An improved approximation scheme for the centrifugal term and the Hulthen potential
aa r X i v : . [ qu a n t - ph ] J a n An improved approximation scheme for the centrifugal term andthe Hulth´en potential
Sameer M. Ikhdair ∗ Department of Physics, Near East University, Nicosia, North Cyprus, Turkey (Dated: December 16, 2018)
Abstract
We present a new approximation scheme for the centrifugal term to solve the Schr¨odinger equa-tion with the Hulth´en potential for any arbitrary l state by means of a mathematical Nikiforov-Uvarov (NU) method. We obtain the bound state energy eigenvalues and the normalized corre-sponding eigenfunctions expressed in terms of the Jacobi polynomials or hypergeometric functionsfor a particle exposed to this potential field. Our numerical results of the energy eigenvalues arefound to be in high agreement with those results obtained by using the program based on a nu-merical integration procedure. The s -wave ( l = 0) analytic solution for the binding energies andeigenfunctions of a particle are also calculated. The physical meaning of the approximate analyticalsolution is discussed. The present approximation scheme is systematic and accurate.Keywords: Bound states, Hulth´en potential, NU method, approximation schemes. PACS numbers: 03.65.Ge, 12.39.Jh ∗ E-mail: [email protected] . INTRODUCTION It is necessary to obtain the exact bound state energy spectrum of the Schr¨odinger equa-tions for some physical potential models. Therefore, much works have been done to solve thewave equation for various radial and angular potentials. Unfortunately, the exact analyticsolutions (EAS) of idealized quantum systems (QS), under consideration, are possible onlyin the s -wave case with angular quantum number l = 0 for some exponential-type poten-tial models. On the other hand, the Schr¨odinger equation cannot be solved analyticallyfor l = 0 because of the centrifugal term potential l ( l +1) r . Over the past years, some au-thors [1-16] have used the approximation l ( l +1) r ≈ l ( l +1) δ e δr ( e δr − ) for the centrifugal term potentialproposed by Greene and Aldrich [1] to obtain the l = 0 analytic bound-states [2,4,5] andscattering states [7] solutions of the non-relativistic [2,5] and relativistic [6] wave equationswith some exponential-type potentials such as Hulth´en potential [2-7], Eckart potential [10-13], Manning-Rosen potential [14-16] and diatomic molecular hyperbolical potential [17].However, this approximation is valid only for small values of the screening parameter δ andit breaks down for large values of δ [5]. Therefore, there have been broad interest and im-pressive efforts in order to find a new approximation scheme which deals with the centrifugalterm potential.The Hulth´en potential [2,5,18] is the special case of the multiparameter exponential-typepotential model [19,20]. It takes the form V ( r ) = − V e δr − , V = Ze δ, (1)where V is a constant and δ is the screening parameter that determines the range of thepotential. If the potential is used for atoms, then V = Zδ (in units ¯ h = c = e = 1), where Z is identified as the atomic number. The Hulth´en potential behaves like the Coulombpotential near the origin (i.e., r → V C ( r ) = − Ze /r , but decreases exponentially in theasymptotic region when r ≫ , so its capacity for bound states is smaller than the Coulombpotential [6,21-24]. This potential has been applied to a number of areas such as nuclearand particle physics [25-27], atomic physics [28-31], molecular physics [32,33] and chemicalphysics [34,35], etc.The bound-state EAS of the Schr¨odinger equation with the Hulth´en potential can besolved in a closed form for s -waves (states with zero orbital angular momentum l ) [36].2owever, for the case l = 0 , this quantum system cannot be exactly solved. For imple-menting approximate schemes economically and profitably; while dealing with practicalquantum mechanical problems, EAS of the Hulth´en potential is desirable although non-perturbative and numerical solutions of