The Influence of Deformation Phase-Space on Spectra of Heavy Quarkonia in Improved Energy Potential at Finite Temperature Model of Shrodinger Equation Via the Generalized Boob’s Shift Method and Standard Perturbation Theory
Abstract
In this work, we obtain solutions of the deformed Schrödinger equation (DSE) with improved internal energy potential at a finite temperature model in a 3-dimensional nonrelativistic noncommutative phase-space (3D-NRNCPS) symmetries framework, using the generalized Bopp’s shift method in the case of perturbed nonrelativistic quantum chromodynamics (pNRQCD). The modified bound state energy spectra are obtained for the heavy quarkonium system such as charmonium cc- and bottomonium bb- at finite temperature. It is found that the perturbative solutions of the discrete spectrum are sensible to the discreet atomic quantum numbers (j,l,s,m) of the ( QQ- (Q=c,b)) state, the parameters of internal energy potential (T,αs(T), mD (T),β,c), which are the Debye screening mass mD (T), the running coupling constant αs(T) the critical temperature β, the free parameter c in addition to noncommutativity parameters (Θ,θ-). The new Hamiltonian operator in 3D-NRNCPS symmetries is composed of the corresponding operator in commutative phase-space and three additive parts for spin-orbit interaction, the new magnetic interaction, and the rotational Fermi-term. The obtained energy eigenvalues are applied to obtain the mass spectra of heavy quarkonium systems (cc- and bb-). The total complete degeneracy of the new energy levels of the improved internal energy potential changed to become equal to the new value 3n2 in 3D-NRNCPS symmetries instead of the value n2 in the symmetries of 3D-NRQM. Our non-relativistic results obtained from DSE will possibly be compared with the Dirac equation in high-energy physics.
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X.Y. Wu, B.J. Zhang, X.J. Liu, Y.H. Wu, Q.C. Wang and Y. Wang, “Finite temperature Schrödinger equation”, Int. J. Theor. Phys., 50(8), 2546 (2017). https://doi.org/10.1007/s10773-011-0745-7
C.Y. Wong, “Heavy quarkonia in quark-gluon plasma”, Phys. Rev. C., 72(3), 034906 (2005). https://doi.org/10.1103/PhysRevC.72.034906
T. Matsui and H. Satz, “J/ψ suppression by quark-gluon plasma formation”, Phys. Lett. B., 178(4), 416 (1986). https://doi.org/10.1016/0370-2693(86)91404-8
V. Mateu, P. G. Ortega, D. R. Entem and F. Fernández, “Calibrating the naïve Cornell model with NRQCD”, Eur. Phys. J. C, 79 (4), 323 (2019). https://doi.org/10.1140/epjc/s10052-019-6808-2
J. Fingberg, “Heavy quarkonia at high temperature”, Physics Letters B, 424(3-4), 343 (1998). https://doi.org/10.1016/s0370-2693(98)00205-6
A.I. Ahmadov, C. Aydin and O. Uzun, “Bound state solution of the Schrödinger equation at finite temperature”, Journal of Physics: Conference Series, 1194, 012001(2019). https://doi.org/10.1088/1742-6596/1194/1/012001
M. Abu-Shady, “Multidimensional Schrödinger Equation and Spectral Properties of Heavy-Quarkonium Mesons at Finite Temperature”, Advances in Mathematical Physics, 2016, 1 (2016). https://doi.org/10.1155/2016/4935940
M.E. Naggar, N.I. Abou Salem, L.G. A. Shalaby, and A.M. Bourham, “The Equation of State for Non-ideal Quark Gluon Plasma”, Physical Science International Journal, 4(7), 912 (2014). https://doi.org/10.9734/PSIJ/2014/10296
W. Heisenberg, Letter to R. Peierls, in ’Wolfgang Pauli, Scientific Correspondence’, Vol. III, p.15, Ed. K. von Meyenn - 1985. Springer: Verlag) (1930).
