Exact solution of Schrodinger equation for modified Kratzer's molecular potential with the position-dependent mass
aa r X i v : . [ qu a n t - ph ] D ec Exact solution of Schr¨odinger equation for modifiedKratzer’s molecular potential with the position-dependentmass
Ramazan Sever , Cevdet Tezcan Middle East Technical University,Department of Physics, 06531 Ankara, Turkey Faculty of Engineering, Ba¸skent University, Ba˜glıca Campus, Ankara, Turkey
October 27, 2018
Abstract
Exact solutions of Schr¨odinger equation are obtained for the modified Kratzer and thecorrected Morse potentials with the position-dependent effective mass. The bound stateenergy eigenvalues and the corresponding eigenfunctions are calculated for any angularmomentum for target potentials. Various forms of point canonical transformations areapplied.PACS numbers: 03.65.-w; 03.65.Ge; 12.39.FdKeywords: Morse potential, Kratzer potential, Position-dependent mass, Point canonicaltransformation, Effective mass Schr¨odinger equation. Introduction
Solutions of Schr¨odinger equation for a given potential with any angular momentum have muchattention in chemical physics systems. Energy eigenvalues and the corresponding eigenfunctionsprovide a complete information about the diatomic molecules.Morse and Kratzer potentials [1,2] are one of the well-known diatomic potentials. Themethod used in the Schr¨odinger equation for vibration-rotation states are mostly based on thewave function expansion and exact solution for a single state with some restrictions on thecoupling constants [3-7]. On the other hand solutions of the position-dependent effective-massSchr¨odinger equation are very interesting chemical potential problem.They have also found important applications in the fields of material science and condensedmatter physics such as semiconductors[8], quantum well and quantum dots[9], H , clusters[10],quantum liquids[11], graded alloys and semiconductor heterostructures]12,13].Recently, number of exact solutions on these topics increased[14-31]. Various methods areused in the calculations. The point canonical transformations (PCT) is one of these methodsproviding exact solutions of energy eigenvalues and corresponding eigenfunctions [24-27]. It isalso used for solving the Schr¨odinger equation with position-dependent effective mass for somepotentials [8-13].In the present work, we solve two different potentials with the three mass distributions. Thepoint canonical transformation is taken in the more general form introducing a free parameter.This general form of the transformation will provide us a set of solutions for different valuesof free parameter. In this work, the exact solution of Schr¨odinger equation is obtained or themodified Kratzer type of molecular potential [31] and the corrected Morse potential [32].The contents of the paper is as follows. In section 2, we present briefly the solution ofthe Schr¨odinger by using point canonical transformation. In section 3, we introduce someapplications for the specific mass distributions. Results are discussed in section 4. To introduce the PCT, we start from a time independent Schr¨odinger equation for a potential V ( y ) − d dx + V ( y ) ! φ ( y ) = Eφ ( y ) (1)where the atomic unit ¯ h = 1 and the constant mass M = 1 are taken. Defining a transformation y → x for a mapping y = f ( x ), the wave function can be rewritten as φ ( y ) = m ( x ) ψ ( x ) . (2)The transformed Schr¨odinger equation takes − d dx − m ′ m − f ′′ f ′ ! ddx − m ′′ m ′ + ( α − m ′ m ! − m ′ m ! f ′′ f ′ + ( f ′ ) V ( f ( x )) o ψ ( x ) = ( f ′ ) E ψ ( x ) , (3)2here the prime