different potentials may lead to new physical ideasand/or calculational techniques in quantum physics. For instance, the numerical integrationof the Schr¨odinger equation [37] is used to obtain the energy eigenvalues numerically for theHulth´en potential case, this provides a probe and/or test for the exactness of any analyticsolution. One-parameter variational calculations are carried out in such numerical integra-tion methods. The variational results are practically identical to the exact energies, exceptin the high-screening region. These variational calculations turn to become sophisticated inthe solution of Schr¨odinger equation with multi-parameter potentials. However, no ”exact”values obtained from a numerical integration of the Schr¨odinger equation have been availableto assess the accuracy of the various methods [37]. Hence, it is important to note that theanalytic solution of any quantum potential model, even if it is an approximated solution,is indispensable since the obtained expressions for energy eigenvalues and eigen functionscontain all the necessary information regarding the quantum system under consideration.In the non-relativistic case, for l = 0 , several techniques have been used to obtain approx-imate analytic solutions, some authors have obtained the bound-state energy eigenvalues byusing the numerical integration approach [37,38], quasi-analytical variational [37,39], pertur-bation [40], SUSYQM [3], shifted 1 /N expansion [41], AIM [5] and Nikiforov-Uvarov (NU)[2] methods. The results obtained by some of these methods [3,5] are in good agreementwith the numerical integration approach [37] for low-screening region (small values of thescreening parameter δ ) but the agreement becomes poor in the high-screening regime [5].Recently, Haouat and Chetouani [42] have solved the Klein-Gordon and Dirac equationsin the presence of the Hulth´en potential, where the energy spectrum and the scatteringwavefunctions are obtained for spin-0 and spin- particles, making a slight modificationto the usual approximation scheme, r ≈ α e − γαr (1 − e − αr ) where γ is a dimensionless parameter( γ = 0 , α and the dimensionless parameter γ are taken as α = 0 . γ =1, respectively, which is simply the case of the normal approximation usedin the literature. Also, Jia and collaborators [43] have recently proposed a new alternativeapproximation scheme, r ≈ α (cid:16) ωe αr − + e αr − (cid:17) where ω is a dimensionless parameter3 ω = 1 . , for the centrifugal potential to improve the numerical energy eigenvalues of theHulth´en potential. When taking ω = 1 , their approximation can be reduced to the usualapproximation [1-16]. However, the accuracy of their numerical results [43] is still in pooragreement with the other numerical integration and variational methods [37] especially inhigh-screening δ regime. This problem could be solved by making a better approximationfor the centrifugal term potential. In this work, for any arbitrary l -state, we aim to ob-tain approximate energy eigenvalues and corresponding normalized wave functions for theHulth´en potential in high agreement with the numerical method [37]. Hence, we present analternative effective approximation that gives highly accurate numerical energy eigenvaluesof the Hulth´en potential as a function of screening parameter for all states with Z = 1 . This paper is organized as follows: In the next Section, the NU method is briefly intro-duced. In Section 3, the l -states Schr¨odinger equation for the Hulth´en potential is solvedwithin the new effective approximation scheme and using the NU method. The calculatedenergy eigenvalues and wave functions are compared with the other ones found by usingdifferent analytical and numerical methods. The normalized wave functions are obtained inSection 4. Finally, the relevant conclusions are given in Section 5. II. NU METHOD