H.S. Snyder, “Quantized Space-Time”, Phys. Rev., 71, 38 (1947). https://doi.org/10.1103/PhysRev.71.38,“The Electromagnetic Field in Quantized Space-Time”, Phys. Rev., 72, 68 (1947). https://doi.org/10.1103/PhysRev.72.68
A. Connes, M.R. Douglas and A. Schwarz, Noncommutative geometry and Matrix theory”, Journal of High Energy Physics, 02, 003 (1998). https://doi.org/10.1088/1126-6708/1998/02/003
N. Seiberg and E. Witten, String theory and noncommutative geometry”, Journal of High Energy Physics, 1999(09) 032 (1999). https://doi.org/10.1088/1126-6708/1999/09/032
S. Capozziello, G. Lambiase and G. Scarpetta, “Generalized Uncertainty Principle from Quantum Geometry”, Int. J. Theor. Phys., 39, 15 (2000). https://doi.org/10.1023/A:1003634814685
S. Doplicher, K., Fredenhagen, and J.E., “Spacetime quantization induced by classical gravity, Roberts”, Phys. Lett. B, 331(1- 2), 39 (1994). https://doi.org/10.1016/0370-2693(94)90940-7
E. Witten, “Reflections on the Fate of Spacetime”, Phys. Today, 49(4), 24 (1996). https://doi.org/10.1063/1.881493
A. Kempf, G. Mangano, and R.B. Mann, “Hilbert space representation of the minimal length uncertainty relation”, Phys. Rev. D, 52(2), 1108 (1995). https://doi.org/10.1103/physrevd.52.1108
F. Scardigli, “Some Heuristic Semi-classical Derivations of the Planck the Hawking effect and the Unruh effect. Il Nuovo Cimento B Series, 11 B (9), 1029 (1995). https://doi.org/10.1007/bf02726152
R.J. Adler, and D.I. Santiago, On gravity and the uncertainty principal”, Mod. Phys. Lett. A, 14(20), 1371 (1999). https://doi.org/10.1142/s0217732399001462
H. Benzaira, M. Meradc , T. Boudjedaad and A. Makhlouf “Relativistic Oscillators in a Noncommutative Space: a Path Integral Approach”, Z. Naturforsch. 67a, 77 (2012). https://doi.org/10.5560/ZNA.2011-0060
F. Scardigli, “Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment”, Phys. Lett. B, 452(1-2), 39 (1999). HTTPS://DOI.ORG/ 10.1016/s0370-2693(99)00167-7
P.M. Ho and H.C. Kao, Noncommutative Quantum Mechanics from Noncommutative Quantum Field Theory”, Phys. Rev. Lett., 88(15), 151602-1(2002). https://doi.org/10.1103/physrevlett.88.151602
O. Bertolami, G.J. Rosa, C.M.L. Dearagao, P. Castorina and D. Zappala, “Scaling variables and the relation between noncommutative parameters in noncommutative quantum mechanics”, Mod. Phys. Lett. A, 21(10), 795 (2006). https://doi.org/10.1142/s0217732306019840
P. Gnatenko, “Parameters of noncommutativity in Lie-algebraic noncommutative space”, Phys. Rev. D., 99(2), 026009-1(2019). https://doi.org/10.1103/physrevd.99.026009
E.E. N’Dolo, D.O. Samary, B. Ezinvi and M.N. Hounkonnou, Noncommutative Dirac and Klein–Gordon oscillators in the background of cosmic string: Spectrum and dynamics. International Journal of Geometric Methods in Modern Physics, 17(05), 2050078 (2020). https://doi.org/10.1142/S0219887820500784
O. Bertolami and P. Leal, Aspects of phase-space noncommutative quantum mechanics”, Phys. Lett. B., 750, 6 (2015). https://doi.org/10.1016/j.physletb.2015.08.024
M.A. De Andrade and C. Neves, “Noncommutative mapping from the symplectic formalism”, J. Math. Phys., 59(1), 012105 (2018). https://doi.org/10.1063/1.4986964
K.P. Gnatenko and V.M.Tkachuk, “Upper bound on the momentum scale in noncommutative phase space of canonical type”, EPL, 127(2), 20008 (2019). https://doi.org/ 10.1209/0295-5075/127/20008
K.P. Gnatenko and V.M. Tkachuk, “Composite system in rotationally invariant noncommutative phase space”, Int. J. Mod. Phys. A, 33(07), 1850037(2018). https://doi.org/10.1142/s0217751x18500379
A., Maireche, “Bound-state solutions of the modified Klien-Gordon and Schrödinger for arbitrary noncommutative quantum mechanics”, J. Phys. Stud. 25(1), 1002 (2021). https://doi.org/10.30970/jps.25.1002
A. Maireche, “A Theoretical Model of Deformed Klein-Gordon Equation with Generalized Modified Screened Coulomb Plus Inversely Quadratic Yukawa Potential in RNCQM Symmetries”, Few-Body Syst. 62, 12 (2021). https://doi.org/10.1007/s00601-021-01596-2
A., Maireche, “ The Relativistic and Nonrelativistic Solutions for the Modified Unequal Mixture of Scalar and Time-Like Vector Cornell Potentials in the Symmetries of Noncommutative Quantum Mechanics”, Jordan Journal of Physics, 14(1), 59 (2021). https://doi.org/10.47011/14.1.6
J. Gamboa, M. Loewe and J.C. Rojas, Noncommutative quantum mechanics Phys. Rev. D., 64, 067901(2001). https://doi.org/10.1103/PhysRevD.64.067901.
