Solution of Effective-Mass Dirac Equation with Scalar-Vector and Pseudoscalar Terms for Generalized Hulthén Potential
aa r X i v : . [ qu a n t - ph ] D ec Solution of Effective-Mass Dirac Equationwith Scalar-Vector and Pseudoscalar Termsfor Generalized Hulth´en Potential
Altu˘g Arda a ∗ c † a Department of Physics Education, Hacettepe University , Abstract
We find the exact bound-state solutions and normalization constant for the Diracequation with scalar-vector-pseudoscalar interaction terms for the generalized Hulth´enpotential in the case where we have a particular mass function m ( x ). We also searchthe solutions for the constant mass where the obtained results correspond to theones when the Dirac equation has spin and pseudospin symmetry, respectively. Aftergiving the obtained results for the non-relativistic case, we search then the energyspectra and corresponding upper and lower components of Dirac spinor for the caseof P T -symmetric forms of the present potential.PACS: 03.65.-w, 03.65.Pm, 03.65.GeKeywords: Dirac equation, scalar-vector-pseudoscalar term, Hulth´en potential,
P T -symmetry, Nikiforov-Uvarov method ∗ Present adress: Department of Mathematical Science, City University London, UK † [email protected] Introduction
The Hulth´en potential [1] is one of the best known potentials in physics, as a short-rangepotential [2]. In the present work, we deal with the following form [3, 4] V ( x ) = V e − βx qe − βx − , (1.1)where the parameter V can be written as Z (2 β ) with the constant Z , and β is the screeningparameter (in atomic units) [2] with deformation parameter q . The constant Z is relatedwith the atomic number if one uses this potential in atomic physics. Basically, the Hulth´enpotential is a special form of the Eckart potential [2, 5].As a short-range potential, the Hulth´en potential has a great advantage because theSchr¨odinger equation can be solved exactly for this potential with ℓ = 0. With this advan-tage, the Hulth´en potential has been used in different areas of physics, such as in solid-statephysics [6], nuclear and particle physics [7], atomic physics [8], and chemical physics [9],and investigated with various techniques [2, 10-17].In this work, we search the bound state solutions of the generalized Hulth´en potentialwhich can be written also in a complex form identifying the P T -symmetric case in a closedform for the case where the mass depends on spatially coordinate, and extend the Diracequation including the scalar, vector and pseudoscalar interaction terms to this case. Afterthe works by von Roos [18], and Levy-Leblond [19], the solutions of relativistic and non-relativistic wave equations with a position-dependent mass have received great attention inliterature [3, references therein]. In Ref. [20], the bound state solutions of the Klein-Gordon(KG) and Dirac equations with the Hulth´en potential by using the approach proposed byBiedenharn have been worked where the scattering state solutions have been also presented.In Ref. [21], the analytical results for the bound states of the Dirac equation with thegeneralized Hulth´en potential as a tensor term have been studied within the concept of theSUSYQM. In the present work, we extend the search including the solutions of the Diracequation having a pseudoscalar interaction term, as in Refs. [33-36], for the q -parameterHulth´en potential within the position-dependent mass (PDM) formalism. This formalismgives an opportunity such as writing the analytical results for the case where the massis constant. This means that our results are also available for the cases where the Dirac2quation has pseudospin and spin symmetry. Our generic results will be given in belowmakes it possible to give the ”wave functions” with their normalization constants both forthe cases of PDM formalism and constant mass. We search here also the analytical resultsfor the P T -symmetric/non-Hermitian and