The NU method is briefly introduced here and the details can be found in Nikiforov-Uvarov handbook [44]. This method was proposed to solve the second-order differentialwave equation of the hypergeometric-type: ψ ′′ n ( s ) + e τ ( s ) σ ( s ) ψ ′ n ( s ) + e σ ( s ) σ ( s ) ψ n ( s ) = 0 , (2)where σ ( s ) and e σ ( s ) are polynomials, at most of second-degree, and e τ ( s ) is a first-degreepolynomial. The prime sign denotes the differentiation with respect to s. To find a particularsolution of Eq. (2), one can decompose the wavefunction ψ n ( z ) as follows: ψ n ( s ) = φ n ( s ) y n ( s ) , (3)leading to a hypergeometric type equation σ ( s ) y ′′ n ( s ) + τ ( s ) y ′ n ( s ) + λy n ( s ) = 0 . (4)4he first part of the wavefunctions ψ n ( s ) is the solution of the differential equation, σ ( s ) φ ′ ( s ) − π ( s ) φ ( s ) = 0 , (5)where τ ( s ) = e τ ( s ) + 2 π ( s ) , (6)and λ in (4) is a parameter defined as λ = λ n = − nτ ′ ( s ) − n ( n − σ ′′ ( s ) , n = 0 , , , · · · . (7)The τ ( s ) is a polynomial function of the parameter s whose first derivative τ ′ ( s ) must benegative which is the essential condition in choosing the proper solutions. The second partof the wavefunctions (3) is a hypergeometric-type function obtained by Rodrigues relation: y n ( s ) = B n ρ ( s ) d n ds n [ σ n ( s ) ρ ( s )] , (8)where B n is a constant related to normalization and the weight function ρ ( s ) can be foundby [44] σ ( s ) ρ ′ ( s ) + [ σ ′ ( s ) − τ ( s )] ρ ( s ) = 0 , (9)The function π ( s ) and the parameter λ are defined as π ( s ) = σ ′ ( s ) − e τ ( s )2 ± s(cid:18) σ ′ ( s ) − e τ ( s )2 (cid:19) − e σ ( s ) + kσ ( s ) , (10) λ = k + π ′ ( s ) , (11)where π ( s ) has to be a polynomial of degree at most one. The discriminant under the squareroot sign in Eq. (10) must be set to zero and then has to be solved for k [44]. Finally, theenergy eigenvalue equation is simply found by solving Eqs. (7) and (11). III. BOUND STATE SOLUTIONS
The Schr¨odinger equation for a central molecular potential V ( r ) can be written as (cid:18) ¯ h µ ∇ + E nl − V ( r ) (cid:19) ψ nlm ( r ,θ, φ ) = 0 , (12)where the representation of the Laplacian operator ∇ , in spherical coordinates, is ∇ = ∂ ∂r + 2 r ∂∂r + 1 r (cid:18) θ ∂∂θ sin θ ∂∂θ + 1sin θ ∂ ∂φ (cid:19) . (13)5ere the wave functions ψ nlm ( r ,θ, φ ) belong to the energy eigenvalues E nl and V ( r ) standsfor the molecular potential in the configuration space and r represents the three-dimensionalintermolecular distance (cid:0)P i =1 x i (cid:1) / . Let us decompose the wave function in (12) as follows: ψ nlm ( r ,θ, φ ) = r − u nl ( r ) Y lm ( θ, φ ) , (14)where Y lm ( θ, φ ) represents contribution from the hyperspherical harmonics that arise inhigher dimensions. Substituting the wave functions (14) into Eq. (12), the result is [45,46] (cid:26) d dr − l ( l + 1) r + 2 µ ¯ h [ E nl − V ( r )] (cid:27) u n,l ( r ) = 0 , (15)where E nl is the bound-state energy of the system under consideration, i.e., E nl < l ( l +1) r is known as the centrifugal term . We also should be careful about the behavior ofthe wave function u nl ( r ) near r = 0 and r → ∞ . Furthermore, u nl ( r ) should be normalizable[47].We can rewrite Eq. (15) for the Hulth´en potential as d u nl ( r ) dr + (cid:20) µE nl ¯ h + 2 µZe δ ¯ h e − δr − e − δr − l ( l + 1) r (cid:21) u nl ( r ) = 0 , (16)where E nl is the bound state energy of the system and n and l signify the radial and angularquantum numbers, respectively. When l = 0 ( s -wave), Eq. (16) with the Hulth´en potentialcan be exactly solved [36,48-50], but for the case l = 0 , Eq. (16) cannot be exactly solved. Sowe must find a new approximation to the entrifugal term to solve the equation analytically.The new proposed approximation is based on the expansion of the centrifugal term in aseries of exponentials depending on the intermolecular distance r and keeping terms up tosecond order. For small 0 . ≤ δr ≤ . δ ) , Eq. (16)is very well approximated to centrifugal term. However, for large screening parameter, abetter approximation to the centrifugal term should be made. Hence, instead of employingthe usual approximation given