Bound state energies and wave functions of spherical quantum dots in presence of a confining potential model
aa r X i v : . [ qu a n t - ph ] O c t Bound state energies and wave functions of spherical quantumdots in presence of a confining potential model
Sameer M. Ikhdair ∗ Physics Department, Near East University, Nicosia, Mersin 10, Turkey (Dated: October 4, 2011)
Abstract
We obtain the exact energy spectra and corresponding wave functions of the radial Schr¨odingerequation (RSE) for any ( n, l ) state in the presence of a combination of psudoharmonic, Coulomband linear confining potential terms using an exact analytical iteration method. The interactionpotential model under consideration is Cornell-modified plus harmonic (CMpH) type which is acorrection form to the harmonic, Coulomb and linear confining potential terms. It is used toinvestigates the energy of electron in spherical quantum dot and the heavy quarkonia (QQ-onia).Keywords: Schr¨odinger equation, confining potentials, spherical quantum dots, Cornell-modifiedpotential, pseudoharmonic oscillator
PACS numbers: 03.65.Fd; 03.65.Ge; 68.65.Hb ∗ E-mail: [email protected] . INTRODUCTION The problem of the inverse-power potential, 1 /r n , has been used on the level of bothclassical and quantum mechanics. Some series of inverse power potentials are applicableto the interatomic interaction in molecular physics [1-3]. The interaction in one-electronatoms, muonic, hadronic and Rydberg atoms takes into account inverse-power potentials[4]. Indeed, it has also been used for the magnetic interaction between spin-1 / V ( r ) = Ar − + Br − + Cr − + Dr − , A > , were presented by Barut et al. [6] and ¨Oz¸celik and S¸im¸sek [7] by making an available ansatz for the eigenfunctions. TheLaurent series solutions of the Schr¨odinger equation for power and inverse-power potentialswith two coupling constants V ( r ) = Ar + Br − and three coupling constants V ( r ) = Ar + Br − + Cr − are obtained [8,9].The analytic exact iteration method (AEIM) which demands making a trial ansatz for thewave function [7] is general enough to be applicable to a large number of power and inverse-power potentials [10]. Recently, this method is applied to a class of power and inverse-powerconfining potentials of three coupling constants and containing harmonic oscillator, linearand Coulomb confining terms [11]. This kind of Cornell plus Harmonic (CpH) confiningpotential of the form V ( r ) = ar + br − cr − is mostly used to study individual sphericalquantum dots in semiconductors [12] and heavy quarkonia (QQ-nia) [13,14]. So far, suchpotentials containing quadratic, linear and Coulomb terms have been studied [15,16].The present work considers the the following confining interaction potential consisting ofa sum of pseudoharmonic, linear and Coulombic potential terms: V ( r ) = V har ( r ) + V Corn- mod ( r ) = ar + br − cr − dr , a > , (1)where a, b, c and d are arbitrary constant parameters to be determined later. The abovepotential includes the well-known funnel or Cornell potential, i.e., a Coulomb plus Linearstatic potential (CpH), V Corn ( r ) = br − c/r [13], and a term − d/r is incorporated into thequarkonium potential for the sake of coherence [14]. We will refer to the potential model(1) as a Cornell-modified plus harmonic (CMpH) potential, since the functional form hasbeen improved by the additional − d/r piece; besides the contribution from the additionalterm also alters the value of b and c [14,17]. The authors of Refs. [14,18] did not considerthe harmonic or power-law as the results are expected to be similar. The CMpH potential2s plotted in Figure 1 for the values of parameters: a = 1 eV.f m − , b = 0 . eV . f m − ,c = 0 . eV.f m and d = 0 . eV.f m . We will apply the AEIM used in [7,11] to obtain the exact energy eigenvalues and wavefunctions of the radial Schr¨odinger radial equation (RSE) for the CMpH potential for anyarbitrary ( n, l ) state.The paper is structured as follows: In Sect. 2, we obtain the exact energy eigenvaluesand wave functions of the RSE in three-dimensions (3D) for the confining CMpH potentialmodel by proposing asuitable form for the wave function. In Sect. 3, we apply our resultsto an electron in spherical quantum dot of InGaAs semiconductor. The relevant conclusionsare given in Sect. 4.
