Solution of Effective-Mass Dirac Equation with Scalar-Vector and Pseudoscalar Terms for Generalized Hulthén Potential

Arda, Altuğ

doi:https://doi.org/10.1155/2017/6340409

Advances in High Energy Physics

On this page

Abstract Introduction Conclusions Appendix Disclosure Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 6340409 | https://doi.org/10.1155/2017/6340409

Solution of Effective-Mass Dirac Equation with Scalar-Vector and Pseudoscalar Terms for Generalized Hulthén Potential

Altuğ Arda¹

Academic Editor: Edward Sarkisyan-Grinbaum

Received28 Sept 2016

Revised14 Dec 2016

Accepted29 Dec 2016

Published24 Jan 2017

Abstract

We find the exact bound state solutions and normalization constant for the Dirac equation with scalar-vector-pseudoscalar interaction terms for the generalized Hulthén potential in the case where we have a particular mass function . We also search the solutions for the constant mass where the obtained results correspond to the ones when the Dirac equation has spin and pseudospin symmetry, respectively. After giving the obtained results for the nonrelativistic case, we search then the energy spectra and corresponding upper and lower components of Dirac spinor for the case of PT-symmetric forms of the present potential.

1. Introduction

The Hulthén potential [1] is one of the best known potentials in physics, as a short-range potential [2]. In the present work, we deal with the following form [3, 4]:where the parameter can be written as with the constant and is the screening parameter (in atomic units) [2] with deformation parameter . The constant is related to the atomic number if one uses this potential in atomic physics. Basically, the Hulthén potential is a special form of the Eckart potential [2, 5].

As a short-range potential, the Hulthén potential has a great advantage because the Schrödinger equation can be solved exactly for this potential with . With this advantage, the Hulthén potential has been used in different areas of physics, such as in solid-state physics [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 Hulthén potential which can be written also in a complex form identifying the PT-symmetric case in a closed form for the case where the mass depends on spatial coordinate and extend the Dirac equation including the scalar, vector, and pseudoscalar interaction terms to this case. After the works by Von Roos [18], and Lévy-Leblond [19], the solutions of relativistic and nonrelativistic wave equations with a position-dependent mass have received great attention in literature [3, references therein]. In [20], the bound state solutions of the Klein-Gordon (KG) and Dirac equations with the Hulthén potential by using the approach proposed by Biedenharn have been worked where the scattering state solutions have been also presented. In [21], the analytical results for the bound states of the Dirac equation with the generalized Hulthén potential as a tensor term have been studied within the concept of the SUSYQM. In the present work, we extend the search including the solutions of the Dirac equation having a pseudoscalar interaction term, as in [22–25], for the -parameter Hulthén potential within the position-dependent mass (PDM) formalism. This formalism gives an opportunity such as writing the analytical results for the case where the mass is constant. This means that our results are also available for the cases where the Dirac equation has pseudospin and spin symmetry. Our generic results which will be given below makes it possible to give the “wave functions” with their normalization constants for both the cases of PDM formalism and constant mass. We search here also the analytical results for the PT-symmetric/non-Hermitian and PT-symmetric/pseudo-Hermitian form of the Hulthén potential for both of upper and lower component which are presented again within the PDM formalism. These give us also the results for the case where the mass is constant if necessary. Among the above results, because of the -parameter in potential, we apply our results for three different forms of the potential as special cases.

The organization of this work is as follows. In Section 2, we write the Dirac equation with scalar (), vector (), and pseudoscalar () potentials in dimension for the case where the mass is a spatially coordinate function. In Section 3, we search the 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 function and the potentials to reduce the Dirac equation to an analytically solvable form of second-order 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 the spin and pseudospin symmetry occur in Dirac equation. The spin symmetry appears when the difference of the scalar and vector potentials is constant; that is, , and the pseudospin symmetry appears when the sum of the scalar and vector potentials is constant; that is, [20, 21, 26]. Finally, we obtain the nonrelativistic result for the bound state solution for the generalized Hulthén potential. The present work can also be seen as an application of the parametric generalization of the Nikiforov-Uvarov method which will be given in Appendix briefly [27, 28]. In Section 4, we write the generalized Hulthén potential in a complex form which corresponds to the PT-symmetric form of the potential and find the energy levels for upper and lower component of the Dirac spinor with normalized wave functions. The PT-symmetric formulation with non-Hermitian Hamiltonians having real or complex spectra of quantum mechanics has received a great attention in literature after the work by Bender and Boettcher [29–31]. In Section 5, we collect briefly our analytical results for special values of the parameter corresponding to the standard Hulthén potential (), to the Woods-Saxon potential () and to the exponential potential () while the mass depends on spatially coordinate function. We give our conclusions in Section 6.