in [1-16], we propose an alternative approximation schemecasted in the form: 1 r ≈ δ (cid:2) d + v ( r ) + v ( r ) (cid:3) , v ( r ) = e − δr − e − δr , r ≈ δ (cid:20) d + 1 e δr − e δr − (cid:21) , (17)6or the centrifugal term which takes a similar ans¨atze like the Hulth´en potential. Underthe coordinate transformation r → x, it is convenient to shift the origin by defining x =( r − r ) /r , we obtain(1 + x ) − = γ " d + 1 e γ (1+ x ) − e γ (1+ x ) − , γ = r δ. (18)Further, expanding Eq. (17) around r = r ( x = 0) , we obtain the following expansion:1 − x + O ( x ) = γ (cid:18) d + 1 e γ − e γ − (cid:19) − γ (cid:18) e γ − e γ − + 2( e γ − (cid:19) x + O ( x ) , (19)from which we have γ (cid:20) d + 1 e γ − e γ − (cid:21) = 1 ,γ (cid:18) e γ − e γ − + 2( e γ − (cid:19) = 2 . (20)Therefore, the shifting parameter d is to be found from the solution of the above twoequations as d = 1 γ − e γ − − e γ − = 0 . , (21)where e is the base of the natural logarithms, e = 2 . γ = 0 . . Therefore, we may cast the centrifugal term aslim δ → δ " γ − e γ − − e γ − + e − δr − e − δr + (cid:18) e − δr − e − δr (cid:19) = 1 r . (22)To conclude, it is important to note that when d = 0 , the approximation expression (17) isreduced to the usual approximation used in literature [1-16]. The variation of the centrifugalterm potential l ( l + 1) /r and the proposed approximation expression given in (17) versus δr are plotted in Figure 1. Obviously, the approximate centrifugal term potential (17) and l ( l + 1) /r are similar and coincide in both high-screening as well as in the low-screeningregimes as shown in Figure 1.Inserting the approximation expression (17) into Eq. (16) and changing the variables r → s = e − δr through the mapping function s = f ( r ) , where r ∈ [0 , ∞ ) or s ∈ [1 , , leadsus to obtain the following equation 7 nl ′′ ( s ) + (1 − s ) s (1 − s ) u ′ nl ( s ) + 1[ s (1 − s )] (cid:2) − ε nl + ( c − c + 2 ε nl ) s − ( c + ε nl ) s (cid:3) u nl ( s ) = 0 , (23)where ε nl = r ∆ E l − µE nl ¯ h δ , ∆ E l = l ( l + 1) d , c = 2 µZe ¯ h δ , c = l ( l + 1) . (24)In the present work, we will deal with bound state solutions, i.e., the radial part of thewavefunction ψ nlm ( r ,θ, φ ) must satisfy the boundary condition that u nl ( r ) /r becomes zerowhen r → ∞ , and u nl ( r ) /r is finite at r = 0 . In addition, we require E nl ≤ ¯ h δ µ ∆ E l , i.e., ε nl ≥ e τ ( s ) = 1 − s, σ ( s ) = s (1 − s ) , e σ ( r ) = − ε nl + ( c − c + 2 ε nl ) s − ( c + ε nl ) s . (25)Inserting the polynomials given by Eq. (25) into Eq. (10) gives the polynomial: π ( s ) = − s ± qe as + e bs + e c, (26)where e a = 1 + 4( c + ε nl − k ) , e b = 4( k − c + c − ε nl ) and e c = 4 ε nl . The equation of quadraticform under the square root sign of Eq. (26) must be solved by setting the discriminant ofthis quadratic equal to zero: ∆ = e b − e a e c = 0 . This discriminant gives a new quadraticequation can be solved for the constant k to obtain the two roots: k , = c − c ± ε nl √ c . (27)When the two values of k given in Eq. (27) are substituted into Eq. (26), the four possibleforms of π ( s ) are obtained as π ( s ) = − s ± (cid:2)(cid:0) ε nl − √ c (cid:1) s − ε nl (cid:3) for k = c − c + ε nl √ c , (cid:2)(cid:0) ε nl + √ c (cid:1) s − ε nl (cid:3) for k = c − c − ε nl √ c . (28)One of the four values of the polynomial π ( s ) is just proper to obtain the bound state energystates because τ ( s ) given by Eq. (6) has a negative derivative for this value of π ( s ) [44].Therefore, the most suitable expression of π ( s ) is chosen as π ( s ) = − s − (cid:20)(cid:18) ε nl + 12 √ c (cid:19) s − ε nl (cid:21) , (29)for k = c − c − ε nl √ c . Hence, τ ( s ) and τ ′ ( s ) are obtained τ ( s ) = 1 + 2 ε nl − (cid:20) ε nl + 12 √ c (cid:21) s, ′ ( s ) = − (cid:20) ε nl + 12 √ c (cid:21) , (30)where τ ′ ( s ) represents the derivative of τ ( s ) . Using Eqs. (25), (29) and (30), the followingexpressions for λ and λ n are obtained, respectively, λ = λ n = n + (cid:2) ε nl + √ c (cid:3) n, ( n = 0 , , , · · · ) , (31) λ = c − c −