P T -symmetric/pseudo-Hermitian form of theHulth´en potential for both of upper and lower component which are presented again withinthe PDM formalism. These give us also the results for the case where the mass is constantif necessary. Among the above results, because of the q -parameter in potential, we applyour results for three different form of the potential as special cases.The organization of this work is as follows. In Section 2, we write the Dirac equationwith scalar ( V S ( x )), vector ( V V ( x )), and pseudoscalar ( V P ( x )) potentials in 1 + 1 dimensionfor the case where the mass is a function of spatially coordinate. In Section 3, we searchthe bound-state solutions for upper and lower component of the Dirac spinor separately,and give the normalization constant. We construct a relation between the mass functionand the potentials to reduce the Dirac equation to an analytically solvable form of secondorder differential equation. We give also the results for the case where the mass is constant,and observe that the results obtained for this case correspond to the solutions when thespin and pseudospin symmetry occur in Dirac equation. The spin symmetry appears whenthe difference of the scalar and vector potentials is constant, i.e., ∆( x ) = const. , and thepseudospin symmetry appears when the sum of the scalar and vector potentials is constant,i.e., Σ( x ) = const. [20-22]. Finally, we obtain the non-relativistic result for the bound-statesolution for the generalized Hulth´en potential. The present work can also be seen as anapplication of the parametric generalization of the Nikiforov-Uvarov method which will begiven in Appendix briefly [23, 24]. In Section 4, we write the generalized Hulth´en potentialin a complex form which corresponds to the P T -symmetric form of the potential, and findthe energy levels for upper and lower component of the Dirac spinor with normalized wavefunctions. The
P T -symmetric formulation with non-Hermitian Hamiltonians having realor complex spectra of quantum mechanics has received a great attention in literature afterthe work by Bender and Boettcher [25-27]. In Section 5, we collect briefly our analyticalresults for special values of the parameter q corresponding to the standard Hulth´en potential( q = 1), to the Woods-Saxon potential ( q = − q = 0)3hile the mass depends on spatially coordinate. We give our conclusions in last Section. Dimension
The time-independent Dirac equation for a spin-1 / x ) = V V ( x ) + V S ( x ), and ∆( x ) = V V ( x ) − V S ( x )is given by ( ~ = c = 1) [28-34] (cid:18) σ p + σ m ( x ) + 1 + σ x ) + 1 − σ x ) + σ V P ( x ) (cid:19) Ψ( x ) = E Ψ( x ) , (2.1)where σ , σ and σ are the Pauli spin matrices, and we write the mass as m ( x ). By takingthe Dirac spinor as Ψ = ( φ , φ ) t where t indicates the transpose, we obtain the followingfirst order coupled equations for upper and lower components − i dφ ( x ) dx − [ m ( x ) − ∆( x ) + E ] φ ( x ) + iV P φ ( x ) = 0 , (2.2) − i dφ ( x ) dx + [ m ( x ) + Σ( x ) − E ] φ ( x ) − iV P φ ( x ) = 0 , (2.3)Writing φ ( x ) in terms of φ ( x ) with the help of Eq. (2.2), and inserting it into Eq.