2. Dirac Equation in Dimension

The time-independent Dirac equation for a spin- particle subjected to scalar, vector, and pseudoscalar potentials in terms of the , and is given by () [22, 23, 32–36] where , , and are the Pauli spin matrices, and we write the mass as . By taking the Dirac spinor as , where indicates the transpose, we obtain the following first-order coupled equations for upper and lower components:

Writing in terms of with the help of (3) and inserting it into (4) give us We write here the equality between the mass function and the potentials to reduce the above complicated equation to a simpler one which can be solved analytically. This relation gives us also the opportunity about finding the mass function explicitly and the second-order equation for upper component as

By following similar steps and using the equality for the mass function as , we obtain the second-order equation for lower component as

Equations (6) and (7) can be solved by using the parametric generalization of the Nikiforov-Uvarov method which is given in Appendix briefly. In the next section, we solve the above equations for the scalar, vector, and pseudoscalar potentials by identifying them in terms of the Hulthén potential given in (1). First, we find the appropriate mass function by using the equalities, and then we write the bound state solutions with the corresponding normalized wave functions.

3. Bound States for Generalized Hulthén Potential

We are now in a position to identify the potentials in terms of the generalized Hulthén potential. We tend to write them as follows [3, 4]:where , is for the upper component , and is for the lower component . We obtain (7) for by replacement in (6). In addition, we can handle the explicit form of the wave function for by doing in .

From (8), we haveBy using (8) and (9), we write the mass function from the equality obtained for as , where the parameter is basically the integral constant, and we denote it as “constant mass”; the other parameter is obtained as . This means that the mass parameter contains the contributions coming from vector and scalar potentials. The equality obtained for gives us the mass function as with including the contributions coming from vector and scalar potentials. So, we can combine these two mass functions in a single form as which will be used in computation below.

By using a new variable as , using (8)-(9), and inserting the mass function , we have the following representative equation for both components: wherewithand . Equation (10) can be solved by using the parametric Nikiforov-Uvarov method. For this aim, we compare (10) with (A.1) in Appendix, and with the help of (A.3) we obtain the parameter set

With the help of (A.2) in Appendix, we write the energy spectrum of the Dirac equation with scalar-vector-pseudoscalar generalized Hulthén potential within the position-dependent mass formalism as which can be solved numerically to get the energy eigenvalues.

In order to handle the generic wave function for the upper and lower component of the Dirac spinor, we use (A.4) in Appendix which gives with , , and the normalization constant . Let us now find the normalization constant. Using a new variable (), and writing the normalization condition asWe get By using the following representation of the Jacobi polynomials [22] and (17) is written as where is the Pochhammer symbol [24]. With the help of the following integral equation including a Jacobi polynomial written in terms of the hypergeometric function [24] with the conditions and ; the normalization constant is computed whereThe condition to be satisfied in (20) gives an upper limit for as which can be used to determine the greatest integer value for the quantum number as .

We obtain the formal analytical solutions for the problem under consideration giving the results in terms of representative equations (14) and (15). Now we move on to consider the upper and lower components of the Dirac spinor separately and summarize the results for the case where the mass is constant and the case of nonrelativistic limit for the present problem.

3.1. Results

For the upper component (), we write the potential parameter as which gives the following energy eigenvalue equation from (14) aswith and . The corresponding wave functions are given by with , and .

For the lower component (), we have the following energy eigenvalue equation from (14) with the potential parameter with and . The corresponding wave functions are written as with Before going further, we tend to give some numerical results obtained from (23) and (25) in Table 1 where one observes that the energy values for upper component are larger than the ones for lower component, and numerical values for both components decrease while quantum number increases.

Now we can modify our results to the case where the mass is constant. Let us first write which means . This situation corresponds to the spin symmetric case for the Dirac equation in dimension [20, 21, 26]. We write the energy eigenvalue equation for the Dirac equation with the generalized Hulthén potential as with , and . Here, one has to choose the positive eigenvalues because in the case of the spin symmetry only the bound states with positive energy occur [26]. For the constant mass, the wave functions with normalization constant given in (21) are with , and . The case where giving corresponds to the pseudospin symmetric situation for the Dirac equation [20, 21, 26], and the energy eigenvalue equation becomes with , and . The last equation can give negative or positive eigenvalues, but one uses only negative energy eigenvalues because negative energy states can exist in the case of pseudospin symmetry [26]. The corresponding wave functions are given as with The pseudospin symmetry, as a hidden symmetry in atomic nuclei, has been suggested firstly by Arima and coworkers [29, 30]. After the pseudospin symmetry has found a place as a relativistic symmetry in literature, some special features, spin symmetry, for example, have been studied [31]. There have been many efforts about the recent progress on pseudospin and 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 different perspectives such as perturbative study of the pseudospin symmetry and SUSY approach to hidden symmetries combining with similarity renormalization group and studying the source of some particular states which intrude from the major shell above to the shell below forming the nuclear magic numbers , and so forth [31].