12 (1 + 2 ε nl ) (cid:2) √ c (cid:3) , (32)where n is the number of nodes of the radial wave function u n,l ( r ). When λ = λ n , anexpression for ε nl is obtained as ε nl = c n + l + 1) − ( n + l + 1)2 , ( n, l = 0 , , , · · · ) . (33)Also, with the aid of Eq. (24), the previous energy equation gives the following bound stateenergy eigenvalue equation: E nl = ¯ h δ µ ( l ( l + 1) d − (cid:20) µZe ¯ h δ ( n + l + 1) − ( n + l + 1)2 (cid:21) ) . (34)In the case of the s -wave ( l = 0), the previous equation turns to be E n = − ¯ h δ µ (cid:18) µZe ¯ h δ ( n + 1) − n + 12 (cid:19) , (35)which is identical to the ones obtained before using the factorization method [36], SUSYQMapproach [3,28,54], quasi-linearization method [55] and NU method [2,6,32]. Further, if wetake the shift parameter d = 0 in the present approximation, Eq. (34) reduces to E nl = − ¯ h δ µ (cid:20) µZe ¯ h δ ( n + l + 1) − ( n + l + 1)2 (cid:21) , (36)which is also identical with the energy eigenvalues formula given in Eq. (32) of Ref. [5], Eq.(24) of Ref. [43] and Eq. (28) of Ref. [2].Let us turn to the calculations of the wave function y n ( s ) , which is the first part solutionof hypergeometric-type equation, we need to multiply Eq. (4) by the weight function ρ ( s )so that it can be rewritten in self-adjoint form [44][ ω ( s ) y ′ n ( s )] ′ + λρ ( s ) y n ( s ) = 0 . (37)9he weight function ρ ( s ) that satisfies Eqs. (9) and (37) is found as ρ ( s ) = s ε nl (1 − s ) (2 l +1) , (38)which gives the Rodrigues relation (8): y nl ( s ) = B nl s − ε nl (1 − s ) − (2 l +1) d n ds n (cid:2) s n +2 ε nl (1 − s ) n +2 l +1 (cid:3) = B nl P (2 ε nl , l +1) n (1 − s ) . (39)Further, inserting the values of σ ( s ) and π ( s ) given in Eqs. (25) and (29) into Eq. (5), onecan find the other part of the wave function as φ ( s ) = s ε nl (1 − s ) ( l +1) . (40)Hence, the wave functions in Eq. (3) become u nl ( s ) = N nl s ε nl (1 − s ) l +1 P (2 ε nl , l +1) n (1 − s ) , s ∈ [1 ,
0) (41)where N nl is the normalization constant to be determined in the next section. Finally, theunnormalized radial wave functions are obtained as ψ nlm ( r ,θ, φ ) = N nl r − (cid:0) e − δr (cid:1) ε nl (1 − e − δr ) l +12 F ( − n, n +2 ( ε nl + l + 1) ; 2 ε nl +1; e − δr ) Y lm ( θ, φ ) . (42)Thus, the Jacobi polynomials can be expressed in terms of the hypergeometric functions[56]: P ( a,b ) n (1 − x ) = F ( − n, n + a + b + 1; a + 1; x ) , (43)where F ( a, b ; c ; x ) = Γ( c )Γ( a )Γ( b ) ∞ X k =0 Γ( a + k )Γ( b + k )Γ( c + k ) x k k ! . The hypergeometric function F ( a, b ; c ; x )is a special case of the generalized hypergeometric function [56] p F q ( α , α , · · · , α p ; β , β , · · · , β q ; x ) = ∞ X k =0 ( α ) k ( α ) k · · · ( α p )( β ) k ( β ) k · · · (cid:0) β q (cid:1) x k k ! , (44)where the Pochhammer symbol is defined by ( y ) k = Γ( y + k ) / Γ( y ) . In the case l = 0 , the above wave functions become ψ n ( r ) = D n r − (cid:0) e − δr (cid:1) ε n (1 − e − δr ) F ( − n, n + 2 ( ε n + 1) ; 2 ε n + 1; e − δr ) , (45)with ε n = µZe ¯ h δ ( n +1) − n +12 and D n is another normalization factor . This result is consistentwith the NU method [2]. Further, if we take the shift parameter d = 0 in the present10pproximation, Eq. (42) reduces to the form ψ nlm ( r ,θ, φ ) = D nl r − (cid:0) e − δr (cid:1) ε nl (1 − e − δr ) l +12 F ( − n, n +2 ( ε nl + l + 1) ; 2 ε nl +1; e − δr ) Y lm ( θ, φ ) , (46)with ε nl = µZe ¯ h δ ( n + l +1) − n + l +12 and D nl is a normalization factor . The critical screen-ing δ c = µZe ¯ h ( n + l +1) at which E nl = 0 has wave functions: ψ nlm ( r ,θ, φ ) = D nl r − (1 − e − δ c r ) l +1 P (0 , l +1) n (1 − e − δ c r ) Y lm ( θ, φ ) . In order to show the accuracy of our analytical results, we present the numerical data insupport of the results obtained in Eqs. (34) and (42) which are the main analytic resultsobtained in this work. Therefore, we calculate the energy eigenvalues for Z = 1 , n and l ar-bitrary quantum numbers as a function of the