E.F. Djemaï and H. Smail, “On Quantum Mechanics on Noncommutative Quantum Phase Space”, Commun. Theor. Phys. (Beijing, China), 41(6), 837 (2004). https://doi.org/10.1088/0253-6102/41/6/837
Y. Yi, K. Kang, W. Jian-Hua and C. Chi-Yi, Spin-1/2 relativistic particle in a magnetic field in NC phase space”, Chin. Phys. C., 34(5), 543 (2010). https://doi.org/10.1088/1674-1137/34/5/005G.
G. Valencia-Ortega and L.A. Arias-Hernandez, “Thermodynamic properties of diatomic molecule systems under SO (2,1)-anharmonic Eckart potential International”, Journal of Quantum Chemistry, 118(14), e25589(2018). https://doi.org/10.1002/qua.25589
A. Maireche, “Solutions of Klein-Gordon equation for the modified central complex potential in the symmetries of noncommutative quantum mechanics”, Sri Lankan Journal of Physics, 22(1), 1 (2021). http://doi.org/10.4038/sljp.v22i1.8079
A. Maireche, “Theoretical Investigation of the Modified Screened cosine Kratzer potential via Relativistic and nonrelativistic treatment in the NCQM symmetries”, Lat. Am. J. Phys. Educ., 14 (3), 3310-1(2020).
A. Maireche, “Modified Unequal Mixture Scalar Vector Hulthén–Yukawa Potentials Model as a Quark–Antiquark Interaction and Neutral Atoms via Relativistic Treatment Using the Improved Approximation of the Centrifugal Term and Bopp’s Shift Method”, Few-Body Syst., 61, 30 (2020). https://doi.org/10.1007/s00601-020-01559-z
O. Bertolami, J.G. Rosa, C.M.L. De Aragão, P. Castorina and D. Zappalà, “Noncommutative gravitational quantum well Phys”, Rev. D., 72(2), 025010-1(2005). https://doi.org/10.1103/PhysRevD.72.025010
J. Zhang, “Coherent states of the deformed Heisenberg–Weyl algebra in non-commutative space”, Phys. Lett. B, 584(1-2), 204 (2004). https://doi.org/10.1016/j.physletb.2005.03.040
J. F. G. Santos, “Heat flow and noncommutative quantum mechanics in phase-space,” J. Math. Phys., 61, 122101, (2020). https://doi.org/10.1063/5.0010076.
E.M.C. Abreu, C. Neves and OW. liveira, “Noncommutativity from the symplectic point of view”, Int. J. Mod. Phys. A., 21, 5359 (2006). https://doi.org/10.1142/S0217751X06034094.