denotes differentiation with respect to x . On the other hand the one dimen-sional Schr¨odinger equation with position dependent mass can be written as − ddx " M ( x ) dψ ( x ) dx + ˜ V ( x ) ψ ( x ) = ˜ Eψ ( x ) , (4)where M ( x ) = m m ( x ), and the dimensionless mass distribution m ( x ) is real function. Forsimplicity, we take m = 1. Thus, Eq. (4) takes the form − d dx + m ′ m ddx + m ˜ V ( x ) ! ψ ( x ) = m ˜ Eψ ( x ) . (5)Comparing Eqs. (3) and (5), we get the following identities f ′′ f ′ − m ′ m = m ′ m (6)and ˜ V ( x ) − ˜ E = f ′ m [ V ( f ( x )) − E ] − m " m ′′ m − m ′ m ! f ′′ f ′ ! (7)From Eq. (6), one gets f ′ = m / (8)Substituting f ′ into Eq. (7), the new potential can be obtained as˜ V ( x ) = V ( f ( x )) − m m ′′ m − m ′ m ! . (9)Therefore, the energy eigenvalues and corresponding wave functions for the potential V ( y ) as E n and φ n ( y ) become ˜ E n = E n (10)and ψ n ( x ) = 1 m ( x ) φ n ( y ) (11) We solve Schr¨odinger equation exactly for two potentials the rotationally corrected Morse po-tential[30] and the modified Kratzer molecular potential[31]. We consider three kinds of theposition dependent mass distributions. Two of them are used before[19], and the third one isthe exponentially decreasing mass distribution with a free parameter q.3 .1 Modified Kratzer Potential V ( r ) = D e y − y e y ! , (12)where D e is the dissociation energy and y e is the equilibrium internuclear separation. Energyspectrum and the wave functions are E nℓ ( n ) = D e − ¯ h µ (cid:18) µD e y e ¯ h (cid:19) n + vuut µD e y e ¯ h + ℓ ( ℓ + 1) ! − , (13) R nℓ = A nℓ (2 iεy ) − (1 − η ) e − iεy L √ γn (2 iεy ) (14)where η = q γ, (15) γ = 2 µ (cid:16) D e y e + ℓ ( ℓ +1)¯ h µ (cid:17) ¯ h , (16) A nℓ = µD e y e ¯ h (2 n + η + 1) ! / " n !(2 n + η + 1)( n + η )! / , (17) ǫ = iβ (2 n + η + 1) , (18)and β = − µD e y e ¯ h , (19) m ( x ) = a q + x y = f ( x ) = Z m ( x ) / dx = aℓn ( x + q q + x ) , (20)The target potential is˜ V ( x ) = D e " aℓn ( x + √ q + x ) − y e aℓn ( x + √ q + x ) − a q + x q + x , (21)Energy eigenvalues and the normalized radial wave function for the target potential ˜ V ( x ) are˜ E n = E nℓ ( n ) , (22) R nℓ ( x ) = A nℓ (2 iεℓn ( x + q q + x )) − (1 − η ) e − iεℓn ( x + √ q + x ) L √ γn (2 iaεℓn ( x + q q + x )) (23) A nℓ = 4 an !(1 + n + ε ) (2 √ ε ) ε (1 + n + 2 ε )! . (24)4 .1.2 Asymptotically vanishing mass distribution m ( x ) = a ( b + x ) y = f ( x ) = Z m ( x ) / dx = a √ b tan − x √ b , (25)The target potential ˜ V ( x ) = D e a √ b tan − x √ b − y ea √ b tan − x √ b − a ( b + 2 x ) , (26)and the corresponding energy spectrum and the wave function are˜ E n = E nℓ ( n ) , (27) R nℓ ( x ) = A nℓ iε a √ b tan − x √ b ! − (1 − η ) e − iε a √ b tan − x √ b L √ γn (2 iε a √ b tan − x √ b ) . (28) m ( x ) = e − qx y = f ( x ) = Z m ( x ) / dx = − q e − q x , (29)the target potential ˜ V ( x ) = D e (cid:18) qr e e q x (cid:19) + 9128 q e − qx , (30)and the corresponding energy spectrum and the wave function are˜ E n = E nℓ ( n ) , (31) R nℓ ( x ) = A nℓ (cid:18) − α iεe − α x (cid:19) − (1 − η ) e α iεe − α x L √ γn ( − α iεe − α x ) . (32) V ( y ) = D ( e − αy − e − αy ) + γ ( D + D e − αy + D e − αy ) , (33) α = ar , (34) γ = ¯ h ℓ ( ℓ + 1)2 µr , (35) r is the equilibrium intermolecular distance, a is a parameter controlling the width of thepotential wall. D is the dissociation energy and D = 1 − α + 3 α , (36)5 = 4 α − α , (37) D = − α + 3 α , (38)Energy spectrum and the radial wave function are E nℓ = ¯ h ℓ ( ℓ + 1)2 µr − ar + 3 a r ! − ¯ h a µ " ε √ ε − ( n + 12 ) , (39)where ε √ ε = 1 a √ ε " µD ¯ h − ℓ ( ℓ + 