screening parameters δ. The results calculatedin Tables 1 and 2 by using Eq. (34) are compared with those obtained with the help of thenumerical integration [37], asymptotic iteration [5], variational [37], SUSY [3] and the re-cently proposed approximation [43] methods. Tables 1 and 2 show that our results obtainedwith the new approximation scheme with the NU method are in high agreement with thoseobtained by numerical integration method [37] for short potential range (small screeningparameter δ ). However, the slight differences in the energy eigenvalues from the numericalintegration method [37] are observed for long potential range (large screening parameter δ ). Therefore, our approximated numerical results are closer to the numerical integrationresults than the results obtained via AIM [5] using Eq. (32) and also the recently proposedapproximation scheme [43] using Eq. (19) for small and large screening parameter δ values.Thus, the present approximation form (22) to the centrifugal term can highly improve theaccuracy of calculating the energy eigenvalues for the Hulth´en potential than the recentlyproposed approximation (5) given in [43]. IV. NORMALIZATION OF THE RADIAL WAVE FUNCTION
Using s ( r ) = e − δr and Eq. (41), we are able to express normalization condition R ∞ u ( r ) dr = 1 as N nl δ Z s ε nl − (1 − s ) l +2 (cid:2) P (2 ε nl , l +1) n (1 − s ) (cid:3) ds = 1 . (47)Unfortunately, there is no formula available to calculate this key integration. Neveretheless,we can find the explicit normalization constant N nl . For this purpose, it is not difficult to11btain the results of the above integral by using the following formulas [56-59], P ( α,β ) n ( x ) = ( n + α )! ( n + β )! n X p =0 p !( n + α − p )! ( β + p )! ( n + p )! (cid:18) x − (cid:19) n − p (cid:18) x + 12 (cid:19) p , (48)and B ( x, y ) = Z t x − (1 − t ) y − dt = Γ( x )Γ( y )Γ( x + y ) , Re( x ) , Re( y ) > . (49)Thus, the normalization constant N nl is now obtained as N nl = 1( n + 2 l + 1)!Γ(2 ε nl + n + 1) vuuut δ Γ(2 ε nl + 2 n + 2 l + 4)Γ(2 ε nl + 2 n + 1) n P p,q =0 ( f p f q f p,q ) − , where f p = ( − p p !Γ(2 ε nl + n − p + 1)(2 l + p + 1)! ( n + p )! ,f q = ( − q q !Γ(2 ε nl + n − q + 1) (2 l + q + 1)! ( n + q )! ,f p,q = (2 l + p + q + 2)! . (50) V. CONCLUSIONS
In this work, we have proposed an alternative improved approximation scheme for thecentrifugal term and used this approximation scheme together with the NU method to solvethe Schr¨odinger equation with any orbital angular momentum number l for the Hulth´enpotential. The bound state energy eigenvalues and the unnormalized radial wavefunctionshave been calculated in analytical and numerical way. The analytic expressions for the energyeigenvalues and wavefunctions have been reduced to the s -wave case and the d = 0 case(usual approximation) [1-16]. Our numerical results obtained by the approximation schemegiven in expression (22) for the centrifugal term has been found to be more effective than thenumerical results of the recently proposed approximation (5) of Ref. [43] and the commonlyused approximation in generating the energy spectrum of the Hulth´en potential. Our resultsin Tables 1 and 2 for small screening δ values show that the present approximation is in highagreement with the numerical integration and variational methods [37] whereas it is in quitegood agreement for large screening δ values. The present approximation method is simple,practical and powerful than the other known methods [2,5,43]. This new method can be used12or many quantum models to improve the accuracy of energy eigenvalues for few potentialmodels of the exponential-type like the hyperbolical and Manning-Rosen potentials (cf. e.g.,Refs. [60,61].) Acknowledgments
The author thanks the kind referees for their useful suggestions. This work was partiallysupported by the Scientific and Technological Research Council (T ¨UB˙ITAK) of Turkey.13
