Approximate Pseudospin and Spin Solutions of the Dirac Equation for a Class of Exponential Potentials
aa r X i v : . [ m a t h - ph ] S e p Approximate Pseudospin and Spin Solutions of the DiracEquation for a Class of Exponential Potentials
Altu˘g Arda, ∗ Ramazan Sever, † and Cevdet Tezcan ‡ Department of Physics Education, Hacettepe University, 06800, Ankara,Turkey Department of Physics, Middle East Technical University, 06800, Ankara,Turkey Faculty of Engineering, Ba¸skent University, Baglıca Campus, Ankara,Turkey
Abstract
Dirac equation is solved for some exponential potentials, hypergeometric-type potential, gener-alized Morse potential and Poschl-Teller potential with any spin-orbit quantum number κ in thecase of spin and pseudospin symmetry, respectively. We have approximated for non s-waves thecentrifugal term by an exponential form. The energy eigenvalue equations, and the correspondingwave functions are obtained by using the generalization of the Nikiforov-Uvarov method.Keywords: Pseudospin symmetry, Spin symmetry, Dirac Equation, Hypergeometric Potential, gen-eralized Morse Potential, Poschl-Teller Potential, Nikiforov-Uvarov Method PACS numbers: 03.65.Fd, 03.65.Ge, 12.39.Fd ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] . INTRODUCTION The solutions of the Dirac equation having the pseudospin, and spin symmetry have beenstrongly studied in the last years. The concept of the pseudospin symmetry [1, 2, 3] is animportant theme in nuclear theory because of its features related to construct an effectiveshell-model coupling scheme [4], to study the structures of the deformed nuclei [5, 6]. Thepseudopin symmetry occurs in nuclei when the magnitude of the scalar and vector potentialsare nearly equal, but opposite sign, i.e., V v ( r ) ∼ − V s ( r ) ,where the scalar potential is negative(attractive) and the vector potential is positive (repulsive) in the relativistic region. Further,the spin symmetry appears when the magnitude of the scalar and vector potentials are nearlyequal, i.e., V v ( r ) ∼ V s ( r ) [7-12]. The Dirac equation under the exact pseudospin and/or spinsymmetry has been studied by with different type of potentials such as the Hulth´en potential[13], the Morse potential [14], the Woods-Saxon potential [15, 16], and harmonic oscillator[17-21].In the present work, we deal with the solutions of the Dirac equation if the exact pseu-dospin, and spin symmetry occur in the theory under the effect of a class of potentials, whichhave exponential form, i.e. the hypergeometric-type potential [22], the generalized Morsepotential and the P¨oschl-Teller potential [23].In order to obtain the energy eigenvalue equation and the corresponding wave functions,we apply a new approximation scheme [24], which is the parametric generalization of theNikiforov-Uvarov (NU) method [25], by using an approximation to the centrifugal-like term.So, we obtain the energy spectra of the above potentials for the spin-orbit quantum number κ = 0, or for any κ -value for the case of pseudospin and spin symmetry, respectively.The organization of the present work is as follows. Firstly, we give briefly the equationsfor the Dirac spinors including the centrifugal term, and the spin-orbit quantum number κ .Than we present the basics of the parametric generalization of the NU method. We find thebound states and the corresponding wave functions of the Dirac equation with the abovepotentials in the case of pseudospin and spin symmetry, respectively. Finally, we give ourconclusions. 2 I. DIRAC EQUATION