(2.3) gives us (cid:18) d φ ( x ) dx − dV P ( x ) dx φ ( x ) − V P ( x ) dφ ( x ) dx (cid:19) [ m ( x ) − ∆( x ) + E ] − (cid:18) dm ( x ) dx − d ∆( x ) dx (cid:19) (cid:18) dφ ( x ) dx − V P ( x ) φ ( x ) (cid:19) + V P ( x ) (cid:18) dφ ( x ) dx − V P ( x ) φ ( x ) (cid:19) [ m ( x ) − ∆( x ) + E ] − [ m ( x ) − ∆( x ) + E ] [ m ( x ) + Σ( x ) − E ] φ ( x ) = 0 , We write here the equality dm ( x ) /dx = d ∆( x ) /dx between the mass function and thepotentials to reduce the above complicated equation to a simpler one which can be solvedanalytically. This relation gives also us the opportunity about finding the mass functionexplicitly, and the second order equation for upper component φ ( x ) as (cid:26) d dx − dV P ( x ) dx − V P ( x ) − [ m ( x ) + E − ∆( x )][ m ( x ) + Σ( x ) − E ] (cid:27) φ ( x ) = 0 , (2.4)4y following similar steps, and using the equality for the mass function as dm ( x ) /dx = − d Σ( x ) /dx , we obtain the second order equation for lower component φ ( x ) as (cid:26) d dx + dV P ( x ) dx − V P ( x ) − [ m ( x ) + E − ∆( x )][ m ( x ) + Σ( x ) − E ] (cid:27) φ ( x ) = 0 . (2.5)Eqs. (2.4) and (2.5) can be solved by using the parametric generalization of theNikiforov-Uvarov method which is given in Appendix briefly. In the next Section, wesolve the above equations for the scalar, vector and pseudoscalar potentials by identifyingthem in terms of the Hulth´en potential given in Eq. (1.1). First, we find the appropriatemass function by using the equalities, and then we write the bound-state solutions with thecorresponding normalized wave functions. We are now in a position to identify the potentials in terms of the generalized Hulth´enpotential. We tend to write them as following [3, 4] V V ( x ) = V e − βx qe − βx − ,V S ( x ) = − S e − βx qe − βx − ,V P ( x ) = V i e − βx qe − βx − , (3.1)where i = 1 ,
2, and V for the upper component φ ( x ), V for the lower component φ ( x ).We obtain Eq. (2.5) for φ ( x ) by replacement V ↔ − V in Eq. (2.4). In addition, we canhandle the explicit form of the wave function for φ ( x ) by doing β ↔ − β in φ ( x ).From Eq. (3.1), we haveΣ( x ) = ( V − S ) e − βx qe − βx − x ) = ( V + S ) e − βx qe − βx − , (3.2)By using Eqs. (3.1) and (3.2), we write the mass function from the equality obtainedfor φ ( x ) as m ( x ) = m + m e − βx qe − βx − , where the parameter m is basically the integralconstant, and we denote it as ’constant mass’, the other parameter m is obtained as m = V + S . This means that the mass parameter m contains the contributions coming5rom vector and scalar potentials. The equality obtained for φ ( x ) gives us the mass functionas m ( x ) = m + m e − βx qe − βx − with m = V − S including the contributions coming fromvector and scalar potentials. So, we can combine these two mass functions in a single formas m ( x ) = m + m i e − βx qe − βx − which will be used in computation below.By using a new variable as s = 1 / (1 − qe − βx ) ( −∞ < x < + ∞ → s m ( x ), we have the following representativeequation for both components (cid:26) d ds + 1 − ss (1 − s ) dds − s (1 − s ) (cid:20) A β + 2 B β s + C β s (cid:21)(cid:27) φ i ( s ) = 0 , (3.3)where A = m − E + 2 QV ( E + m ) + a i ,B = − QV ( E + m ) − b i ,C = 2 βQV i + Q V i + Q [( m i − S ) − V ] , (3.4)with a i = 2 Qm m i + Q V i + Q [( m i − S ) − V ] − Qm ( V + S ) ,b i = QV i ( β + QV i ) + Q [( m i − S ) − V ] − Qm ( S + V ) + Qm m i . (3.5)and Q = 1 /q . Eq. (3.3) can be solved by using the parametric Nikiforov-Uvarov method.For this aim, we compare Eq. (3.3) with Eq. (A.1) in Appendix, and with the help of Eq.