II. EXACT SOLUTION OF RSE FOR THE CONFINING POTENTIAL MODEL
The three-dimensional (3 D ) Schr¨odinger equation takes the form [19] (cid:20) − ¯ h m ∆ + V ( r ) (cid:21) ψ ( r, θ, ϕ ) = E nl ψ ( r, θ, ϕ ) , (2)with ∆ = ∂ ∂r + 2 r ∂∂r − L ( θ, ϕ )¯ h r , where m is the isotropic effective mass and E nl is the total energy of the particle. For anyarbitrary state, the complete wave function, ψ ( r, θ, ϕ ) , can be written as ψ ( r, θ, ϕ ) = X n,l N l ψ nl ( r ) Y lm ( θ, ϕ ) , (3)where spherical harmonic Y lm ( θ, ϕ ) is the eigenfunction of L ( θ, ϕ ) satisfying L ( θ, ϕ ) Y lm ( θ, ϕ ) = l ( l + 1)¯ h Y lm ( θ, ϕ ) , (4)and the radial wave function ψ nl ( r ) is the solution of the equation (cid:18) d dr + 2 r ddr − l ( l + 1) r (cid:19) ψ nl ( r ) + 2 m ¯ h [ E nl − V ( r )] ψ nl ( r ) = 0 , (5)where r stands for the relative radial coordinates. The radial wave function ψ nl ( r ) is well-behaved at the boundaries (the finiteness of the solution requires that ψ nl (0) = ψ nl ( r →∞ ) = 0) . Now, the transformation ψ nl ( r ) = 1 r φ nl ( r ) , (6)3educes Eq. (5) to the simple form φ ′′ nl ( r ) + (cid:20) ε n,l − a r − b r + c r + d − l ( l + 1) r (cid:21) φ nl ( r ) = 0 , (7)where φ nl ( r ) is the reduced radial wave function and ε nl = 2 m ¯ h E nl , a = 2 m ¯ h a, b = 2 m ¯ h b, c = 2 m ¯ h c, d = 2 m ¯ h d. (8)The analytic exact iteration method (AEIM) requires making the following ansatze for thewave function [9], φ nl ( r ) = f n ( r ) exp [ g l ( r )] , (9)with f n ( r ) = , n = 0 , n Π i =1 (cid:16) r − α ( n ) i (cid:17) , n = 1 , , · · · , (10a) g l ( r ) = − αr − βr + δ ln r, α > , β > . (10b)It is clear that f n ( r ) are equivalent to the Laguerre polynomials [20]. Substituting Eq. (9)into Eq. (5) we obtain φ ′′ nl ( r ) = (cid:18) g ′′ l ( r ) + g ′ l ( r ) + f ′′ n ( r ) + 2 g ′ l ( r ) f ′ n ( r ) f n ( r ) (cid:19) φ nl ( r ) . (11)and comparing Eq. (11) and Eq. (7) yields a r + b r − c r + l ( l + 1) − d r − ε nl = g ′′ l ( r ) + g ′ l ( r ) + f ′′ n ( r ) + 2 g ′ l ( r ) f ′ n ( r ) f n ( r ) . (12)First of all, for n = 0 , let us take f ( r ) and g l ( r ) given in Eq. (10b) to solve Eq. (12), a r + b r − ε l − c r + l ( l + 1) − d r = α r +2 αβr − α [1 + 2 ( δ + 0)]+ β − β ( δ + 0) r + δ ( δ − r . (13)By comparing the corresponding powers of r on both sides of Eq. (13) we find the followingcorresponding energy and the restrictions on the potential parameters, α = √ a , (14a) β = b √ a , a > , (14b) c = 2 β ( δ + 0) , (14c)4 = 12 (1 ± l ′ ) , where l ′ = r (2 l + 1) − m ¯ h d (14d) ε l = α [1 + 2 ( δ + 0)] − β . (14e)Actually, to have well-behaved solutions of the radial wave function at boundaries, namelythe origin and the infinity, we need to take δ from Eq. (14d) as δ = 12 (1 + l ′ ) . (15)Therefore, the lowest (ground) state energy from Eq. (14e) together with Eqs. (14a)-(14c),Eq. (15) and Eq. (8) is given as follows E l = s ¯ h a m (2 + l ′ ) − mc ¯ h (1 + l ′ ) , (16)where the parameter c of potential (1) should satisfy the following restriction: c = b q ma ¯ h r (2 l + 1) − m ¯ h d ! . (17)Furthermore, the substitution of α, β and δ from Eqs. (14a), (14b) and (15), respectively,together with the parameters given in Eq. (8) into Eqs. (9) and (10), we finally obtain thefollowing ground state wave function: ψ l ( r ) = N l r ( − l ′ ) / exp − r ma ¯ h r − mc ¯ h (1 + l ′ ) r ! , (18)with N l = 1 r Γ( l ′ ) D − l ′ (cid:16) mc ¯ h (1+ l ′ ) q ¯ h √ ma (cid:17) r ma ¯ h ! l ′ / exp − r m ¯ h a mc ¯ h (1 + l ′ ) ! , where D ν ( z ) are the parabolic cylinder functions [21]. It should be noted that theabove solutions are well-behaved at the boundaries, i.e., a regular solution near the ori-gin could be φ nl ( r → → r (1+ l ′ ) / and