The Dirac equation for a particle with rest mass m in the absence of the scalar and vectorpotential is written (¯ h = c = 1) [15] n ~α.~p + β [ m + V s ( r )] o Ψ( r ) = ( E − V v ( r ))Ψ( r ) , (1)where ~p is the momentum operator, ~α , and β are 4 × ~α = ~σ i ~σ i ,β = I − I , I is the 2 × ~σ i ( i = 1 , ,
3) are the Pauli matrices. Theoperator ˆ K is the spin-orbit matrix operator and written in terms of the orbital angularmomentum operator ˆ L as ˆ K = − β (ˆ σ. ˆ L + 1) ,which commute with the Dirac Hamiltonian.The Dirac spinors can be labelled by the quantum number set ( n, κ ) ,where κ is the eigenvalueof the spin-orbit operator, and written asΨ nκ ( r ) = f nκ g nκ , (2)where f nκ = [ F nκ ( r ) /r ] Y ℓjm ( θ, φ ) is the upper, and g nκ = [ iG nκ ( r ) /r ] Y ˜ ℓjm ( θ, φ ) is the lowercomponent, and Y ℓjm ( θ, φ ) , and Y ˜ ℓjm ( θ, φ ) are the spherical harmonics, respectively. The totalangular momentum, the orbital angular momentum, and pseudo-orbital angular momentumcan be written in terms of the spin-orbit quantum number κ = ± , ± , . . . , such as j = | κ | − / ℓ = | κ + 1 / | − / ℓ = | κ − / | − / F nκ ( r ), we obtain two uncoupleddifferential equations for the lower, and upper components of the Dirac equation n d dr − κ ( κ − r − M Σ ( r ) d Σ( r ) dr (cid:16) ddr − κr (cid:17)o G nκ ( r ) = M ∆ ( r ) M Σ ( r ) G nκ ( r ) , (3)and n d dr − κ ( κ + 1) r + 1 M ∆ ( r ) d ∆( r ) dr (cid:16) ddr + κr (cid:17)o F nκ ( r ) = M ∆ ( r ) M Σ ( r ) F nκ ( r ) . (4)where M ∆ ( r ) = m + E − ∆( r ) , ∆( r ) = V v ( r ) − V s ( r ) , M Σ ( r ) = m − E + Σ( r ) , andΣ( r ) = V v ( r ) + V s ( r ) . 3 II. PARAMETRIC GENERALIZATION OF THE METHOD
Let us give briefly the parametric generalization of the NU method. By using an ap-propriate coordinate transformation, the Schr¨odinger equation can be transformed into thefollowing form σ ( z ) d Ψ( z ) dz + σ ( z )˜ τ ( z ) d Ψ( z ) dz + ˜ σ ( z )Ψ( z ) = 0 , (5)where σ ( z ) , and ˜ σ ( z ) are polynomials, at most, second degree, and ˜ τ ( z ) is a first-degreepolynomial. By writing the general solution as Ψ( z ) = ψ ( z ) ϕ ( z ), we obtain a hypergeometrictype equation [24] d ϕ ( z ) dz + τ ( z ) σ ( z ) dϕ ( z ) dz + λσ ( z ) ϕ ( z ) = 0 , (6)where ψ ( z ) and ϕ n ( z ) are defined as [24]1 ψ ( z ) dψ ( z ) dz = π ( z ) σ ( z ) , (7) ϕ n ( z ) = a n ρ ( z ) d n dz n [ σ n ( z ) ρ ( z )] , (8)where a n is a normalization constant, and ρ ( z ) is the weight function satisfying the followingequation [24] dσ ( z ) dz + σ ( z ) ρ ( z ) dρ ( z ) dz = τ ( z ) . (9)The function π ( z ) , and the parameter λ in the above equation are defined as π ( z ) = 12 [ σ ′ ( z ) − ˜ τ ( z )] ± "