(A.3) we obtain the parameter set α = 1 , α = 2 , α = 1 , ξ = C β , ξ = − B β , ξ = A β ,α = α = 0 , α = ξ , α = − ξ , α = ξ , α = ξ − ξ + ξ ,α = 1 + 2 p ξ , α = 2 + 2( p ξ − ξ + ξ + p ξ ) ,α = p ξ , α = − ( p ξ − ξ + ξ + p ξ ) , (3.6)With the help of Eq. (A.2) in Appendix, we write the energy spectrum of the Diracequation with scalar-vector-pseudosclar generalized Hulth´en potential within the position-dependent mass formalism as hp m − E + 2 QV ( E + m ) + a i + p m − E + β (2 n + 1) i − [( β + QV i ) + Q [( m i − S ) − V ]] = 0 , (3.7)6hich can be solved numerically to get the energy eigenvalues.In order to handle the generic wave function for the upper and lower component of theDirac spinor, we use Eq. (A.4) in Appendix which gives φ i ( s ) = N i s α ′ / (1 − s ) β ′ / P ( α ′ ,β ′ ) n (1 − s ) . (3.8)with α ′ = 2 √ ξ , β ′ = 2 √ ξ − ξ + ξ , and the normalization constant N i . Let us now findthe normalization constant. Using a new variable z = 1 − s (0 s → +1 z − R + ∞−∞ | φ i ( x ) | dx = 1 as Z +1 − | φ i ( z ) | dzβ (1 − z )(1 + z ) = 1 , (3.9)we get | N i | β α ′ + β ′ Z +1 − (1 − z ) α ′ − (1 + z ) β ′ − h P ( α ′ ,β ′ ) n ( z ) i dz = 1 , (3.10)By using the following representation of the Jacobi polynomials [33] P ( α ′ ,β ′ ) n ( z ) = 1 n ! n X ℓ =0 ℓ ! ( − n ) ℓ ( n + α ′ + β ′ + 1) ℓ ( n + α ′ + 1) ℓ (cid:18) − z (cid:19) ℓ , (3.11)Eq. (3.10) is written as | N i | β α ′ + β ′ + ℓ n ! n X ℓ =0 ℓ ! ( − n ) ℓ ( n + α ′ + β ′ + 1) ℓ ( n + α ′ + 1) ℓ × Z +1 − (1 − z ) α ′ − ℓ (1 + z ) β ′ − P ( α ′ ,β ′ ) n ( z ) dz = 1 , (3.12)where ( n ) r is the Pochammer symbol [35]. With the help of the following integral equationincluding a Jacobi polynomial written in terms of the hypergeometric function F ( − n, a, b ; c, d ; y )[35] Z +1 − (1 − y ) ρ (1 + y ) σ P ( α ′ ,β ′ ) n ( y ) dy = 2 ρ + σ +1 Γ( ρ + 1)Γ( σ + 1)Γ( n + 1 + α ′ ) n !Γ( ρ + σ + 2)Γ( α ′ + 1) × F ( − n, n + α ′ + β ′ + 1 , ρ + 1; α ′ + 1 , ρ + σ + 2; 1) , (3.13)with the conditions Reρ > −
1, and
Reσ > −
1, the normalization constant is computed N i = √ Γ ′ Γ ′′ , (3.14)7here Γ ′ = 2 β ( n !) Γ( α ′ + β ′ )Γ( α ′ + 1)Γ( β )Γ( n + α ′ + 1) , Γ ′′ = " n X ℓ =0 ℓ ! ( − n ) ℓ ( n + α ′ + β ′ + 1) ℓ ( n + α ′ + 1) ℓ F ( − n, n + α ′ + β ′ + 1 , α ′ + ℓ ; α ′ + 1 , α ′ + β ′ + ℓ ; 1)] − . The condition to be satisfied in Eq. (3.13) gives an upper limit for ℓ as ℓ < Aβ which canbe used to determining the greatest integer value for the quantum number n as n < h Aβ i .We obtain the formal analytical solutions for the problem under consideration givingthe results in terms of representative equations (3.7) and (3.8). Now we move on to considerthe upper and lower components of the Dirac spinor separately, and summarize the resultsfor the case where the mass is constant, and the case of non-relativistic limit for the presentproblem. For the upper component ( i = 1), we write the potential parameter as m = V + S whichgives the following energy eigenvalue equation from (3.7) as (cid:20)q m − E + 2 QV ( E + m ) + a + q m − E + β (2 n + 1) (cid:21) − ( β + QV ) = 0 , (3.15)with a = Q V , and b = QV ( β + QV ). The corresponding wave functions are given by φ ( s ) = N s α ′ / (1 − s ) β ′ / P ( α ′ ,β ′ ) n (1 − s ) , (3.16)with α ′ = β p m − E + 2 QV ( E + m ) + Q V , and β ′ = β p m − E .For the lower component ( i = 2), we have the following energy eigenvalue equation from(3.7) with the potential parameter m = V − S (cid:20)q m − E + 2 QV ( E + m ) + a + q m − E + β (2 n + 1) (cid:21) − (cid:2) ( − β + QV ) − Q S ( V − S ) (cid:3) = 0 , (3.17)with a = − Qm S + Q V − Q S ( V − S ), and b = − Qm − Q S ( V − S ) + QV ( β + QV ). The corresponding wave functions are written as φ ( s ) = N s α ′ / (1 − s ) β ′ / P ( α ′ ,β ′ ) n (1 − s ) , (3.18)8ith α ′ = β p m − E + 2 QV ( E + m ) − Qm S + Q V − Q S ( V − S ) , and β ′ = β p m − E . Before going further, we tend to give some numerical results obtained fromEqs. (3.15) and (3.17) in Table 1 where one observes