asymptotically at infinity as φ nl ( r → ∞ ) → exp ( − αr − βr ) → . When b = 0 ( c = 0) , the problem turns to become the com-moly known pseudoharmonic oscillator (p.h.o.) interaction ( a = mω / , and consequently α = mω, β = b/ω and c = ( bδ/mω ) yielding E l = (2 + l ′ ) ¯ hω − mc ¯ h (1+ l ′ ) and wave function ψ l ( r ) = N l r ( − l ′ ) / exp (cid:16) − mω ¯ h r − mc ¯ h (1+ l ′ ) r (cid:17) , where N l = 1 r Γ( l ′ ) D − l ′ (cid:16) mc ¯ h (1+ l ′ ) q ¯ h mω (cid:17) (cid:18) mω ¯ h (cid:19) l ′ / exp (cid:18) − mc h ω (1 + l ′ ) (cid:19) . a, b, c and d. There-fore, the solutions (16) and (18) are valid for the potential parameters satisfying the re-striction (17). Moreover, the relation between the potential parameters (17) depends on theorbital quantum number l which means that the potential has to be different for differentquantum numbers. In applying the AEIM, the obtained solution for any potential is alwaysfound to be subjected to certain restrictions on potential parameters as can be traced inother works (see, for example, [7-9,11]).Secondly, for the first node ( n = 1), using f ( r ) = ( r − α (1)1 ) and g l ( r ) from Eq. (10b) tosolve Eq. (12), a r + b r − ε l − c r + l ( l + 1) − d r = α r + 2 αβr − α [1 + 2 ( δ + 1)] + β − h β ( δ + 1) + αα (1)1 i r + δ ( δ − r . (19)The relations between the potential parameters and the coefficients α, β, δ and α (1)1 are α = √ a , β = b √ a , δ = 12 (1 + l ′ ) , ε l = α [1 + 2 ( δ + 1)] − β .c − β ( δ + 1) = 2 αα (1)1 , ( c − βδ ) α (1)1 = 2 δ, (20)where c and α (1)1 are found from the constraint relations, c = b q ma ¯ h (2 + l ′ ) + vuut b ma ¯ h + ¯ h m s ¯ h a m (1 + l ′ ) , (21a) αα (1)21 + βα (1)1 − δ = 0 → α (1)1 = − b a + vuut b a + (1 + l ′ )2 q ma ¯ h . (21b)The energy eigenvalue is E l = s ¯ h a m (4 + l ′ ) − b a ,b = 2 r ma ¯ h (2 + l ′ ) c (1 + l ′ ) (3 + l ′ ) vuuut ¯ h mc s ¯ h a m (1 + l ′ ) − (1 + l ′ ) (3 + l ′ )(2 + l ′ ) , (22)and the wave function is ψ l ( r ) = N l (cid:16) r − α (1)1 (cid:17) r ( − l ′ ) / exp − r ma ¯ h r − r m h a br ! , (23)6ith N l = (cid:16) q ma ¯ h (cid:17) l ′ / exp (cid:16) − q ma ¯ h b (cid:17)r(cid:16) q ma ¯ h (cid:17) − Γ( l ′ + 2) S + α (1)21 Γ( l ′ ) S − (cid:16) q ma ¯ h (cid:17) − / α (1)1 Γ( l ′ + 1) S , where S = D − ( l ′ +2) s ¯ h a r ma b ¯ h , S = D − l ′ s ¯ h a r ma b ¯ h , S = D − ( l ′ +1) s ¯ h a r ma b ¯ h , and α (1)1 is given in Eq. (21b). If there is a p.h.o. interaction, the energy becomes E l = (4 + l ′ ) ¯ hω − b mω , (24)and the wave function ψ l ( r ) = N l (cid:16) r − α (1)1 (cid:17) r ( − l ′ ) / exp (cid:18) − mωr − bω r (cid:19) , (25)with N l = (cid:0) mω ¯ h (cid:1) l ′ / exp (cid:16) −
14 ¯ h b mω (cid:17)r ¯ h mω Γ( l ′ + 2) S + α (1)21 Γ( l ′ ) S − α (1)1 q ¯ h mω Γ( l ′ + 1) S ,S = D − ( l ′ +2) r hmω b ¯ hω ! , S = D − l ′ r hmω b ¯ hω ! , S = D − ( l ′ +1) r hmω b ¯ hω ! , where b = 2 mω ¯ h (2 + l ′ ) c (1 + l ′ ) (3 + l ′ ) " s (cid:18) ¯ h ω mc (1 + l ′ ) − (cid:19) (1 + l ′ ) (3 + l ′ )(2 + l ′ ) , and α (1)1 = ( l ′ +1)2 mω . Following the analytic iteration procedures for the second node ( n = 2) with f ( r ) =( r − α (2)1 )( r − α (2)2 ) and g l ( r ) as defined in Eq. (10b), we obtain a r + b r − ε ,l − c r + l ( l + 1) − d r = α r + 2 αβr − α [1 + 2 ( δ + 2)] + β − " β ( δ + 2) + α X i =1 α (2) i r + δ ( δ − r , (26)7he relations between the potential parameters and the coefficients α, β, δ, α (2)1 and α (2)2 are α = √ a , β = b √ a , δ = 12 (1 + l ′ ) , ε ,l = α [1 + 2 ( δ + 2)] − β .c − β ( δ + 2) = 2 α X i =1 α (2) i , ( c − βδ ) X i 1) (2 δ + n − X i =1 α (2) i , ( n = 2)( c − βδ ) n X i 1) (2 δ + n − n X i Now, we consider a special case of potential (1) and an application to our results. Forexample, when b = 0 then leads to c = 0, then we have the p.h.o potential, i.e., V ph ( r ) = mω r − dr , hence, the energy difference between the ground state and the excited statesis ∆ E = E l − E l = (4 + l ′ ) ¯ hω − (2 + l ′ ) ¯ hω hω, (39)which can be used to calculate the values of the potential parameters for the desired system.We now apply the present results to describe a realistic physical system called indiumgallium arsenide (InGaAs) quantum dot, i.e., a piece of this material of a spherical formwhich is considered as a semiconductor composed of indium, gallium and arsenic [11]. Itis used in high-power and high-frequency (say, ω ∼ Hz ) electronics because of itssuperior electron velocity with respect to the more common semiconductors silicon andgallium arsenide. InGaAs bandgap also makes it the detector material of choice in opticalfiber communication at 1300 and 1550 nm . The gallium indium arsenide (GaInAs) is analternative name for InGaAs. In Fig. 2, we plot the ground state electron energy E l ( ω ) = r (2 l + 1) − m ¯ h d ! ¯ hω − mc ¯ h r (2 l + 1) − m ¯ h d ! − , (40)versus ω in the interval 2 × ≤ ω ≤ × Hz taking the value of c = 0 . eV.f m and d = 0 eV.f m for the cases l = 0 and l = 1 , respectively (harmonic, Coulomb and linearcombination terms). In Fig. 3, we take instead the value of the parameter d = 0 . eV.f m . The effective mass of electronin the InGaAs semiconductor has been chosen as m = 0 . m e and ¯ h = 6 . × − eV.s. It is seen from Fig. 2 and Fig. 3 how the increase in the value of ω leads to an increase inthe energy of electron. The flexibility in the adjustment of the parameter d allows one tofit the spectrum of the desired model properly (cf. Fig. 2 and Fig. 3). The parameter d should satisfy the condition d ≤ (2 l + 1) ¯ h / (8 m ) . In Fig. 4 we plot the ground state wavefunction ψ ,l ( r ) of the CpH potential for the cases l = 0 and l = 1 , respectively, using thevalues of potential parameter c = 0 . eV.nm for an electron with effective mass m = 0 . m e and frequency ω = 10 × Hz . Further, in Fig. 5 we plot the ground state wavefunction ψ ,l ( r ) of the CMpH potential for the cases l = 0 and l = 1 , respectively, using thevalues of potential parameters c = 0 . eV.nm and d = 0 . eV.nm for an electron witheffective mass m = 0 . m e and frequency ω = 10 × Hz . In Figs. 6 and 7, we showelectron energy as a function of parameter c in the interval 6 × − ≤ c ≤ × − eV.nm and d = 0 . eV.nm for frequency ω = 8 × Hz and effective mass m = 0 . m e for thecases l = 0 and l = 1 , respectively. From Fig. 5, the increase in c leads in the decrease inthe electron energy in the InGaAs semiconductor. In Fig. 8, we plot the first excited stateelectron energy E l ( ω ) = (4 + l ′ ) ¯ hω − l ′ ) (1 + l ′ ) (3 + l ′ ) mc ¯ h × " s l ′ ) (3 + l ′ )(2 + l ′ ) (cid:18) ¯ h ω mc (1 + l ′ ) − (cid:19) , (41)versus ω in the interval 2 × ≤ ω ≤ × Hz taking the value of c = 0 . eV.f m and d = 0 eV.f m for the cases l = 0 and l = 1 , respectively. In Fig. 9, we take the value ofthe parameter d = 0 . eV.f m . We remark that the strongly attractive singular part − d/r is physically incorporated into the quarkonium Cornell potential as the first perturbativeterm for the sake of coherence to describe the heavy quarkonia (QQ-nia) (see, for example,[13,14] and the references therein). It also resemles the centrifugal barrier term l ( l + 1) /r in the Schr¨odinger equation. This attractive term − d/r together with the h.o. part ar constitute the so-called p.h.o. when b = 0 ( β = 0) in Eq. (14b) leading to c = 0 in Eq.