoia for the quartic m / perturbatio give as K x ad λ = perturbatio coefficiet. The Schrodiger equatio for the system uder cosideratio ad properly defied i terms of lowerig, or aihilatio operator â ad â + is the raisig or creatio operator, is give as ˆ o o () H o E E.. (3) I Dirac otatio, equatio () becomes ˆ Hˆ Hˆ Hˆ. () H 0 where Hˆ ( aˆ o aˆ ( o) ), E ( The eergy fuctio is the give as E ) x. (5) If x ( aˆ aˆ ), the the results of equatio (5) is / K E (6) Factorizig, we have that 3 / K E.... (7). 8 From equatio (6), we get that m ( / ) the the eergy equatio (7) is fially give as E 3 / K (8) 8m / With the followig ormalizatio uits: m =.0, =.0, w =.0 ad K = 0.0 ad the various values deoted by 0,,, 3..., the various eergy levels ca be obtaied. The values of the eergy levels from the groud state to excited states are put together i table.0.. Computatio of Eigevalues usig umerov Approach umerov method was desiged to solve umerical secod order differetial equatios of the form d y F( x, y) (9) dx The values for equatio (8) are determied usig both the Predictor ad Corrector iterative expressios respectively give as 0

4 ISS -79X (Paper) ISS (Olie) Vol.7, 0 Predictor h Y k Yk 0Fk Fk. (0) Corrector h Y k Yk 0Fk Fk Fk.... () I this study, Schrodiger Equatio for Quartic Perturbatio i its trasformed form, as expressed i equatio (3) is reduced to the form of equatio (9) so that the iterative procedures i equatios (0) ad () ca be used effectively to obtai the solutio to the Schrodiger Equatio for Quartic Perturbatio i equatio (). By a simple trasformatio techique, equatio (3) reduces to d ( ) ( )... d () where λ is a adjustable parameter. O expasio, we have that d ( ) ( ) ( )..... (3) d The solutio to equatio (3) usig the predictor equatio yields.. () ( ) 0( ) ( ) Applyig the corrector equatio () yields FK 0F where =,,3,. k Fk (5) Thus the expressio to iterated becomes K 0( ) ( The results geerated from these two approaches are tabulated i table.0., ). (6) 5. Results ad Discussio The calculated values for the eigevalues from the two methods are put together i table.0. The eigevalues obtaied by Dirac Operator Techiques are i the first colum. The secod colum cotais the eigevalues obtaied by umerov Approach. The ormalized Eige values from umerov o Dirac Operatio Techiques are put together i colum three ad the percet differece are compiled i colum four for the various eergy levels computed i this study. The ormalized eigevalues from the umerov approach compared well with the eigevalues obtaied by Dirac Operator Techiques as show by the percetage differece i the values obtaied from these two methods for each of the eergy levels. The covergece show i the results preseted i the table is made possible by the choice of Zeta, ( ). The choice of Zeta axis which satisfies the related covergece of the two results are foud to be withi the boudary coditios -< ζ <+. The results show a cosistece icrease i the values of the ormalized eigevalues as icreases for each state regardless of the chage i the value of Zeta. Also, the

5 ISS -79X (Paper) ISS (Olie) Vol.7, 0 percetage differeces betwee the two results at the groud, first ad secod eergy state levels are less tha %. 6. Coclusio umerov ad Dirac operatio techiques have bee used to calculate the groud eergy state ad the first ie eigevalues eergy states for a aharmoic oscillator potetial with the quartic perturbatio potetial. The percetage differeces for the ie states are less tha % whe the results from the two methods are compared. It was observed that results from these two methods agree whe the perturbig potetial is weak ad particularly at the low eergy states. As the perturbig potetial gets stroger at the higher excited states, the results from the Dirac operator techiques start to diverge from those obtaied from the umerical schemes. The methods used i this study provide a simple techique for solvig aharmoic oscillator potetial with the quartic perturbatio potetial ad of higher perturbatio potetial, because the mathematical method ca easily be solved usig simple computer simulatios. Table.0. Values of the Eige values obtaied from the Dirac operator techiques ad from the umerov iteratio method. Eergy Level Eigevalues obtaied by Dirac Operator Techiques Eigevalues obtaied by umerov Approach Groud State First Excited State Secod Excited State Third Excited State Fourth Excited State Fifth Excited State Sixth Excited State Seveth Excited State Eighth Excited State ith Excited State Eigevalues by umerov ormalized o Dirac Operatio Techique Percet Differece 0.505% 0.006% 0.69% 0.00% 0.06% 0.09% 0.073% 0.088% 0.07% 0.07%

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