1] R.L. Greene and C. Aldrich, Phys. Rev. A 14 (1976) 2363.[2] S.M. Ikhdair and R. Sever, J. Math. Chem. 42 (2007) 461.[3] B. G¨on¨ul, O. ¨Ozer, Y. Can¸celik and M. Kocak, Phys. Lett. A 275 (2000) 238; B. G¨on¨ul, Chin.Phys. Lett. 21 (2004) 1685.[4] M. Akta¸s and R. Sever, J. Mol. Struct. 710 (2004) 219.[5] O. Bayrak, G. Kocak and I. Boztosun, J. Phys. A: Math. Gen. 39 (2006) 11521.[6] S.M. Ikhdair, arXiv:0810.1590, to appear in Int. J. Mod. Phys. C 20 (1) (2009).[7] C.-Y. Chen, F.-L. Lu and D.-S. Sun, Cent. Eur. J. Phys. 6 (2008) 884.[8] U. Myhrman, J. Phys. A: Math. Gen. 16 (1983) 263.[9] A. Bechlert and W. B¨uhring, J. Phys. B: At. Mol. Opt. Phys. 21 (1988) 817.[10] S.H. Dong, W.C. Qiang, G.H. Sun and V.B. Bezerra, J. Phys. A: Math. Teeor. 40 (2007)10535.[11] G.F. Wei, C.Y. Long, X.Y. Duan and S.H. Dong, Phys. Scr. 77 (2008) 035001.[12] C.Y. Chen, D.S. Sun and F.L. Lu, J. Phys. A: Math. Theor. 41 (2008) 035302.[13] L.H. Zhang, X.P. Li and C.S. Jia, Phys. Lett. A 372 (2008) 2201.[14] W.C. Qiang and S.H. Dong, Phys. Lett. A 368 (2007) 13.[15] G.F. Wei, C.Y. Long and S.H. Dong, Phys. Lett. A 372 (2008) 2592.[16] S.M. Ikhdair and R. Sever, Ann. phys. (Berlin) 17 (11) (2008) 897. .[17] S.S. Dong, J. Garcia-Ravelo and S.H. Dong, Phys. Scr. 76 (2007) 393.[18] L. Hulth´en, Ark. Mat. Astron. Fys. A 28 (1942) 5.[19] C.-S. Jia, X.-L. Zeng and L.-T. Sun, Phys. Lett. A 294 (2002) 185.[20] C.-S. Jia, Y. Li, Y. Sun, J.-Y. Liu and L.-T. Sun, Phys. Lett. A 311 (2003) 115.[21] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. 46 (2007) 1643.[22] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (2007) 218.[23] S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. E 17 (6) (2008) 1107.[24] M. S¸im¸sek and H. E˘grifes, J. Phys. A: Math. Gen. 37 (2004) 4379.[25] C.S. Lam and Y.P. Varshni, Phys. Rev. A 4 (1971) 1874.[26] B. Durand and L. Durand, Phys. Rev. D 23 (1981) 1092.[27] R.L. Hall, Phys. Rev. A 32 (1985) 14.
28] R. Dutt, K. Chowdhury and Y.P. Varshni, J. Phys. A: Math. Gen. 18 (1985) 1379; T. Xu,Z.Q. Cao, Y.C. Ou, Q.S. Shen and G.L. Zhu, Chin. Phys. 15 (2006) 1172.[29] T. Tietz, J. Chem. Phys. 35 (1961) 1917; K. Szalcwicz and H.J. Mokhorst, J. Chem. Phys. 75(1981) 5785.[30] G. Malli, Chem. Phys. Lett. 26 (1981) 578.[31] J. Lindhard and P.G. Hansen, Phys. Rev. Lett. 57 (1986) 965.[32] I.S. Bitensky, V.K. Ferleger and I.A. Wojciechowski, Nucl. Instrum. Meth. B 125 (1997) 201.[33] C.-S. Jia, J.Y. Wang, S. He and L.-T.Sun, J. Phys. A: Math. Gen. 33 (2000) 6993.[34] P. Pyykko and J. Jokisaari, Chem. Phys. 10 (1975) 293.[35] J.A. Olson and D.A. Micha, J. Chem. Phys. 68 (1978) 4352.[36] S. Fl¨ugge, Practical Quantum Mechanics, Springer, Berlin, 1974.[37] Y.P. Varshni, Phys. Rev. A 41 (1990) 4682.[38] M.A. Nunez, Phys. Rev. A 47 (1993) 3620.[39] S.H. Patil, J. Phys. A: Math. Gen. 34 (2001) 3153.[40] P. Matthys and H.D. Meyer, Phys. Rev. A 38 (1988) 1168.[41] A.Z. Tang and F.T. Chan, Phys. Rev. A 35 (1987) 911; B. Roy and R. Roychoudhury, J. Phys.A: Math. Gen. 20 (1987) 3051.[42] S. Haouat and L. Chetouani, Phys. Scr. 77 (2008) 025005.[43] C.-S. Jia, J.-Y. Liu and P.-Q. Wang, Phys. Lett. 372 (2008) 4779.[44] A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathematical Physics (Birkhauser,Bassel, 1988).[45] S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; Z. Phys. C 58 (1993) 153; Z. Phys. D 28(1993) 1; Hadronic J. 15 (1992) 389; Int. J. Mod. Phys. A 18 (2003) 4215; Int. J. Mod. Phys.A 19 (2004) 1771; Int. J. Mod. Phys. A 20 (2005) 4035; Int. J. Mod. Phys. A 20 (2005) 6509;Int. J. Mod. Phys. A 21 (2006) 2191; Int. J. Mod. Phys. A 21 (2006) 3989; Int. J. Mod. Phys.A 21 (2006) 6699; Int. J. Mod. Phys. E 17 (2008) 669.[46] S.M. Ikhdair, Chin. J. Phys. 46 (2008) 291; S.M. Ikhdair and R. Sever, Int. J. Mod. Phys.19 (2008) 1425; S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. C 19 (2008) 221; Int. J. Mod.Phys. C 18 (2007) 1571; Cent. Eur. J. Phys. 5 (2007) 516; Cent. Eur. J. Phys. 6 (2008) 141;Cent. Eur. J. Phys. 6 (2008) 685; Cent. Eur. J. Phys. 6 (2008) 697; S.M. Ikhdair and R. Sever,DOI:10.1007/s10910-008-9438-8 to appear in J. Math. Chem. (2009).