(14c) . In Table 1, we calculate the lowest ( n = 0) energy states ( l = 0 , . − . . IV. CONCLUSIONS AND OUTLOOK In this work, we explored the analytical exact solution for the energy eigenvalues andtheir associated wave functions of a particle in the field of Cornell-modified plus harmonicconfining potential. We have used the analytical exact iteration method (AEIM) whichrequired making a trial ansatz for the wave function. The general equation for the energyeigenvalues is given by Eq. (36) with some restrictions on the potential parameters. If onetakes b = 0 then c = 0 , hence, the potential (1) turns to the p.h.o. potential with energyeigenvalues: E nl = s ¯ h a m n + r (2 l + 1) − m ¯ h d ! (42)and wave functions: ψ nl ( r ) = N l n Π i =1 (cid:16) r − α ( n ) i (cid:17) r ( − l ′ ) / exp − r ma ¯ h r ! . (43)The present results in Eqs. (42) and (43) coincide with Eqs. (15) and (16) of Ref. [22]obtained by the exact polynomial method, Eqs. (72) and (78) of Ref. [23] obtained by theNikiforov-Uvarov method and Eqs. (30) and (31) of Ref. [24] obtained by the wave functionansatz method after setting D /r = a, D r = − d and 2 D = 0 . The model solved in thepresent work can be used in modeling the quarkonium [14] perturbed by the field of p.h.o.or electron confined in spherical quantum dots [11]. Finally, our solution to this confiningpotential is being considered important in many different fields of physics, such as atomicand molecular physics [25,26], particle physics [13,27,28], plasma physics and solid-statephysics [29-33]. Acknowledgments The partial support provided by the Scientific and Technological Research Council ofTurkey is highly appreciated. 12 1] G.C. Maitland, M. Rigby, E.B. smith and W.A. Wakeham, Intermolecular forces (OxfordUniv. Press, Oxford, 1987).[2] R.J. Le Roy and W. Lam, Chem. Phys. Lett. 71, 544 (1980); R.J. Le Roy and R.B. Bernstein,J. Chem. Phys. 52, 3869 (1970).[3] S.M. Ikhdair and R. Sever, J. Molec. Struct. :Theochem 855, 13 (2008).[4] B.H. Bransden and C.J. Joachain, Physics of atoms and molecules (Longman, London, 1983).[5] A.O. Barut, J. Math. Phys. 21, 568 (1980).[6] A.O. Barut, M. Berrondo and G. Garcia-Calderon, J. Math. Phys. 21, 1851 (1980).[7] S. ¨Oz¸celik and M. S¸im¸sek, Phys. Lett. A, 152 (1991).[8] M. Znojil, J. Math. Phys. 30, 23 (1989).[9] M. Znojil, J. Math. Phys. 31, 108, 1955 (1990).[10] S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. C 18 (10), 1571 (2007).[11] H. Hassanabadi and A.A. Rajabi, Phys. Lett. A 373, 679 (2008).[12] F. Geerinckx, F.M. Peeters and J.T. Devreese, J. Appl. Phys. 68 (7), 3435 (1990).[13] N.V. Maksimenko and S.M. Kuchin, Russ. Phys. J. 54, 57 (2011); E. Eichten, K. Gottfried,T. Kinoshita, K.D. Lane and T.M. Yan, Phys. Rev. D 21, 203 (1980).[14] J.-L. Domenech-Garret, M.-A. Sanchis-Lozano, Phys. Lett. B 669, 52 (2008).[15] B. Szafran and F.M. Peeters, Phys. Rev. B 72, 155316 (2005).[16] G.A. Farias and F.M. Peeters, Solid State Commun. 100, 711 (1996).[17] Y. Koma, M. Koma and H. Wittig, arXiv: 0711.2322 [hep-lat].[18] S. Deoghuria and S. Chakrabarty, Z. Phys. C 53, 293 (1992).[19] R.L. Liboff, Introductory Quantum Mechanics, 4th edn. (Pearson Education, Inc., San Fran-cisco, CA, 2003).[20] A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, Beteman Manuscript Project,Higher transcendental functions (McGraw-Hill, New York, 1953).[21] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series, and Products, 5th ed. (Academic,New York, 1994).[22] S.M. Ikhdair and R. Sever, J. Molec. Struct. :Theochem 806, 155 (2007).[23] S.M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 6(3), 685 (2008). 24] S.M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 6(3), 697 (2008).[25] E.J. Austin, Mol. Phys. 40, 893 (1980).[26] J. Killingbeck, J. Phys. A. Math. Gen. 19, 2903 (1986); 13, L393 (1980).[27] E.R. Vrscay, Int. J. Quantum Chem. 32, 613 (1987); Phys. Rev. A 31, 2054 (1985).