14 [ σ ′ ( z ) − ˜ τ ( z )] − ˜ σ ( z ) + kσ ( z ) / , (10) λ = k + π ′ ( z ) . (11)In the NU method, the square root in Eq. (10) must be the square of the polynomial, sothe parameter k can be determined. Thus, a new eigenvalue equation becomes4 = λ n = − nτ ′ ( z ) −
12 ( n − n ) σ ′′ ( z ) . (12)where prime denotes the derivative and the derivative of the function τ ( z ) = ˜ τ ( z ) + 2 π ( z )should be negative.Now, in order to clarify the parametric generalization of the NU method [25], let us takethe following Schr¨odinger-like equation written for any potential z (1 − α z ) d Ψ( z ) dz + z (1 − α z )( α − α z ) Ψ( z ) dz + [ − ξ z + ξ z − ξ ]Ψ( z ) = 0 . (13)Comparing Eq. (13) with Eq. (5), we obtain˜ τ ( z ) = α − α z ; σ ( z ) = z (1 − α z ) ; ˜ σ ( z ) = − ξ z + ξ z − ξ . (14)Substituting these into Eq. (10), we obtain π ( z ) = α + α z ± " ( α − kα ) z + ( α + k ) z + α / , (15)with the following parameters α = (1 − α ) , α = ( α − α ) ,α = α + ξ , α = 2 α α − ξ ,α = α + ξ . (16)We obtain the parameter k from the condition that the function under the square rootshould be the square of a polynomial k , = − ( α + 2 α α ) ± √ α α , (17)where α = α α + α α + α . The function π ( z ) becomes π ( z ) = α + α z − [( √ α + α √ α ) z − √ α ] . (18)5or the k -value k = − ( α + 2 α α ) − √ α α . We also have from τ ( z ) = ˜ τ ( z ) + 2 π ( z ) , τ ( z ) = α + 2 α − ( α − α ) z − √ α + α √ α ) z − √ α ] . (19)Thus, we impose the following condition to fix the k -value τ ′ ( z ) = − ( α − α ) − √ α + α √ α )= − α − √ α + α √ α ) < . (20)From Eqs. (11), (18) and (19) and by using τ ( z ) = ˜ τ ( z ) + 2 π ( z ) and equating Eq. (11)with the condition that λ must satisfy given by Eq. (12), we obtain the energy eigenvalueequation for the potential under the consideration n [( n − α + α − α ] − α + (2 n + 1)( √ α + α √ α )+ α + 2 α α + 2 √ α α = 0 . (21)By using Eq. (9) ρ ( z ) = z α − (1 − α z ) α α − α − , (22)and together with Eq. (8), we obtain ϕ n ( z ) = P ( α − , α α − α − n (1 − α z ) , (23)where α = α + 2 α + 2 √ α , α = α − α + 2( √ α + α √ α , and P ( α,β ) n (1 − α z ) areJacobi polynomials. By using Eq. (7), we obtain ψ ( z ) = z α (1 − α z ) − α − α α , (24)and the total wave function become 6( z ) = z α (1 − α z ) − α − α α P ( α − , α α − α − n (1 − α z ) , (25)where α = α + √ α , α = α − ( √ α + α √ α ).In some problems the situation appears where α = 0. For this type of the problems, thesolution given in Eq. (25) becomes asΨ( z ) = z α e α z L α − n ( α z ) , (26)and the energy spectrum is α n − α n + (2 n + 1)( √ α − α √ α ) + n ( n − α + α + 2 α α − √ α α + α = 0 . (27)when the limits become lim α → P ( α − , α α − α − n (1 − α z ) = L α − n ( α z ) and lim α → (1 − α z ) − α − α α = e α z . IV. BOUND STATESA. The Hypergeometric-Type Potential
The Dirac equation has the exact pseudospin symmetry if Σ( r ) = C = const. , so Eq. (3)becomes under that condition n d dr − κ ( κ − r − ( m − E + C ) M ∆ ( r ) o G nκ ( r ) = 0 , (28)where κ = ˜ ℓ + 1 for κ > κ = − ˜ ℓ for κ < V ( r ) = D [1 − σ coth( αr )] = D + D e − αr − e − αr ! , (29)where the real parameters D , σ , and α represent the potential [22], and D = √ D (1 − σ ) ,and D = √ D (1 + σ ). 