E.M.C. Abreu, J.A. Neto, A.C.R Mendes, C. Neves, W. Oliveira and M.V. Marcial, “Lagrangian formulation for noncommutative nonlinear systems Int. J. Mod. Phys. A, 27, 1250053(2012). https://doi.org/10.1142/S0217751X12500534
J. Wang and K. Li, “The HMW effect in noncommutative quantum mechanics”, J. Phys. A Math. Theor., 40(9), 2197 (2007). https://doi.org/10.1088/1751-8113/40/9/021
K. Lin and J. Wang, “The topological AC effect on non-commutative phase space”, Eur. Phys. J. C, 50(4), 1007 (2007). https://doi.org/10.1140/epjc/s10052-007-0256-0
A. Maireche, A theoretical investigation of nonrelativistic bound state solution at finite temperature using the sum of modified Cornell plus inverse quadratic potential”, Sri Lankan Journal of Physics, 21, 11-35(2020). http://doi.org/10.4038/sljp.v21i1.8069
A. Maireche, “Extended of the Schrödinger equation with new Coulomb potentials plus linear and Harmonic radial terms in the symmetries of noncommutative quantum mechanics”, J. Nano-Electron. Phys., 10(6), 06015-1 (2018). https://doi.org/10.21272/jnep.10(6).06015
A. Maireche, “The Klein-Gordon equation with modified Coulomb plus inverse-square potential in the noncommutative three-dimensional space”, Mod. Phys. Lett. A, 35(5), 052050015 (2020). https://doi.org/10.1142/s0217732320500157
A. Maireche,“The Klein–Gordon equation with modified Coulomb potential plus inverse-square–root potential in three-dimensional noncommutative space”, To Physics Journal, 3, 186 (2019).
A. Maireche, “Bound-state solutions of Klien-Gordon and Schrödinger equations for arbitrary l-state with combination of Hulthén and Kratzer potentials”, Afr. Rev Phys. 15 :003, 19 (2020).
H. Motavalli and A.R. Akbarieh, “Klien-Gordon equation for the Coulomb potential in noncommutative space”, Phys. Lett. A, 25(29), 2523 (2010). https://doi.org/10.1142/s0217732310033529
M. Darroodi, H. Mehraban and H. Hassanabadi, “The Klein–Gordon equation with the Kratzer potential in the noncommutative space”, Mod. Phys. Lett. A, 33 (35), 1850203 (2018). https://doi.org/10.1142/s0217732318502036
A. Maireche, “A New Theoretical Investigation of the Modified Equal Scalar and Vector Manning-Rosen Plus Quadratic Yukawa Potential Within the Deformed Klein-Gordon and Schrödinger Equations Using the Improved Approximation of the Centrifugal Term and Bopp’s Shift Method in RNCQM and NRNCQM Symmetries”, Spin J. 11, No. 4, 2150029 (2021). https://doi.org/10.1142/S2010324721500296
A. Maireche, “Heavy quarkonium systems for the deformed unequal scalar and vector Coulomb–Hulthén potential within the deformed effective mass Klein–Gordon equation using the improved approximation of the centrifugal term and Bopp’s shift method in RNCQM symmetries”, International Journal of Geometric Methods in Modern Physics, 18(13), 2150214 (2021). https://doi.org/10.1142/S0219887821502133
A. Maireche, “A new theoretical study of the deformed unequal scalar and vector Hellmann plus modified Kratzer potentials within the deformed Klein–Gordon equation in RNCQM symmetries”, Modern Physics Letters A, 36(33), 2150232. (2021). https://doi.org/10.1142/S0217732321502321
A. Maireche, “A theoretical study of the modified equal scalar and vector Manning-Rosen potential within the deformed Klein-Gordon and Schrödinger in RNCQM and NRNCQM symmetries”, Rev. Mex. Fís., 67, no. 5 050702 1–18, (2021). https://doi.org/10.31349/revmexfis.67.050702
A. Maireche, “The Investigation of Approximate Solutions of Deformed Klein–Gordon and Schrödinger Equations Under Modified More General Exponential Screened Coulomb Potential Plus Yukawa Potential in NCQM Symmetries”, Few-Body Syst 62, 66 (2021). https://doi.org/10.1007/s00601-021-01639-8
A. Maireche, “The Relativistic and Nonrelativistic Solutions for the Modified Unequal Mixture of Scalar and Time-Like Vector Cornell Potentials in the Symmetries of Noncommutative Quantum Mechanics”, Jordan Journal of Physics, 14(1), 59 (2021). https://doi.org/10.47011/14.1.6
A. Maireche, “A Theoretical Model of Deformed Klein–Gordon Equation with Generalized Modified Screened Coulomb Plus Inversely Quadratic Yukawa Potential in RNCQM Symmetries”, Few-Body Syst, 62, 12 (2021). https://doi.org/10.1007/s00601-021-01596-2
A. Maireche, “New bound-state solutions of the deformed Kilien-Gordon and Schrödinger equations for arbitrary l-state with modified equal vector and scalar Manning-Rosen plus a class of Yukawa potential in RNCQM and NRNCQM symmetries”, Journal of Physical Studies, 25(4). 4301 (2021). https://doi.org/10.30970/jps.25.4301
A. Maireche, “Bound-state solutions of the modified Klein-Gordon and Schrödinger equations for arbitrary l-state with the modified morse potential in the symmetries of noncommutative quantum mechanics”, Journal of Physical Studies, 25(1), 1002 (2021). https://doi.org/10.30970/jps.25.1002
A. Saidi and M.B. Sedra, “Spin-one (1 + 3)-dimensional DKP equation with modified Kratzer potential in the non-commutative space”, Phys. Lett. A, 35(5), 2050014(2020). https://doi.org/10.1142/s0217732320500145
A. Maireche, “A model of modified Klien-Gordon equation with modified scalar-vector Yukawa potential”, Afr. Rev Phys. 15: 0001, 1 (2020).