Finally, we tend to give only the eigenvalue equation for the nonrelativistic limit which can be obtained by using and in (14) () The last equation gives two different results for energy eigenvalues, and one should choose the appropriate one.

4. Bound States for PT-Symmetric Forms

Let us now study the case where the potential parameter is pure imaginary which means that the potential has a complex form as follows: with . This form of the potential in (1) is PT-symmetric because it satisfies which is non-Hermitian [4]. The bound state spectra of the generalized, PT-symmetric Hulthén potential can be found from (14), and we write it explicitly as withwhere . The obtained result says that four different solutions can be possible, and we expect that one of them, at least, gives a real spectra for the PT-symmetric Hulthén potential [4]. The corresponding upper and lower components of the Dirac spinor are written with the help of (15) as where , and . We write the upper and lower spinor component without the normalization constant, but it can be computed in a similar way given in the above section by using a modified normalization condition written for the non-Hermitian quantum systems [25, 37, 38].

An interesting form of the potential can be obtained if all potential parameters are taken as pure imaginary; namely, , , , giving which is PT-symmetric but non-Hermitian (and also pseudo-Hermitian) [4, 25]. The energy spectra for this form of the potential are written as withIt is worthwhile to say that one has to choose the result giving a real spectrum obtained from (40) for the above form of the generalized Hulthén potential.

5. Solutions for Specific -Values

The value of corresponds to the standard Hulthén potential for which the energy equation is obtained from (14), and the upper and lower components of Dirac spinor from (15). For , the generalized Hulthén potential gives which is the Woods-Saxon potential. The energy levels and upper and lower components of Dirac spinor for this form are obtained from (14) and (15), respectively.

For , we have which is the exponential potential, and it is known that there is no explicit expression for the bound states for nonrelativistic, and relativistic wave equations [39–41]. Hence we have to reconsider the problem by using the new variable giving with We compare (44) with (A.1) in Appendix, and with the help of (A.3) we obtain the parameter set

Equation (A.10) gives the upper and lower component of Dirac spinor for exponential potential in (43)

6. Conclusions

We have analyzed the analytical solutions of the Dirac equation with scalar-vector-pseudoscalar generalized Hulthén potential in dimension within the position-dependent mass formalism. We have reduced the two extended effective-mass versions of coupled equations written for the upper and lower component to a form of analytical solvable equations by relating the mass function with the potentials. We have given both energy eigenvalue equations and normalized wave functions in closed forms. We have also computed the results for the case where the mass is constant which correspond to spin and pseudospin symmetric cases in Dirac equation. We have written the results for the bound states in the nonrelativistic case. We have studied the bound state spectrum and the corresponding normalized upper and lower component of Dirac spinor for the complex, generalized Hulthén potential which are PT-symmetric, non-Hermitian forms of the potential.

Appendix

The general form of a second-order differential equation which is solved by using the parametric generalization of the Nikiforov-Uvarov method [27] is as follows: with the quantization rule where .

The parameters within this approach are defined as

The corresponding wave functions are given in terms of the parameters [27] wherewith the Jacobi polynomials and a normalization constant .

For the second independent solution the quantization condition is given by with the corresponding wave functions where

If a situation is appearing in the problem such as , then the quantization rule in (A.2) becomes with the corresponding wave functions when the limits become and with generalized Laguerre polynomials .

Disclosure

Present address for the author is Department of Mathematical Science, City University London, UK.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks Professor Dr. Andreas Fring from City University London and the Department of Mathematics for hospitality. This research was partially supported through a fund provided by University of Hacettepe.