7q. (28) can not be solved analytically for any κ values because of κ ( κ − /r term, sowe use the approximation 1 /r ≃ α e − αr / (1 − e − αr ) [26] to solve the equation for anyspin-orbit quantum number κ .By using this approximation to centrifugal-like term, setting ∆( r ) to the potential givenin Eq. (29), and inserting into Eq. (28), we obtain n d dr − α κ ( κ − e − αr (1 − e − αr ) + µ ( D + D e − αr ) (1 − e − αr ) − ǫ o G nκ ( r ) = 0 , (30)where µ = m − E + C , and ǫ = m ( m + C ) + E ( C − E ) . At this point, it is worthwhile tonote that Eq. (29) becomes for σ = 1( D = 0) V ( r ) = D e − αr − e − αr ! , (31)This form of the potential corresponds to the Manning-Rosen potential for A = 0 [27] if weset q κb α ( α − → D (here, κ and α are the parameters in Ref. [27]), and b → α . Thismeans that we could also obtain the energy eigenvalue equation of the Dirac equation forthe Manning-Rosen potential in the case of the exact spin symmetry, if we set the parameter D = 0 in the equations.By using the new variable z = e − αr (0 < z <
1) , we obtain from Eq. (30) d G nκ ( z ) dz + 1 − zz (1 − z ) dG nκ ( z ) dz + 1[ z (1 − z )] n β ( µD − ǫ )+ 2 β [ D D µ − ǫ − α κ ( κ − z + β ( µD − ǫ ) z o G nκ ( z ) = 0 . (32)By comparing Eq. (32) with Eq. (13), we get the parameter set α = 1 , α = 1 , ξ = − β ( µD − ǫ ) ,α = 1 , α = 0 , ξ = 2 β [ D D µ − ǫ − α κ ( κ − ,α = − , α = ξ + , ξ = − β ( µD − ǫ ) ,α = − ξ , α = ξ , α = ξ − ξ + ξ + ,α = 1 + 2 √ ξ , α = 2 + 2( q ξ − ξ + ξ + + √ ξ ) ,α = √ ξ , α = − − ( q ξ − ξ + ξ + + √ ξ ) . (33)8he energy eigenvalue equation becomes s β [4 α κ ( κ − − µ ( D + D ) ] + 14 + β q ǫ − µD !(cid:16) n + 1 + 2 β q ǫ − µD (cid:17) + β h α κ ( κ − − ǫ + µD D ) i = − n ( n + 1) − / . (34)In this case, we use only the negative energy eigenvalues, because negative energy statesexist in the pseudospin symmetry [28].Now, let us give the corresponding Dirac spinors from Eq. (25) G nκ ( z ) = z β √ ǫ − µD × (1 − z ) + √ − µβ ( D + D ) + κ ( κ − P (2 β √ ǫµD , √ − µβ ( D + D ) + κ ( κ − ) n (1 − z ) . (35)Finally, we briefly give the energy eigenvalue equation for the special case σ = 1 , whichgives the energy spectra of the Manning-Rosen potential with A = 0 s β [ α κ ( κ − − Dµ ] + 14 + β √ ǫ !(cid:16) n + 1 + 2 β √ ǫ (cid:17) +4 β h α κ ( κ − − ǫ i + 14 ((2 n + 1) + 1) = 0 . (36)Under the exact spin symmetry, i.e. ∆( r ) = C = const. , Eq. (4) becomes n d dr − κ ( κ + 1) r − ( m + E − C ) M Σ ( r ) o F nκ ( r ) = 0 , (37)where κ = − ( ℓ + 1) for κ > κ = ℓ for κ < r ) to the potential givenin Eq. (30), using the above approximation to the κ ( κ + 1) /r term, and using the newvariable z = e − αr (0 < z < d F nκ ( z ) dz + 1 − zz (1 − z ) dF nκ ( z ) dz + 1[ z (1 − z )] n β ( ǫ ′ − µ ′ D ) − β [2 ǫ ′ + 2 D D µ ′ + 4 α κ ( κ + 1)] z + β ( ǫ ′ − µ ′ D ) z o F nκ ( r ) = 0 . (38)9here µ ′ = m + E − C , ǫ ′ = m ( C − m ) + E ( E − C ), and β = 1 / α . By comparing Eq. (39)with Eq. (13), we obtain the parameter set given in Eq. (34) where ξ = − β ( ǫ ′ − µ ′ D ), ξ = − β [ ǫ ′ + D D µ ′ + 2 α κ ( κ + 1)], and ξ = − β ( ǫ ′ − µ ′ D ). The energy eigenvalueequation of the hypergeometric potential for the exact spin symmetry is written from Eq.(21) as s β µ ′ ( D + D ) + κ ( κ + 1) + 14 + β q µ ′ D − ǫ ′ !