A. Maireche,“ The relativistic treatment of hydrogen-like and neutral atoms subjected to the generalized perturbed Yukawa potential with centrifugal barrier in the symmetries of noncommutative quantum mechanics”, International Journal of Geometric Methods in Modern Physics, 17(5), 2050067 (2020). https://doi.org/10.1142/S021988782050067X
A. Maireche, “A Recent Study of Excited Energy Levels of Diatomics for Modified more General Exponential Screened Coulomb Potential: Extended Quantum Mechanics”, J. Nano- Electron. Phys., 9 No 3, 03021 (2017). https://doi.org/10.21272/jnep.9(3).03021
A. Maireche, “A new study of energy levels of hydrogenic atoms and some molecules for new more general exponential screened coulomb potential”, Open Acc J Math Theor Phy.,1(6), 232 (2018). https://doi.org/10.15406/oajmtp.2018.01.00040
A. Maireche, “A New Approach to the Approximate Analytic Solution of the Three-Dimensional Schrödinger Equation for Hydrogenic and Neutral Atoms in the Generalized Hellmann Potential Model”, Ukr. J. Phys., 65(11) 987 (2020). https://doi.org/10.15407/ujpe65.11.987
A. Maireche,“Effects of Two-Dimensional Noncommutative Theories on Bound States Schrödinger Diatomic Molecules under New Modified Kratzer-Type Interactions”, International Letters of Chemistry, Physics and Astronomy, 76, 1 (2017). https://doi.org/10.18052/www.scipress.com/ILCPA.76.1
A. Maireche, “Any L-states solutions of the modified Schrödinger equation with generalized Hellmann–Kratzer potential model in the symmetries of NRNCQM”, To Physics Journal, 4, 16 (2019).
L. Mezincescu, “Star Operation in Quantum Mechanics”, (2000). https://arxiv.org/abs/hep-th/0007046
L. Gouba, “A comparative review of four formulations of noncommutative quantum mechanics”, Int. J. Mod. Phys. A, 31(19), 1630025 (2016). https://doi.org/10.1142/s0217751x16300258
F. Bopp, “La mécanique quantique est-elle une mécanique statistique classique particulière”, Ann. Inst. Henri Poincaré, 15(2), 81 (1956). http://www.numdam.org/item/AIHP_1956__15_2_81_0.pdf
A. Maireche, “New Relativistic Bound States for Modified Pseudoharmonic Potential of Dirac Equation with Spin and Pseudo-Spin Symmetry in One-electron Atoms”, Afr. Rev Phys., 12: 0018, 130 (2017).
A. Maireche, “New Relativistic Atomic Mass Spectra of Quark (u, d and s) for Extended Modified Cornell Potential in Nano and Plank’s Scales J. Nano- Electron. Phys., 8 (1), 01020 (2016). https://doi.org/10.21272/jnep.8(1).01020
A. Maireche, “A New Relativistic Study for Interactions in One-electron atoms (Spin ½ Particles) with Modified Mie-type Potential J. Nano-Electron. Phys., 8(4), 04027 (2016). https://doi.org/10.21272/jnep.8(4(1)).04027
A. Maireche, “Investigations on the Relativistic Interactions in One-Electron Atoms with Modified Yukawa Potential for Spin 1/2 Particles”, International Frontier Science Letters, 11, 29 (2017). https://doi.org/10.18052/www.scipress.com/IFSL.11.29
A. Maireche, “On the interaction of an improved Schiöberg potential within the Yukawa tensor interaction under the background of deformed Dirac and Schrödinger equations”, Indian J Phys, 96(10), (2022). https://doi.org/10.1007/s12648-022-02433-w
A. Maireche, “New Approximate Solutions to a Spatially-Dependent Mass Dirac Equation for Modified Hylleraas Plus Eckart Potential with Improved Yukawa Potential as a Tensor in the DQM Framework”, Few-Body Syst, 63, 63 (2022). https://doi.org/10.1007/s00601-022-01766-w
A. Maireche, “Approximate Arbitrary k State Solutions of Dirac Equation with Improved Inversely Quadratic Yukawa Potential within Improved Coulomb-like Tensor Interaction in Deformation Quantum Mechanics Symmetries”, Few-Body Syst, 63, 54 (2022). https://doi.org/10.1007/s00601-022-01755-zS.