that the energy values for upper com-ponent larger than the ones for lower component, and numerical values for both componentsdecrease while quantum number n increase.Now we can modify our results to the case where the mass is constant. Let us first write m = 0 which means V = − S . This situation corresponds to the spin symmetric case forthe Dirac equation in 3 + 1 dimension [20-22]. We write the energy eigenvalue equation forthe Dirac equation with the generalized Hulth´en potential as (cid:20)q m − E + 2 QV ( E + m ) + a + q m − E + β (2 n + 1) (cid:21) − ( β + QV ) = 0 , (3.19)with a = Q V , and b = QV ( β + QV ). Here, one has to choose the positive eigenvaluesbecause in the case of the spin symmetry occurs only the bound states with positive energy[22]. For the constant mass, the wave functions with normalization constant given in Eq.(3.14) are φ ( s ) = N s α ′ / (1 − s ) β ′ / P ( α ′ ,β ′ ) n (1 − s ) . (3.20)with α ′ = β p m − E + 2 QV ( E + m ) + Q V , and β ′ = β p m − E . The case where m = 0 giving V = + S corresponds to the pseudospin symmetric situation for the Diracequation [20-22], and the energy eigenvalue equation becomes (cid:20)q m − E + 2 QV ( E + m ) + a + q m − E − β (2 n + 1) (cid:21) − ( β + QV ) = 0 , (3.21)with a = − Qm S + Q V , and b = − Qm S + QV ( β + QV ). The last equationcan give negative or positive eigenvalues, but one uses only negative energy eigenvaluesbecause negative energy states can exist in the case of pseudospin symmetry [22]. Thecorresponding wave functions are given as φ ( s ) = N s α ′ / (1 − s ) β ′ / P ( α ′ ,β ′ ) n (1 − s ) . (3.22)with α ′ = β p m − E + 2 QV ( E + m ) − Qm S + Q V , and β ′ = β p m − E . Thepseudospin symmetry, as a hidden symmetry in atomic nuclei, has been suggested firstly9y Arima and co-workers [25, 26]. After the pseudospin symmetry has found a place as arelativistic symmetry in literature, some special features, spin symmetry for example, havebeen studied [27]. There have been many efforts about the recent progress on pseudospinand spin symmetry in different systems such as stable, exotic, deformed and spherical nuclei.These efforts extend the subject of ”hidden symmetries” in atomic nuclei to include differentperspectives such as perturbative study of the pseudospin symmetry, SUSY approach tohidden symmetries combining with similarity renormalization group and studying the sourceof some particular states which intrude from the major shell above to the shell below formingthe nuclear magic numbers 28 , ,
82, etc. [27].Finally, we tend to give only the eigenvalue equation for the non-relativistic limit whichcan be obtained by using E − m ∼ E and E + m ∼ m in (3.7) ( ~ = c = 1) (cid:2) √− m E + 4 Qm V + a i + √− m E + β (2 n + 1) (cid:3) − [( β + QV i ) + Q [( m i − S ) − V ]] = 0 . (3.23)The last equation gives two different results for energy eigenvalues, and one should choosethe appropriate one. P T -symmetric Forms
Let us now study the case where the potential parameter β is pure imaginary which meansthat the potential has a complex form as following V ( x ) = QV cos(2 βx ) + i sin(2 βx )cos(2 βx ) + i sin(2 βx ) − Q , (4.1)with i = √− P T -symmetric because it satisfies[ V ( − x )] ∗ = V ( x ) . (4.2)which is non-Hermitian [4]. The bound state spectra of the generalized, P T -symmetricHulth´en potential can be found from Eq. (3.7), and we write it explicitly as q m − E + 2 QV ( E + m ) + a i + q m − E + iβn ′ + λ p Γ i = 0 , (4.3)10ith n ′ = 2 n + 1 ; Γ i = ( iβ + QV i ) + Q [( m i − S ) − V ] . (4.4)where λ = ±
1. The obtained result says that four different solution can be possible, andwe expect that one of them, at least, gives a real spectra for the