1) r ar − a r ! , (40) R nℓ ( y ) = A nℓ e − αε y e −√ ε e − αy L ε n (2 √ e − αy ) , (41) − ε = 2 µr ( E nℓ − γD )¯ h α , (42) − ε = 2 µr (2 D − γD )¯ h α , (43) − ε = 2 µr ( D + γD )¯ h α , (44) m ( x ) = a q + x y is given in 3.1.1. The target potential˜ V ( x ) = D + γD ( x + √ q + x ) αa + γD − D ( x + √ q + x ) αa + γD (45)Energy spectrum and the wave function are˜ E n = E nℓ ( n ) , (46) R nℓ ( x ) = A nℓ ( x + √ q + x ) αε a e − √ ε x + √ q + x αa L ε n √ ε ( x + √ q + x ) αa ! . (47) m ( x ) = a ( b + x ) The target potential˜ V ( x ) = ( D + γD ) e − αa √ b tan − x √ b + ( γD − D ) e − αa √ b tan − x √ b + γD − q + 2 x a (48)Energy spectrum and the wave function are 6 E n = E nℓ ( n ) , (49) R nℓ ( x ) = A nℓ e − ε P x e −√ ε e − Px L ε n (cid:16) √ ε e − P x (cid:17) . (50) m ( x ) = e − qx The target potential˜ V ( x ) = D (cid:18) e αq e − qx − e αq e − qx (cid:19) + γ (cid:18) D + D e αq e − qx + D e αq e − qx (cid:19) , (51)Energy spectrum and the wave function ˜ E n = E nℓ ( n ) , (52) R nℓ ( x ) = A nℓ [ T ( x )] ǫ e −√ ǫ T ( x ) L ǫ n (cid:16) √ T ( x ) (cid:17) (53)where T ( x ) = e Q ( x ) and Q ( x ) = αq e − q x . We have applied the PCT in a general form by introducing a free parameter to solve theSchr¨odinger equation for the corrected Morse and modified Kratzer potentials with spatiallydependent mass. In the computations, we have used three position dependent mass distribu-tions. Energy eigenvalues and corresponding wave funtions for target potentials are written inthe compact form.
This research was partially supported by the Scientific and Technological Research Council ofTurkey. 7 eferences [1] P. M. Morse, Phys. Rev. , 57 (1929).[2] A. Kratzer, Z. Phys. , 289 (1920).[3] A. O. Barut, J. Math. Phys. , 568 (1980); A. O. Barut, M. Berrondo, G. Garcia-Calderon, J. Math. Phys. , 1851 (1980).[4] A. E. DePristo, J. Chem. Phys. , 5037 (1981).[5] S. Ozcelik, Tr. J. Phys. , 1233 (1996).[6] M. Znojil, J. Mat. Chem. , 157 (1999).[7] J. P. Killingbeck, A. Grosjean, G, Jolierd, J. Chem. Phys. , 447 (2002).[8] Bastard, G. “ Wave Mechanics Applied to Heterostructure ”, (Les Ulis, Les Edition dePhysique, 1989).[9] P. Harrison, “
Quantum Wells, Wires and Dots ” (New York,[10] M. Barranco et al., Phys. Rev.
B56 , 8997 (1997).[11] F. Arias et al., Phys. Rev.
B50 , 4248 (1997).[12] C. Weisbuch and B. Vinter “
Quantum Semiconductor Heterostructure ”, (New York, Aca-demic Press, 1993) and references therein; O. Von Roos, Phys. Rev.
B27 , 7547 (1983); O.Von Roos and H. Mavromatis, Phys. Rev.
B31 , 2294 (1985); R. A. Morrow, Phys. Rev.
B35 , 8074 (1987); V. Trzeciakowski, Phys. Rev.
B38 , 4322 (1988); I. Galbraith and G.Duygan Phys. Rev.
B38 , 10057 (1988); K. Young, Phys. Rev.
B39 , 13434 (1989); G. T.Einvoll et al., Phys. Rev.
B42 , 3485 (1990); G. T. 1Einvoll, Phys. Rev.
B42 , 3497 (1990).[13] J. Yu, S. H. Dong, G. H. Sun, Phys. Lett.
A322 , 290 (1999).[14] S. H. Dong, M. Lozada-Cassou, Phys. Lett. , 313 (2005).[15] A. D. Alhaidari, Phys. Rev.
A66 , 042116; A. D. Alhaidari, Int. J. Theor. Phys. , 2999(2003).[16] B. G¨on¨ul, O. ¨Ozer, B. G¨on¨ul, F. ¨Uzg¨un, Mod. Phys. Lett. A1 , 2453 (2002).[17] J. Yu, S. H. Dong, Phys. Lett. A325 , 194 (2004).[18] G. Chen, Z. D. Chen, Phys. Lett.
A331 , 312 (2004).[19] K. Bencheikh, S. Berkane, S. Bouizane, J. Phys. A: Math. Gen. , 10719 (2004).[20] R. Koc, H. T¨ut¨unc¨u, Ann. Pjys.(leipzig) , 684 (2003).[21] L. Dekar, , T. Chetouani, F. Hammann, J. Phys. A: Math. Gen. , 2551 (1998); L. Dekar,, T. Chetouani, F. Hammann, Phys. Rev. A59 , 107 (1999).822] A. R. Plastino, A. Rigo, M. Casas, A. Plastino, Phys. Rev.