A. Maireche, “Approximate k-state solutions of the deformed Dirac equation in spatially dependent mass for the improved Eckart potential including the improved Yukawa tensor interaction in ERQM symmetries”, International Journal of Geometric Methods in Modern Physics 19, No. 06, 2250085 (2022). https://doi.org/10.1142/S0219887822500852
A. Maireche, “Relativistic symmetries of the deformed Dirac equation through the improved Hulthén plus a class Yukawa potential including a Coulomb-like in deformed quantum mechanics”, Journal of Physical Studies, 26(2), 2001-1 (2022). https://doi.org/10.30970/jps.26.2001
I.S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series and Products, 7th. ed.; Elsevier, edited by Alan Jeffrey (the University of Newcastle upon Tyne, England) and Daniel Zwillinger (Rensselaer Polytechnic Institute USA) 2007.
K. Bencheikh, S. Medjedel, and G. Vignale, “Current reversals in rapidly rotating ultracold Fermi gases”, Phys. Lett. A, 89(6), 063620 (2014). https://doi.org/10.1103/physreva.89.063620
M. Abu-Shady, T.A. Abdel-Karim, and S.Y. Ezz-Alarab, “Masses and thermodynamic properties of heavy mesons in the non-relativistic quark model using the Nikiforov–Uvarov method”, J. Egypt Math. Soc. 27, 14 (2019). https://doi.org/10.1186/s42787-019-0014-0
R. Rani, S.B. Bhardwaj, and F. Chand, “Mass Spectra of Heavy and Light Mesons Using Asymptotic Iteration Method”, Communications in Theoretical Physics, 70(2), 179 (2018). https://doi.org/10.1088/0253-6102/70/2/179
A. Maireche, “A New Asymptotic Study to the 3-Dimensional Radial Schrodinger Equation under Modified Quark-antiquark Interaction Potential J Nanosci Curr Res, 4(1), 131 (2019).
A. Maireche, “Heavy-light mesons in the symmetries of extended nonrelativistic quark model”, Yanbu Journal of Engineering and Science, 17, 51 (2019). https://doi.org/10.53370/001c.23732
A. Maireche, “Analytical Expressions to Energy Eigenvalues of the Hydrogenic Atoms and the Heavy Light Mesons in the Framework of 3D-NCPS Symmetries Using the Generalized Bopp's Shift Method”, Bulg. J. Phys. 49(3), 239 (2022). https://doi.org/10.55318/bgjp.2022.49.3.239
A. Maireche, “The Impact of Deformed Space-Phase Parameters into HAs and HLM Systems with the Improved Hulthén Plus Hellmann Potentials Model in the Presence of Temperature-Dependent Confined Coulomb Potential Within the Framework of DSE”. Rev. Mex. Fis., 68, no. 5 050702 1 (2022), https://doi.org/10.31349/RevMexFis.68.050702.
A. Maireche, “The investigation of approximate solutions of Deformed Schrödinger Equations for the Hydrogenic Atoms, Heavy Quarkonium Systems ( = , ) and Diatomic molecule Bound-State Problem under Improved Exponential, Generalized, Anharmonic Cornell potential Model in NCPS symmetries”, Lat. Am. J. Phys. Educ. 16(2), 2304-1 ( 2022).
A. Maireche, “Relativistic symmetries of bosonic particles and antiparticles in the background of the position-dependent mass for the improved deformed Hulthén plus deformed type-hyperbolic potential in 3D EQM symmetries”, East Eur. J. Phys. 4, 201 (2022). https://doi.org/10.26565/2312-4334-2022-4-21
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