[28] V. Gupta and A. Khare, Phys. Lett. 70, 313 (1979); C. Quigg and J.L. Rosner, Phys. Rep.56, 167 (1979).[29] J.E. Avron, Ann. Phys. 131, 73 (1991).[30] S. Skupsky, Phys. Rev. A 21, 1316 (1980).[31] R. Cauble, M. blaha and J. Davis, Phys. Rev. A 29, 3280 (1984).[32] S. Glasberg et al ., Phys. rev. B 59, R10425 (1999).[33] G. Yusa, H. Shatrikman and I. Bar-Joseph, cond-mat/0103505.[34] R.N. Chaudhury and M. Mondal, Phys. Rev. A 52, 1850 (1995). ABLE I: Lowest ( n = 0) energy spectra (for ¯ h = m = 1). a b c d l Numerical Present SUSYQM [34] − . − . − . − . − . − . a = 1 eV.f m − , b = 0 . eV . f m − , c = 0 . eV.f m and d = 0 . eV.f m . FIG. 3: The ground state electron energy in InGaAs semiconductor versus ω in the field of theCMpH potential with c = 0 . eV.nm and d = 0 . eV.nm for the cases l = 0 and l = 1 , respectively.FIG. 4: Behaviour of the ground state wave function ψ n =0 ,l =0 ( r ) (dashed line) and ψ n =0 ,l =1 ( r )(continuous line) in the field of the CpH potential with the value of c = 0 . eV.nm for an electronwith effective mass m = 0 . m e and frequency ω = 10 × Hz in the InGaAs semiconductor.FIG. 2: The ground state electron energy in InGaAs semiconductor versus ω in the field of CpHpotential with c = 0 . eV.nm for cases l = 0 and l = 1 , respectively . IG. 5: Behaviour of the ground state wave function ψ n =0 ,l =0 ( r ) (dashed line) and ψ n =0 ,l =1 ( r )(continuous line) of the CMpH potential with the values of c = 0 . eV.nm and d = 0 . eV.nm for an electron with an effective mass m = 0 . m e and frequency ω = 10 × Hz in the InGaAssemiconductor.FIG. 6: Ground state energy of electron versus c, for the case l = 0 , ω = 8 × Hz and d = 0 . eV.nm . FIG. 7: Ground state energy of electron versus c, for the case l = 1 , ω = 8 × Hz and d = 0 . eV.nm . FIG. 8: The first excited state electron energy in InGaAs semiconductor versus ω in the field ofCpH potential with c = 0 . eV.nm for cases l = 0 and l = 1 , respectively . FIG. 9: The first excited state electron energy in InGaAs semiconductor versus ω in the field ofthe CMpH potential with c = 0 . eV.nm and d = 0 . eV.nm for the cases l = 0 and l = 1 , respectively. .01 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4−125−100−75−50−250 r (fm) V (r) ( e V ) ω (Hz) E ( e V )) 24] S.M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 6(3), 697 (2008).[25] E.J. Austin, Mol. Phys. 40, 893 (1980).[26] J. Killingbeck, J. Phys. A. Math. Gen. 19, 2903 (1986); 13, L393 (1980).[27] E.R. Vrscay, Int. J. Quantum Chem. 32, 613 (1987); Phys. Rev. A 31, 2054 (1985).[28] V. Gupta and A. Khare, Phys. Lett. 70, 313 (1979); C. Quigg and J.L. Rosner, Phys. Rep.56, 167 (1979).[29] J.E. Avron, Ann. Phys. 131, 73 (1991).[30] S. Skupsky, Phys. Rev. A 21, 1316 (1980).[31] R. Cauble, M. blaha and J. Davis, Phys. Rev. A 29, 3280 (1984).[32] S. Glasberg et al ., Phys. rev. B 59, R10425 (1999).[33] G. Yusa, H. Shatrikman and I. Bar-Joseph, cond-mat/0103505.[34] R.N. Chaudhury and M. Mondal, Phys. Rev. A 52, 1850 (1995). ABLE I: Lowest ( n = 0) energy spectra (for ¯ h = m = 1). a b c d l Numerical Present SUSYQM [34] − . − . − . − . − . − . a = 1 eV.f m − , b = 0 . eV . f m − , c = 0 . eV.f m and d = 0 . eV.f m . FIG. 3: The ground state electron energy in InGaAs semiconductor versus ω in the field of theCMpH potential with c = 0 . eV.nm and d = 0 . eV.nm for the cases l = 0 and l = 1 , respectively.FIG. 4: Behaviour of the ground state wave function ψ n =0 ,l =0 ( r ) (dashed line) and ψ n =0 ,l =1 ( r )(continuous line) in the field of the CpH potential with the value of c = 0 . eV.nm for an electronwith effective mass m = 0 . m e and frequency ω = 10 × Hz in the InGaAs semiconductor.FIG. 