EXPLICIT SOLUTIONS FOR N-DIMENSIONAL SCHRÖDINGER EQUATIONS WITH POSITION-DEPENDENT MASS. B. Gönül and M. Koçak

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1 EXPLICIT SOLUTIONS FOR N-DIENSIONAL SCHRÖDINGER EQUATIONS WITH POSITION-DEPENDENT ASS B. Gönül and. Koçak Depatment of Engineeing Physics, Univesity of Gaziantep, 7310, Gaziantep-Tükiye Abstact With the consideation of spheical symmety fo the potential and mass function, onedimensional solutions of non-elativistic Schödinge equations with spatially vaying effective mass ae successfuly extended to abitay dimensions within the fame of ecently developed elegant non-petubative technique, whee the BenDaniel-Duke effective Hamiltonian in one-dimension is assumed like the unpetubed piece, leading to well-known solutions, wheeas the modification tem due to possible use of othe effective Hamiltonians in one-dimension and, togethe with, the coections coming fom the teatments in highe dimensions ae consideed as an additioanal tem like the petubation. Application of the model and its genealization fo the completeness ae discussed. PACS: Fd Keywods: Schödinge equation, Position-dependent mass, N dimension 1. INTRODUCTION Gaining confidence fom the successful applications [1] of the ecently developed simple model [] in diffeent fields of physics, we have investigated [3] the elation between the solutions of physically acceptable effective mass Hamiltonians poposed in the liteatue fo the teatment of one dimensional poblems. Using the spiit of the pesciption suggested in [3], we aim hee to tackle the moe difficult poblem of geneating exact solutions fo position-dependent mass Schödinge equations (PDSE) in N dimension, as the most of the elated woks in the liteatue have been devoted to one-dimensional systems except the ones in [4]. The concept of PDSE is known to play an impotant ole in diffeent banch of physics. This fomalism has been extensively used in nuclei, quantum liquids, 3 He and metal clustes. Anothe aea wheein the such concepts povide vey useful tool is the study of electonic popeties of many condensed-matte systems, such as semiconductos and quantum dots. In paticula, ecent pogess in cystal-gowth techniques fo poducing non-unifom semiconducto specimens, wheein the caie effective mass depends on position, has consideably enhanced the inteest in the theoetical desciption of semiconducto heteostuctues. It has also ecently been signalled in the apidly gowing field of PTsymmetic o moe geneally pseudo-hemitian quantum mechanics. Fo an excellent ecent eview, leading to the elated efeences, the eade is efeed to [4]. In Section, the systematic teatment of N dimensional PDSE is pesented and closed expessions coesponding to the full wave function and enegy spectum fo exactly solvable

2 potentials ae given. Section 3 contains the application of the model while the genealization of the fomalism is discussed in Section 4. Concluding emaks ae given in the last section.. THEORETICAL CONSIDERATION Tacking down solvable potentials in PDSE has always aoused inteest. Apat fom being useful in undestanding of many physical phenomena, the impotance of seaching fo them also stems fom the fact that they vey often povide a good stating point fo undetaking petubative calculations of moe complex systems. As is well known (see, e.g., [3,5]), the geneal fom of adial PDSE with Hemitian Hamiltonians in one-dimension gives ise to d 1 dφ( z) eff V dz Φ = Φ dz λ, (1) whee the effective potential ( α γ ) V eff = V0 U = V0 ( α γ), () 3 depends on the mass tem and ambiquity paametes. Hee a pime denotes deivative with espect to the vaiable, is the dimensionless fom of the mass function m = m0 and we have set h = m = 1 0. The effective potential is the sum of the eal potential pofile V ( ) and the modification U (z) emeged fom the location dependendence of the 0 z effective mass. A diffeent Hamiltonian leads to a diffeent modification tem. Some of them ae the ones poposed by [6] BenDaniel-Duke ( α =γ = 0), Bastad ( α = 1), Zhu-Koeme ( =γ = 1 ) α and Li-Kuhn ( γ 1, α = 0 ) =. Consideing the woks in [3,7], the adial piece of PDSE in abitay dimensions fo spheically symmetic potentials and mass functions eads d d N 1 d L d ( L N ) ( N 1)( N 3) whee we assume that 1 F G F G F G F G () F() G() Ψ = F G = U F G which leads to The effective potential in highe dimensions ( N U eff eff 4 [ E V () ] Ψ() eff, (3) E. (4) f 1) now is tansfomed to the fom ( N 1) L( L N ) ( N 1)( N 3) 4 = V0 U, (5) in which L is the angula momentum. As the one-dimensional calculations equie N = 1 and L = 0, eq. (5) educes in this case to U = V = V U as in Ref. [3], which povides us a safe testing gound. eff eff o