(cid:16) n + 1 + 2 β q µ ′ D − ǫ ′ (cid:17) + β h α κ ( κ + 1) + 2( ǫ ′ + µ ′ D D ) i = − n ( n + 1) − / , (39)The last equation can give negative, and positive eigenvalues, but we choose only positiveenergy eigenvalues, because in the case of the exact spin symmetry appears only the positiveenergy eigenstates [28]. The corresponding Dirac spinors are obtained from Eq. (25), andgiven F nκ ( z ) = z β √ µ ′ D − ǫ ′ × (1 − z ) + √ µ ′ β ( D + D ) + κ ( κ +1)+ P (2 β √ µ ′ D − ǫ ′ , √ µ ′ β ( D + D ) + κ ( κ +1)+ ) n (1 − z ) . (40)Now, we briefly give the energy eigenvalue equation for the Manning-Rosen potentialwith A = 0 ( σ = 1) s β D ( m + E − C ) + κ ( κ + 1) + 14 + β q m ( m − C ) + E ( C − E ) ! × (cid:16) n + 1 + 2 β q m ( m − C ) + E ( C − E ) (cid:17) + β h α κ ( κ + 1) + 2( m ( C − m ) + E ( E − C )) i + 14 ((2 n + 1) + 1) = 0 . (41) B. The Generalized Morse Potential
Assuming that the potential ∆( r ) = V v ( r )+ V s ( r ) is the generalized Morse potential givenby V ( r ) = V e − αr − V e − αr , (42)10nd substituting Eq. (42) into Eq. (3), and taking Σ( r ) = Σ = const. , we have the followingequation in the exact pseudospin symmetry for κ = 0 ( z = e − αr ) n d dz + 1 s ddz + 1 z h β (cid:16) µ − E + E ( µ + Σ) (cid:17) + 4 β V ( µ − E + Σ) z + 4 β V ( E − µ − Σ) z io G nκ ( z ) = 0 . (43)Comparing Eq. (43) with Eq. (13), we obtain the following parameter set α = 1 , α = 0 , ξ = 4 β V ( µ − E + Σ) ,α = 0 , α = 0 , ξ = 4 β V ( µ − E + Σ) ,α = 0 , α = ξ , ξ = 4 β ( E − µ − E ( µ + Σ)) ,α = − ξ , α = ξ , α = ξ α = 1 + 2 √ ξ , α = 2 √ ξ ,α = √ ξ , α = −√ ξ . (44)From Eq. (27), we obtain the energy eigenvalue equation for κ = 0 E − E ( µ + Σ) − µ = 116 β n + 1 − βV √ V q µ − E + Σ ! , (45)where β = 1 / α . We should choose the negative energy solution in Eq. (46) because thenegative energy states exist only in the exact pseudospin limit. The corresponding lowerspinor component can be obtained from Eq. (26) G nκ ( z ) = z β √ E − µ − E ( µ +Σ) e − β √ V ( µ − E +Σ) z × L β √ E − µ − E ( µ +Σ) n (4 β q V ( µ − E + Σ) z ) , (46)In the case of exact spin symmetry the potential ∆( r ) = V v ( r ) − V s ( r ) is a constant, letsay ∆( r ) = ∆ = const. . We set the potential Σ( r ) as the Morse potential in Eq. (42).Substituting the potential into Eq. (4), and using the same variable z = e − αr , we obtain n d dz + 1 z ddz + 1 z h β (cid:16) E − µ + ∆( µ − E ) (cid:17) + 4 β V ( µ + E − ∆) z + 4 β V ( − E − µ + ∆) z io F nκ ( z ) = 0 , (47)11omparing Eq. (47) with Eq. (13), we obtain the parameter set given in Eq. (44) where ξ = 4 β V ( E + µ − ∆) , ξ = 4 β V ( E + µ − ∆) and ξ = 4 β ( µ − E + ∆( E − µ )). FromEq. (27), we obtain the energy eigenvalue equation in the case of exact spin symmetry for κ = 0 µ − E + ∆( E − µ ) = 116 β n + 1 − βV √ V q E + µ − ∆ ! . (48)where β = 1 / α . We should choose the positive energy solution in Eq. (49) because thepositive energy states exist only in the exact spin limit. The corresponding Dirac spinor canbe written as F nκ ( z ) = z β √ µ − E +∆( E − µ ) e − β √ V ( µ + E − ∆) z × L β √ µ − E +∆( E − µ ) n (4 β q V ( µ + E − ∆) z ) . (49) C. The P¨oschl-Teller Potential