References

L. Hulthén, “Uber die Eigenlösungen der Schrödingergleichung des Deuterons,” Arkiv för Matematik, Astronomi och Fysik, vol. 28A, no. 5, pp. 1–12, 1942.
View at: Google Scholar | MathSciNet
Y. P. Varshni, “Eigenenergies and oscillator strengths for the Hulthén potential,” Physical Review A, vol. 41, no. 9, pp. 4682–4689, 1990.
View at: Publisher Site | Google Scholar
A. Arda, R. Sever, and C. Tezcan, “Approximate analytical solutions of the effective mass Dirac equation for the generalized Hulthén potential with any κ-value,” Central European Journal of Physics, vol. 8, no. 5, pp. 843–849, 2010.
View at: Publisher Site | Google Scholar
H. Egrifes and R. Sever, “Bound states of the Dirac equation for the PT-symmetric generalized Hulthén potential by the Nikiforov–Uvarov method,” Physics Letters. A, vol. 344, no. 2-4, pp. 117–126, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
C. Eckart, “The penetration of a potential barrier by electrons,” Physical Review, vol. 35, no. 11, pp. 1303–1309, 1930.
View at: Publisher Site | Google Scholar
C. S. Lam and Y. P. Varshni, “Bound eigenstates of the exponential cosine screened coulomb potential,” Physical Review A, vol. 6, no. 4, pp. 1391–1399, 1972.
View at: Publisher Site | Google Scholar
B. Durand and L. Durand, “Duality for heavy-quark systems,” Physical Review D, vol. 23, no. 5, pp. 1092–1102, 1981.
View at: Publisher Site | Google Scholar
J. Lindhard and A. Winther, “Transient fields acting on heavy ions during slowing-down in magnetized materials,” Nuclear Physics A, vol. 166, no. 3, pp. 413–435, 1971.
View at: Publisher Site | Google Scholar
P. Pyykko and J. Jokisaari, “Spectral density analysis of nuclear spin-spin coupling: I. A Hulthén potential LCAO model for JX-H in hydrides XH4,” Chemical Physics, vol. 10, no. 2-3, pp. 293–301, 1975.
View at: Publisher Site | Google Scholar
R. Dutt and Y. P. Varshni, “Extension of Fudas off-shell analysis to screened Coulomb potentials for arbitrary ℓ and limiting relations,” Journal of Mathematical Physics, vol. 24, no. 12, pp. 2770–2775, 1983.
View at: Publisher Site | Google Scholar | MathSciNet
F. Domínguez-Adame, “Bound states of the Klein-Gordon equation with vector and scalar Hulthén-type potentials,” Physics Letters A, vol. 136, no. 4-5, pp. 175–177, 1989.
View at: Publisher Site | Google Scholar | MathSciNet
B. Roy and R. Roychoudhury, “The shifted 1=N expansion and the energy eigenvalues of the Hulthén potential for $l \neq 0$ ,” Journal of Physics A, vol. 20, pp. 3051–3055, 1987.
View at: Google Scholar
R. L. Greene and C. Aldrich, “Variational wave functions for a screened Coulomb potential,” Physical Review A, vol. 14, no. 6, pp. 2363–2366, 1976.
View at: Publisher Site | Google Scholar
M. Znojil, “Exact solution of the Schrodinger and Klein-Gordon equations for generalised Hulthen potentials,” Journal of Physics A: Mathematical and General, vol. 14, no. 2, pp. 383–394, 1981.
View at: Publisher Site | Google Scholar | MathSciNet
J. M. Cai, P. Y. Cai, and A. Inomata, “Path-integral treatment of the Hulthén potential,” Physical Review A, vol. 34, no. 6, pp. 4621–4628, 1986.
View at: Publisher Site | Google Scholar | MathSciNet
M. R. Setare and E. Karimi, “Algebraic approach to the Hulthen potential,” International Journal of Theoretical Physics, vol. 46, no. 5, pp. 1381–1388, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. M. Ikhdair and R. Sever, “Approximate eigenvalue and eigenfunction solutions for the generalized Hulthén potential with any angular momentum,” Journal of Mathematical Chemistry, vol. 42, no. 3, pp. 461–471, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
O. Von Roos, “Position-dependent effective masses in semiconductor theory,” Physical Review B, vol. 27, no. 12, pp. 7547–7552, 1983.
View at: Publisher Site | Google Scholar
J.-M. Lévy-Leblond, “Position-dependent effective mass and Galilean invariance,” Physical Review A, vol. 52, no. 3, pp. 1845–1849, 1995.
View at: Publisher Site | Google Scholar | MathSciNet
S. Haouat and L. Chetouani, “Approximate solutions of Klein-Gordon and Dirac equations in the presence of the Hulthén potential,” Physica Scripta, vol. 77, no. 2, Article ID 025005, 6 pages, 2008.
View at: Publisher Site | Google Scholar