3 Keeping in mind the spiit of the technique used simply in [3], we split Eq. (4) in two pats deviating fom the teatments in [4] W F W = V0 ε, W =, F (6) whee ε is the coesponding enegy of the equied quantum state F n ( n = 0,1,,...) fo V 0 which is assumed in this model as an exactly solvable mass-dependent potential, and G W W = V E, = G, (7) whee ( N 1) L( L N ) ( N 1)( N 3) 4 V = U. (8) Note that the total enegy appeaing in (4) is E =ε E and, in one-dimension the modification tem V becomes U as in Ref. [3]. This claifies that the coections due to the highe dimensions aise because of the second and thid tem on RHS of Eq. (8). Fom the pesent theoetical consideation, Eq. (6) has an algebaic solution leading to closed analytical expessions fo the wave functions and enegy eigenvalues, hence one needs to solve Eq. (7) exactly. To poceed futhe, with the consideation of elativistic Diac equations having no ambiquity paametes, we confidently choose ( α γ ) ( N L 1) =, (9) 3 in which the second tem disappeas fo N = 1 as in [3]. Within the fame of Eq. (7), this choice leads us ( α γ )( N 1) E W = ( 1) L α γ, (10) that is the main esult of the pesent Lette. Fom the definition of the effective potential in Eq. (), we also note that the use of Eqs. (7) and (8) natually esticts the choice of some ambiquity paametes yielding diffeent physically acceptable effective mass Hamiltonians, allowing only α =γ = 0 (Ben-Daniel Duke Hamiltonian) and α =γ = 1 (Zhu-Koeme Hamiltonian) cases. This obsevation claifies that the unpetubed pat ( V 0 ) of the effective potential in (6) should coesponds to the case α =γ = 0, having well known solutions in one dimension, while α =γ = 1 is used to calculate U in (8). Obviosly, all the coections coming fom the highe dimensions to the enegy and well-behaved wave function tems can be systematically calculated fo a given with the consideation of Eqs. (7-10) in the light of coesponding W in (6). 3. APPLICATION Recently, some eseaches have been devoted to the analysis of the classification of quantum systems with position-dependent mass egading thei exact solvability [3-5, and the efeences theein]. On a simila basis, Plastino and his co-wokes [8] applied an appoach within the supesymmetic quantum mechanical famewok, fo the case α =γ = 0, to such systems and succeed to show that some one-dimensional systems with non-constant mass have a supesymmetic patne with the same effective mass. They wee also able to solve

4 exactly some paticula cases by constucting the supepotential [ W () ] fom the fom of the effective mass [ ()] and genealize the concept of the shape invaiance fo these systems. Fo illustation, the supepotential expessions given by [8] fo the systems having hamonic oscillato and ose-like specta can be easly used in Eq. (6) to seve explicit expessions fo the coections to the one-dimensional solutions obtained by consideing the Ben-Daniel- Duke effective Hamiltonian in thei [8] calculations. This simple investigation enables us testing ou esults, because all the coections should disappea in case N = 1 and α =γ = 0 leading to the expessions in [8]. Fo claity, this section involves only the application on the hamonic oscillato system. Howeve, the genealization of the pesent model yielding selfconsistent calculations, epoducing W () tem within the model fo any system of inteest will be discussed in the next section. Accoding to Ref. [8], W () tem in Eqs. (6) and (10) is ω 1 1 W () = dz, ω = ε n= 0, (11) fo the systems having hamonic oscillato specta. Hence, use of Eqs. (7) though (10) gives ( α γ )( N 1) ( ) ( N L 1) ω E = L α γ 1 ( ) dz z 1 ( α γ ) 1 ( N L 1) ω dz, (1) which is the explicit fom of the enegy coections fo a given smooth mass. Clealy, it can be seen that fo a constant mass 1, Eq. 1 educes to ( N L 1) w fo abitay dimensions [7] while in one dimension it goes to zeo fo a non-constant mass in case α =γ = 0 [8]. Futhemoe, fom Eqs. (7) and (9), the modification tem fo the coesponding wave function is ( N L1) ( αγ ) G( ) = exp dz =. (13) As Eq. (6) is analytically solvable having a closed expession fo W () given by (11) epoducing explicit expessions fo ε and F, the coesponding total enegy and wave function can easily be calculated though E =ε E and Ψ = FG fo the system of inteest with a location dependent mass. At this stage it is also noted that the fomalism suggested hee seems supeio to the usual teatment in supesymmetic quantum theoy that in pinciple stat with the gound state and builds up excited state wave functions by the use of some ± linea opeatos ( A ) wheeas thee is no such estiction in the pesent theoy poviding flexible investigations. 4. DISCUSSION Although the pocedue used in the fomalism seems easonable, the use of othe woks as in the pevious section fo an appopiate W () tem to solve Eq. (6) may be seen as a dawback of the model. To emove this seeming deficiency, we popose hee a unified teatment within the model consideing the ecent wok in [9].