P T -symmetric Hulth´enpotential [4]. The corresponding upper and lower components of the Dirac spinor arewritten with the help of Eq. (3.8) as φ i ( s ) ∼ s α ′′ / (1 − s ) β ′′ / P ( α ′′ ,β ′′ ) n (1 − s ) . (4.5)where α ′′ = − α ′ , and β ′′ = β p E − m = − β ′ . We write the upper and lower spinorcomponent without the normalization constant, but it can be computed in a similar waygiven in the above Section by using a modified normalization condition written for thenon-Hermitian quantum systems [36-38].An interesting form of the potential can be obtained if all potential parameters are takenpure imaginary, namely, V → iV ( S → iS ) , β → iβ, q → iq , giving V ( x ) = V q − sin( βx ) − i cos( βx ) q − q sin( βx ) + 1 = V ∗ (cid:16) π − x (cid:17) , (4.6)which is P T -symmetric but non-Hermitian (and also pseudo-Hermitian) [4, 36]. The energyspectra for this form of the potential is written as q m − E + 2 QV ( E + m ) + a i + q m − E + iβn ′ + λ p Γ i = 0 , (4.7)with n ′ = 2 n + 1 ; Γ i = − ( β − QV i ) + Q [( m i − iS ) + V ] a i = − m Q ( V + S ) − iQm m i − Q [( m i − iS ) + V ] − Q V i . (4.8)It is worthwhile to say that one has to chose the result giving a real spectrum obtainedfrom Eq. (4.7) for the above form of the generalized Hulth´en potential. q -values The value of q = +1 corresponds to the standard Hulth´en potential for which the energyequation is obtained from Eq. (3.7), and the upper and lower components of Dirac spinor11rom Eq. (3.8). For q = −
1, the generalized Hulth´en potential gives V ( x ) = − V e − βx e − βx + 1 , (5.1)which is the Woods-Saxon potential. The energy levels and upper and lower componentsof Dirac spinor for this form are obtained from Eqs. (3.7) and (3.8), respectively.For q = 0, we have V ( x ) = − V e − β , (5.2)which is the exponential potential, and it is known that there is no explicit expression forthe bound states for non-relativistic, and relativistic wave equations [39-41]. Hence we haveto reconsider the problem by using the new variable s = e − βx giving (cid:26) d ds + 1 s dds − s (cid:20) A ′ β − B ′ β s + C ′ β s (cid:21)(cid:27) φ i ( s ) = 0 , (5.3)with A ′ = m − E ,B ′ = m m i − m S + EV + βV i ,C ′ = V i + ( m i − S ) − V , (5.4)We compare Eq. (5.3) with Eq. (A.1) in Appendix, and with the help of Eq. (A.3) weobtain the parameter set α = 1 , α = α = 0 , ξ = C ′ β , ξ = 2 B ′ β , ξ = A ′ β ,α = α = 0 , α = ξ , α = − ξ , α = ξ , α = ξ ,α = 1 + 2 p ξ , α = 2 p ξ , α = p ξ , α = p ξ , (5.5)Eq. (A.10) gives the upper and lower component of Dirac spinor for exponential poten-tial in Eq. (5.2) φ i ( s ) ∼ s β √ m − E e β √ ( m i − S ) − V + V i L β √ m − E n (2 p ξ s ) . (5.6)12 Conclusions
We have analyzed the analytical solutions of the Dirac equation with scalar-vector-pseudoscalargeneralized Hulth´en potential in 1+1 dimension within the position-dependent mass formal-ism. We have reduced the two extended effective-mass versions of coupled equations writtenfor the upper and lower component to a form of analytical solvable equations by relatingthe mass function with the potentials. We have given both energy eigenvalue equationsand normalized wave functions in closed forms. We have also computed the results for thecase where the mass is constant which correspond to spin and pseudospin symmetric casesin Dirac equation. We have written the results for the bound states in the non-relativisticcase. We have studied the bound state spectrum and the corresponding normalized upperand lower component of Dirac spinor for the complex, generalized Hulth´en potential whichare
P T -symmetric, non-Hermitian forms of the potential.