47] G.T. Einevoll, P.C. Hemmer and J. Thomson, Phys. Rev. B 42 (1990) 3485.[48] F. Dominguez-Adame, Phys. Lett. A 136 (1989) 175.[49] L. Chetouani, L. Guechi, A. Lecheheb, T.F. Hammann and A. Messouber, Physics A 234(1996) 529.[50] B. Talukdar, A. Yunus and M.R. Amin, Phys. Lett. A 141 (1989) 326.[51] H. E˘grifes and R. Sever, Int. J. Theor. Phys. 46 (2007) 935.[52] G. Chen, Z.D. Chen and Z.M. Lou, Phys. Lett. A 331 (2004) 374.[53] X.-C. Zhang, Q.-W. Liu, C.-S. Jia and L.-Z. Wang, Phys. Lett. A 340 (2005) 59.[54] E.D Filho and R.M. Ricotta, Mod. Phys. Lett. A 10 (1995) 1613.[55] V.B. Mandelzweig, Ann. Phys. 321 (2006) 2810.[56] I.S. Gradshteyn and I.M Ryzhik, Tables of Integrals, Series, and Products, 5th edn (NewYork, Academic, 1994).[57] G. Sezgo, Orthogonal Polynomials, (American Mathematical Society, New York, 1939).[58] W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theorems for the Special Functionof Mathematical Physics, 3rd Ed., (Berlin, Springer, 1966).[59] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York,1964); A.P. Prudrinkov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series, (New York,Gordon and Breach, 1986).[60] S.M. Ikhdair and R. Sever, arXiv:0809.2485 submitted to Ann. Phys. (Berlin) (2008).[61] S.M. Ikhdair and R. Sever, arXiv:0807.2085 submitted to Int. J. Mod. Phys. B (2008). IG. 1: A plot of the variation of the centrifugal term, 1 /r and its corresponding approximationexpression δ (cid:20) d + e δr ( e δr − ) (cid:21) versus δr, where the screening parameter δ changes from δ = 0 . δ = 0 .
250 in steps of 0 . . The parameters are in atomic units (¯ h = µ = e = 1) and Z = 1 . ABLE I: Bound energy spectra of the Hulth´en potential as a function of the screening parameter δ for 2 p, p and 3 d states for Z = 1 in atomic units (¯ h = µ = e = 1)State δ Present a Previous [43] b Numerical [37] AIM [5] Variational [37] SUSY [3]2 p .
025 0 . . . . . . .
050 0 . . . . . . .
075 0 . . . . . . .
100 0 . . . . . . .
150 0 . . . . . . .
200 0 . . . . . . .
250 0 . . . . . . .
300 0 . . . . . . .
350 0 . . . . . . p .
025 0 . . . . . . .
050 0 . . . . . . .
075 0 . . . . . . .
100 0 . . . . . . .
150 0 . . . . . . d .
025 0 . . . . . . .
050 0 . . . . . . .
075 0 . . . . . . .
100 0 . . . . . . .
150 0 . . . . . . a The present approximation. b Jia et al approximation. ABLE II: Bound energy spectra of the Hulth´en potential as a function of the screening parameter δ for 4 p, d, f, p, d, f, g, p, d, f and 6 g states for Z = 1 in atomic units (¯ h = µ = e = 1)State δ Present a Previous [43] b Numerical [37] AIM [5] Variational [37] SUSY [3]4 p .
025 0 . . . . . .
050 0 . . . . . .
075 0 . . . . . .
100 0 . . . . . d .
025 0 . . . . . .
050 0 . . . . . .
075 0 . . . . . f .
025 0 . . . . .
050 0 . . . . .
075 0 . . . . p .
025 0 . . . . .
050 0 . . . . d .
025 0 . . . . .
050 0 . . . . f .
025 0 . . . . .
050 0 . . . . g .
025 0 . . . . .
050 0 . . . . p .
025 0 . . . . d .
025 0 . . . . f .
025 0 . . . . g .