H. Hassanabadi, B. H. Yazarloo, M. Mahmoudieh, and S. Zarrinkamar, “Dirac equation under the Deng-Fan potential and the Hulthén potential as a tensor interaction via SUSYQM,” European Physical Journal Plus, vol. 128, article no. 111, 2013.
View at: Publisher Site | Google Scholar
A. D. Alhaidari, “Dirac equation with coupling to 1=r singular vector potentials for all angular momenta,” Foundations of Physics, vol. 40, no. 8, pp. 1088–1095, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
A. D. Alhaidari, “Relativistic coulomb problem for Z larger than 137,” International Journal of Modern Physics A, vol. 25, no. 18-19, pp. 3703–3714, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. D. Alhaidari, “Generalized spin and pseudo-spin symmetry: relativistic extension of supersymmetric quantum mechanics,” Physics Letters. B, vol. 699, no. 4, pp. 309–313, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
L. B. Castro, A. S. de Castro, and M. Hott, “Relativistic effects of mixed vector-scalar-pseudoscalar potentials for fermions in 1 + 1 Dimensions,” International Journal of Modern Physics E, vol. 16, pp. 3002–3005, 2007.
View at: Google Scholar
K. T. Hecht and A. Adler, “Generalized seniority for favored $J \neq 0$ pairs in mixed configurations,” Nuclear Physics A, vol. 137, no. 1, pp. 129–143, 1969.
View at: Publisher Site | Google Scholar
D. Troltenier, C. Bahri, and J. P. Draayer, “Generalized pseudo-SU(3) model and pairing,” Nuclear Physics A, vol. 586, no. 1, pp. 53–72, 1995.
View at: Publisher Site | Google Scholar
J. N. Ginocchio, “Relativistic symmetries in nuclei and hadrons,” Physics Reports A, vol. 414, no. 4-5, pp. 165–261, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
A. Arima, M. Harvey, and K. Shimizu, “Pseudo LS coupling and pseudo SU3 coupling schemes,” Physics Letters B, vol. 30, no. 8, pp. 517–522, 1969.
View at: Google Scholar
K. T. Hect and A. Adler, “Generalized seniority for favored $J \neq 0$ pairs in mixed configurations, Nucl,” Nuclear Physics A, vol. 137, no. 1, pp. 129–143, 1969.
View at: Publisher Site | Google Scholar
H. Liang, J. Meng, and S.-G. Zhou, “Hidden pseudospin and spin symmetries and their origins in atomic nuclei,” Physics Reports, vol. 570, pp. 1–84, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
C. Tezcan and R. Sever, “A general approach for the exact solution of the Schrödinger equation,” International Journal of Theoretical Physics, vol. 48, no. 2, pp. 337–350, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhäuser, Basel, Switzerland, 1988.
View at: Publisher Site | MathSciNet
C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having PT symmetry,” Physical Review Letters, vol. 80, no. 24, pp. 5243–5246, 1998.
View at: Publisher Site | Google Scholar | MathSciNet
C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” Journal of Mathematical Physics, vol. 40, no. 5, pp. 2201–2229, 1999.
View at: Publisher Site | Google Scholar | MathSciNet
C. M. Bender and G. V. Dunne, “Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian,” Journal of Mathematical Physics, vol. 40, no. 10, pp. 4616–4621, 1999.
View at: Publisher Site | Google Scholar | MathSciNet
M. Hamzavi and A. A. Rajabi, “Scalar–vector–pseudoscalar Cornell potential for a spin-1/2 particle under spin and pseudospin symmetries: 1 + 1 dimensions,” Annals of Physics, vol. 334, pp. 316–320, 2013.
View at: Publisher Site | Google Scholar
S. Haouat and L. Chetouani, “Bound states of Dirac particle subjected to the pseudoscalar Hulthén potential,” Journal of Physics A: Mathematical and Theoretical, vol. 40, no. 34, pp. 10541–10547, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Chargui, “Effective mass and pseudoscalar interaction in the dirac equation with woods–saxon potential,” Few-Body Systems, vol. 57, no. 4, pp. 289–306, 2016.
View at: Publisher Site | Google Scholar
M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1965.
S. M. Ikhdair and R. Sever, “Bound-states of a semi-relativistic equation for the PT-symmetric generalized Hulthén potential by the Nikiforov-Uvarov method,” International Journal of Modern Physics E, vol. 17, no. 6, pp. 1107–1124, 2008.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2017 Altuğ Arda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

PDF Download Citation

Download other formats

Order printed copies

Views

1206

Downloads

886

Citations