Competing Interests
The author(s) declare(s) that there is no conflict of interests regarding the publicationof this paper.
The author thanks Prof Dr Andreas Fring from City University London and the Depart-ment of Mathematics for hospitality. This research was partially supported through a fundprovided by University of Hacettepe.
Appendix A
The general form of a second order differential equation which is solved by using the para-metric generalization of the Nikiforov-Uvarov method [23] d F ( s ) ds + α − α ss (1 − α s ) dF ( s ) ds − ξ s − ξ s + ξ [ s (1 − α s )] F ( s ) = 0 , (A.1)13ith the quantization rule α n − (2 n + 1) α + (2 n + 1)( √ α + α √ α ) + n ( n − α + α + 2 α α + 2 √ α α = 0 , (A.2)where n = 0 , , , . . . .The parameters α ′ i s within this approach are defined as α = 12 (1 − α ); α = 12 ( α − α ); α = α + ξ ; α = 2 α α − ξ ; α = α + ξ ; α = α ( α + α α ) + α , (A.3)The corresponding wave functions are given in terms of the parameters α i [23] F ( s ) = N s α (1 − α s ) − α − α α P ( α − , α α − α − n (1 − α s ) , (A.4)where α = α + 2 α + 2 √ α ; α = α − α + 2( √ α + α √ α ); α = α + √ α ; α = α − ( √ α + α √ α ) . (A.5)with the Jacobi polynomials P ( σ ,σ ) n ( s ), and a normalization constant N .For the second independent solution the quantization condition is given by α n + (1 − n ) α + (2 n + 1)( √ α − α √ α ) + n ( n − α + α + 2 α α − √ α α = 0 , (A.6)with the corresponding wave functions F ( s ) = N s α ∗ (1 − α s ) − α ∗ − α ∗ α P ( α ∗ − , α ∗ α − α − n (1 − α s ) , (A.7)where α ∗ = α + 2 α − √ α ; α ∗ = α − α − √ α − α √ α ) ,α ∗ = α − √ α ; α ∗ = α − ( √ α − α √ α ) . (A.8)If a situation appearing in the problem such as α = 0, then the quantization rule in(A.2) becomes( α − α ) n + (2 n + 1)( √ α − α √ α ) + n ( n − α + α + 2 α α − √ α α + α = 0 , (A.9)14ith the corresponding wave functions F ( s ) = N s α e α s L α − n ( α s ) . (A.10)when the limits become lim α → P ( α − , α α − α − n (1 − α s ) = L α − n ( α s ) and lim α → (1 − α s ) − α − α α = e α s with generalized Laguerre polynomials L σ n ( s ).15 eferences [1] L. Hulth´en, ” ¨Uber die Eigenl¨osungen der Schr¨odingergleichung des Deuterons,” Ark.Mat. Astron. Fys. , vol. 28A, p. 1-12, 1942.[2] Y. P. Varshni, ”Eigenenergies and oscillator strengths for the Hulth´en potential,”
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