2: The ground state electron energy in InGaAs semiconductor versus ω in the field of CpHpotential with c = 0 . eV.nm for cases l = 0 and l = 1 , respectively . IG. 5: Behaviour of the ground state wave function ψ n =0 ,l =0 ( r ) (dashed line) and ψ n =0 ,l =1 ( r )(continuous line) of the CMpH potential with the values of c = 0 . eV.nm and d = 0 . eV.nm for an electron with an effective mass m = 0 . m e and frequency ω = 10 × Hz in the InGaAssemiconductor.FIG. 6: Ground state energy of electron versus c, for the case l = 0 , ω = 8 × Hz and d = 0 . eV.nm . FIG. 7: Ground state energy of electron versus c, for the case l = 1 , ω = 8 × Hz and d = 0 . eV.nm . FIG. 8: The first excited state electron energy in InGaAs semiconductor versus ω in the field ofCpH potential with c = 0 . eV.nm for cases l = 0 and l = 1 , respectively . FIG. 9: The first excited state electron energy in InGaAs semiconductor versus ω in the field ofthe CMpH potential with c = 0 . eV.nm and d = 0 . eV.nm for the cases l = 0 and l = 1 , respectively. .01 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4−125−100−75−50−250 r (fm) V (r) ( e V ) ω (Hz) E ( e V )) l=0, d=0l=1, d=0 3 4 5 6 7 8 9 10x 10 ω (Hz) E ( e V )) 24] S.M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 6(3), 697 (2008).[25] E.J. Austin, Mol. Phys. 40, 893 (1980).[26] J. Killingbeck, J. Phys. A. Math. Gen. 19, 2903 (1986); 13, L393 (1980).[27] E.R. Vrscay, Int. J. Quantum Chem. 32, 613 (1987); Phys. Rev. A 31, 2054 (1985).[28] V. Gupta and A. Khare, Phys. Lett. 70, 313 (1979); C. Quigg and J.L. Rosner, Phys. Rep.56, 167 (1979).[29] J.E. Avron, Ann. Phys. 131, 73 (1991).[30] S. Skupsky, Phys. Rev. A 21, 1316 (1980).[31] R. Cauble, M. blaha and J. Davis, Phys. Rev. A 29, 3280 (1984).[32] S. Glasberg et al ., Phys. rev. B 59, R10425 (1999).[33] G. Yusa, H. Shatrikman and I. Bar-Joseph, cond-mat/0103505.[34] R.N. Chaudhury and M. Mondal, Phys. Rev. A 52, 1850 (1995). ABLE I: Lowest ( n = 0) energy spectra (for ¯ h = m = 1). a b c d l Numerical Present SUSYQM [34] − . − . − . − . − . − . a = 1 eV.f m − , b = 0 . eV . f m − , c = 0 . eV.f m and d = 0 . eV.f m . FIG. 3: The ground state electron energy in InGaAs semiconductor versus ω in the field of theCMpH potential with c = 0 . eV.nm and d = 0 . eV.nm for the cases l = 0 and l = 1 , respectively.FIG. 4: Behaviour of the ground state wave function ψ n =0 ,l =0 ( r ) (dashed line) and ψ n =0 ,l =1 ( r )(continuous line) in the field of the CpH potential with the value of c = 0 . eV.nm for an electronwith effective mass m = 0 . m e and frequency ω = 10 × Hz in the InGaAs semiconductor.FIG. 2: The ground state electron energy in InGaAs semiconductor versus ω in the field of CpHpotential with c = 0 . eV.nm for cases l = 0 and l = 1 , respectively . IG. 5: Behaviour of the ground state wave function ψ n =0 ,l =0 ( r ) (dashed line) and ψ n =0 ,l =1 ( r )(continuous line) of the CMpH potential with the values of c = 0 . eV.nm and d = 0 . eV.nm for an electron with an effective mass m = 0 . m e and frequency ω = 10 × Hz in the InGaAssemiconductor.FIG. 6: Ground state energy of electron versus c, for the case l = 0 , ω = 8 × Hz and d = 0 . eV.nm . FIG. 7: Ground state energy of electron versus c, for the case l = 1 , ω = 8 × Hz and d = 0 . eV.nm . FIG. 8: The first excited state electron energy in InGaAs semiconductor versus ω in the field ofCpH potential with c = 0 . eV.nm for cases l = 0 and l = 1 , respectively . FIG. 9: The first excited state electron energy in InGaAs semiconductor versus ω in the field ofthe CMpH potential with c = 0 . eV.nm and d = 0 . eV.nm for the cases l = 0 and l = 1 , respectively. .01 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4−125−100−75−50−250 r (fm) V (r) ( e V ) ω (Hz) E ( e V )) l=0, d=0l=1, d=0 3 4 5 6 7 8 9 10x 10 ω (Hz) E ( e V )) l=0, d=0.1l=1, d=0.1 1 2 3 4 5 600.10.20.30.40.50.60.70.80.91 r(fm) ψ , l (r) ψ (r), d=0 ψ (r), d=0 1 2 3 4 5 600.20.40.60.811.21.4 r(fm) ψ , l (r) ψ (r), d=0.01 ψ (r), d=0.01.