5 any of the special functions H(g) of mathematics epesent solutions to diffeential equations of the fom d H ( g) dh ( g) Q( g) R( g) H ( g) = 0, (14) dg dg whee the functions Q(g) and R(g) ae well defined fo any paticula function [10]. Since in this Lette we ae inteested in bound state wave functions, we should estict ouselves to polynomial solutions of Eq. (14). Beaing in mind Eq. (14), the substitution of Φ( z ) = H g( z) f z) in Eq. (1) leads to the second-ode diffeential equation [ ] ( 1 f H g g H H g f f H g = V eff λ, f H H Hf f H (15) in which pimes denote deivatives with espect to g and z fo the functions H (g), g(z) and f (z) espectively. With the confidince gained by the similaity between Eqs. (15) and (4), one can safely use the pesent teatment splitting Eq. (15) in two pieces W f W = V0 ε, W =, f (16) and H g W W = V E, = H, (17) which ae simila to Eqs. (6) and (7), whee λ =ε E and V V V. Afteall, it can be clealy seen that Eq. (16) is the one equied fo obtaining an explicit expession fo W tem used in Eq. (6) coesponding to an exactly solvable system consideed in one-dimension ( α =γ = 0). Howeve, to poceed futhe, the functions f and g should be solved as H, Q and R ae known in pinciple. Now, equating like tems between the esulting expession in (15) and (14) gives 1 g f 1 f f Q[ g ] =, R[ g ] = ( E V ), (18) g g f g f f whee, fom the definition of Q, ( z ) 1 g 1 f exp ( ) Q g dg. (19) g Consideation of Eqs. (15) though (18) suggests a novel pesciption g V E = R[ g ], (0) which, fo plausible and R functions, povides a eliable expession fo g(z). It is emaked that in the constant mass case 1 this pocedue educes to the well known fomalis which has been thooughly investigated [11] that, togethe with [9], justify ou new poposal in solving PDSE. The moe detailed investigation of this teatment will be discussed elsewhee. 5. CONCLUDING REARKS In this Lette, a geneal method has been pesented to addess the question of coections to the solution in one-dimension fo a lage class of N dimensional and exactly solvable PDSE. We have also descibed how to extend the method to the case whee the necessay eff = 0

6 function W () in (6) geneating algebaically solvable potentials in one dimension ae pesent, which initiates calculations in the model leading to explicit expessions fo the modifications due to both the use of physically plausible Zho-Koeme effective Hamiltonian ( α =γ = 1 ) and highe dimesional teatments. The main esults ae consistent with the othe elated woks in the liteatue, which allow a non-petubative teatment of these issues. Although, fo claity we have illustated an application of the method fo an easily accessible case of inteest, it can be eadily employed in vaious typical situations. In view of the impotance in calculating such coections in physics, we believe that the pesent model would seve as a useful toolbox to teat even moe ealistic situations which now occu in expeimental obsevations with the advent of the quantum technology. REFERENCES [1] Öze O and Gönül B 003 od. Phys. Lett. A ; Gönül B 004 Chin. Phys. Lett ; Gönül B and Koçak 005 od. Phys. Lett. A 0 355; Gönül B, Çelik N and Olğa E 005 od. Phys. Lett. A ; Gönül B and Koçak 005 od. Phys. Lett. A ; Gönül B, Koay K and Bakõ E; pepint, quant-ph/ [] Gönül B 004 Chin. Phys. Lett [3] Gönül B and Koçak 005 Chin. Phys. Lett. 74 [4] Quesne C 005 to appea in Ann. Phys. (N.Y.), quant-ph/ v3; Yu J, Dong S and Sun G 004 Phys. Lett. A 3 90; Chen G 004 Phys. Lett. A 39 ; Chen G and Chen D 004 Phys. Lett. A ; Chen Z and Chen G 005 Physica Scipta [5] Gönül B, Gönül B, Tutcu D and Öze O 00 od. Phys. Lett. A ; Gönül B, Öze O, Gönül B and Üzgün F 00 od. Phys. Lett. A [6] BenDaniel D J and Duke C B 1966 Phys. Rev ; Bastad G 1981 Phys. Rev. B ; Zhu Q G and Koeme H 1983 Phys. Rev. B ; Li T and Kuhn K J 1993 Phys. Rev [7] Gönül B, Öze O, Koçak, Tutcu D and Cançelik Y 001 J. Phys. A: ath. Gen [8] Plastino A R, Rigo A, Casas, Gacias F and Plastino A 1999 Phys. Rev. C [9] Bagchi B, Goain P, Quesne C and Roychoudhuy R 005 Euophys. Lett [10] Abamowitz and Stegun I A 1970 Handbook of athematical Functions (New Yok: Dove) [11] Bhattachajie A and Sudashan E C G 196 Nuovo Cimento 5 864; Natanzon G A 1979 Theo. ath. Phys ; Levai G 1989 J. Phys. A: ath. Gen. 689; Levai G 1991 J. Phys. A: ath. Gen ; Znojil and Levai G 001 J. Phys. A: ath. Gen ; Bagchi B and Ganguly A 003 J. Phys. A: ath. Gen. 36 L161

A Relativistic Electron in a Coulomb Potential

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