By taking the potential ∆( r ) = V v ( r ) + V s ( r ) is the P¨oschl-Teller potential [23] given by V ( r ) = − V e − αr (1 + e − αr ) , (50)and substituting Eq. (50) into Eq. (3), taking into account Σ( r ) = Σ = const. , we have thefollowing equation in the exact pseudospin symmetry for κ = 0 ( z = − e − αr ) d G nκ ( z ) dz + 1 − zz (1 − z ) dG nκ ( z ) dz + 1[ z (1 − z )] n β ( µ − E + Σ) h µ + E − [2 µ + 2 E + 4 V ] z + ( µ + E ) z io G nκ ( z ) = 0 . (51)Following the same procedure, we obtain the parameter set12 = 1 , ξ = β ( µ + E )( − µ − Σ + E ) ,α = 1 , ξ = 2 β ( − µ − Σ + E )[ µ + E + 2 V ] ,α = 1 , ξ = β ( µ + E )( − µ − Σ + E ) ,α = 0 , α = − ,α = ξ + , α = − ξ ,α = ξ , α = ξ − ξ + ξ + ,α = 1 + 2 √ ξ , α = 2 + 2( q ξ − ξ + ξ + + √ ξ ) ,α = √ ξ , α = − − ( q ξ − ξ + ξ + + √ ξ ) . (52)and the energy eigenvalue equation of P¨oschl-Teller potential under the exact pseudospinsymmetry for κ = 0 from Eq. (21) E − µ − Σ( µ + E ) = 14 (cid:18) (2 n + 1) α + q V ( µ − E + Σ) + α (cid:19) . (53)The last energy eigenvalue equation has a quadratic form in terms of energy E . We takethe negative energy vales in the exact pseudospin limit. The corresponding Dirac spinor canbe written in terms of Jacobi polynomials, i.e., P ( α , β ) n ( x ), G nκ ( z ) = z β √ ( µ + E )( − µ − Σ+ E ) (1 − z ) h √ V β ( µ +Σ − E ) i × P (2 β √ ( µ + E )( − µ − Σ+ E ) , √ V β ( µ +Σ − E ) ) n (1 − z ) , (54)In the case of exact spin symmetry, we set the potential Σ( r ) as P¨oschl-Teller potentialgiven in Eq. (50), ∆( r ) = ∆ = const. , and by using the coordinate transformation z = − e − αr , we obtain from Eq. (4) d F nκ ( z ) dz + 1 − zz (1 − z ) dF nκ ( z ) dz + ∆ − µ − E [ z (1 − z )] n β ( µ − E ) − β ( µ − E − V ) z + β ( µ − E ) z o F nκ ( z ) = 0 , (55)which gives the parameter set given in Eq. (52) where ξ = β ( E − µ )(∆ − µ − E ), ξ =2 β (∆ − µ − E )( E − µ + 2 V ) and ξ = β ( E − µ )(∆ − µ − E ). The energy eigenvalue13quation of the P¨oschl-Teller potential under the exact spin symmetry for κ = 0 from Eq.(21) is obtained E + µ − ∆( µ − E ) = 14 (cid:18) (2 n + 1) α + q V ( µ + E − ∆) + α (cid:19) . (56)Finally, we obtain the corresponding Dirac spinor from Eq. (25) F nκ ( z ) = z β √ ( E − µ )(∆ − µ − E ) (1 − z ) h √ V β ( µ + E − ∆) i × P (2 β √ ( E − µ )(∆ − µ − E ) , √ V β ( µ + E − ∆) ) n (1 − z ) . (57) V. CONCLUSION
We have studied the energy eigenvalues and the corresponding eigenfunctions of the Diracequation with the hypergeometric potential, the Morse potential, and the P¨oschl-Teller po-tential in the case of pseudospin, and spin symmetry. We have used the parametric general-ization of the NU method to obtain the results. The energy eigenvalues of all potentials arereal and the wave functions are written in terms of the Laguerre (Jacobi) polynomials. Wehave also investigated the special case σ = 1 in the case of the hypergeometric potential,which corresponds to the case of the Manning-Rosen potential with A = 0 . So we haveobtained the energy eigenvalue equation of the Manning-Rosen potential in the pseudospin,and spin symmetry case, respectively. 14
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