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.1050.7790.77950.780.78050.7810.78150.7820.7825 c (eV.nm) E ( e V ) l=0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.1051.31321.31331.31341.31351.31361.31371.31381.3139 c (eV.nm) E ( e V ) l=1 3 4 5 6 7 8 9 10x 10 ω (Hz) E ( e V )) 24] S.M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 6(3), 697 (2008).[25] E.J. Austin, Mol. Phys. 40, 893 (1980).[26] J. Killingbeck, J. Phys. A. Math. Gen. 19, 2903 (1986); 13, L393 (1980).[27] E.R. Vrscay, Int. J. Quantum Chem. 32, 613 (1987); Phys. Rev. A 31, 2054 (1985).[28] V. Gupta and A. Khare, Phys. Lett. 70, 313 (1979); C. Quigg and J.L. Rosner, Phys. Rep.56, 167 (1979).[29] J.E. Avron, Ann. Phys. 131, 73 (1991).[30] S. Skupsky, Phys. Rev. A 21, 1316 (1980).[31] R. Cauble, M. blaha and J. Davis, Phys. Rev. A 29, 3280 (1984).[32] S. Glasberg et al ., Phys. rev. B 59, R10425 (1999).[33] G. Yusa, H. Shatrikman and I. Bar-Joseph, cond-mat/0103505.[34] R.N. Chaudhury and M. Mondal, Phys. Rev. A 52, 1850 (1995). ABLE I: Lowest ( n = 0) energy spectra (for ¯ h = m = 1). a b c d l Numerical Present SUSYQM [34] − . − . − . − . − . − . a = 1 eV.f m − , b = 0 . eV . f m − , c = 0 . eV.f m and d = 0 . eV.f m . FIG. 3: The ground state electron energy in InGaAs semiconductor versus ω in the field of theCMpH potential with c = 0 . eV.nm and d = 0 . eV.nm for the cases l = 0 and l = 1 , respectively.FIG. 4: Behaviour of the ground state wave function ψ n =0 ,l =0 ( r ) (dashed line) and ψ n =0 ,l =1 ( r )(continuous line) in the field of the CpH potential with the value of c = 0 . eV.nm for an electronwith effective mass m = 0 . m e and frequency ω = 10 × Hz in the InGaAs semiconductor.FIG. 2: The ground state electron energy in InGaAs semiconductor versus ω in the field of CpHpotential with c = 0 . eV.nm for cases l = 0 and l = 1 , respectively . IG. 5: Behaviour of the ground state wave function ψ n =0 ,l =0 ( r ) (dashed line) and ψ n =0 ,l =1 ( r )(continuous line) of the CMpH potential with the values of c = 0 . eV.nm and d = 0 . eV.nm for an electron with an effective mass m = 0 . m e and frequency ω = 10 × Hz in the InGaAssemiconductor.FIG. 6: Ground state energy of electron versus c, for the case l = 0 , ω = 8 × Hz and d = 0 . eV.nm . FIG. 7: Ground state energy of electron versus c, for the case l = 1 , ω = 8 × Hz and d = 0 . eV.nm . FIG. 8: The first excited state electron energy in InGaAs semiconductor versus ω in the field ofCpH potential with c = 0 . eV.nm for cases l = 0 and l = 1 , respectively . FIG. 9: The first excited state electron energy in InGaAs semiconductor versus ω in the field ofthe CMpH potential with c = 0 . eV.nm and d = 0 . eV.nm for the cases l = 0 and l = 1 , respectively. .01 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4−125−100−75−50−250 r (fm) V (r) ( e V ) ω (Hz) E ( e V )) l=0, d=0l=1, d=0 3 4 5 6 7 8 9 10x 10 ω (Hz) E ( e V )) l=0, d=0.1l=1, d=0.1 1 2 3 4 5 600.10.20.30.40.50.60.70.80.91 r(fm) ψ , l (r) ψ (r), d=0 ψ (r), d=0 1 2 3 4 5 600.20.40.60.811.21.4 r(fm) ψ , l (r) ψ (r), d=0.01 ψ (r), d=0.01.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.1050.7790.77950.780.78050.7810.78150.7820.7825 c (eV.nm) E ( e V ) l=0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.1051.31321.31331.31341.31351.31361.31371.31381.3139 c (eV.nm) E ( e V ) l=1 3 4 5 6 7 8 9 10x 10 ω (Hz) E ( e V )) l=0, d=0l=1, d=0 3 4 5 6 7 8 9 10x 10 ω (Hz) E ( e V ))