Bound State Solution of Dirac Equation for Generalized Pöschl-Teller plus Trigomometric Pöschl-Teller Non- Central Potential Using SUSY Quantum Mechanics


  • S. Suparmi Physics Department, Sebelas Maret University Jalan Ir. Sutami 36 A, Surakarta 57126
  • C. Cari Physics Department, Sebelas Maret University Jalan Ir. Sutami 36 A, Surakarta 57126



bound state solution, Dirac equation, generalized Pöschl-Teller potential, non-central potentials, SUSY quantum mechanics, trigonometric Pöschl-Teller potential


The bound state solution of the Dirac equation for generalized Pöschl-
Teller and trigonometric Pöschl-Teller non-central potentials was obtained using
SUSY quantum mechanics and the idea of shape invariance potential. The
approximate relativistic energy spectrum was expressed in the closed form. The
radial and polar wave functions were obtained using raising and lowering of
radial and polar operators. The orbital quantum numbers were found from the
polar Dirac equation, which was solved using SUSY quantum mechanics and the
idea of shape invariance.


Alhaidari, A.D., Relativistic Extension of Shape Invariant Potential, J.Phys. A, 34(46), pp. 9827- 9833, 2001.

Hu, X.Q., Luo, G., Wu, Z.M., Niu, L.B., & Ma, Y., Solving Dirac Equation with New Ring-Shaped Non-Spherical Harmonic Oscillator Potential, Commun. Theor. Phys., 53, pp. 242-249, 2010.

Hamzavi, M. & Rajabi, A.A., Spin and Pseudospin Symmetries with Trigonometric Pschl-Teller Potential including Tensor Coupling, Adv.High En. Phys., 2013, pp. 1-12, 2013. ID 196986.

Soylu, A., Bayrak, O. & Boztosun, I., State Solutions of the Dirac equation for the Eckart potential with pseudospin and spin symmetry, J.Phys. A: Math. Theor., 41, pp. 065308-1-8, 2008.

Onate, C.A., Oyewumi, K.J. & Falaye, B.J., An Approximate Solution of Dirac Equation for Second Pschl-Teller like Scalar and Vector Potentials with a Coulomb Tensor Interaction, Af. Rev. Phys., 8(0020),pp. 129-137, 2013.

Sukumar, C.V., Supersymmetry and the Dirac equation for a central Coulomb field, J. Phys. A: Math. Gen., 18, p. L697, 1985.

Ginocchio, J.N., Relativistic Symmetries in Nuclei and Hadrons, Phys.Rep., 414, pp. 165-262, 2005.

Ginocchio, J.N. & Madland, D.G., Pseudospin Symmetry and Relativistic Single-Nucleon Wave Functions, Phys. Rev. C, 57(3), pp. 1167-1173, 1998.

Alberto, P., Fiolhais, M., Malheiro, M., Delfino, A. & Chiapparini, M., Pseudospinsymmetry as a Relativistic Dynamical Symmetry in the Nucleus,Phys. Rev. C, 65, pp. 034307-1-9, 2002.

Troltenier, D., Nazarewicz, W., Szymanski, Z. & Draayer, J.P., On the Validity of the Pseudo-Spin Concept for Axially Symmetric Deformed Nuclei, Nucl. Phys. A, 567, pp. 591-610, 1994.

Stuchbery, A.E., Magnetic Properties of Rotational States in the Pseudo-Nilsson Model, Nucl. Phys. A, 700, pp. 83-116, 2002.

Zou, X., Yi, L.Y. & Jia, C.S., Bound States of The Dirac Equation with Vector and Scalar Eckart Potentials, Phys. Lett. A, 346(1-3), pp. 54-64, 2005.

Ikhdair, S.M. & Sever, R., Exact Bound States of the D-dimensional Klein-Gordon Equation with Equal Scalar and Vector Ring-Shaped Pseudoharmonic Potential, Int. J. Mod. Phys. C, 19(9), pp.1425-1442, 2008.

Zhou, F., Wu, Y. & Guo, J.Y., Solutions of Dirac Equation for Makarov Potential with Pseudospin Symmetry, Commun. Theor. Phys. (Beijing, China), 52, pp. 813-816, 2009.

Debnath, S. & Biswas, B., Analytical Solutions of the Klein-Gordon Equation for Rosen-Morse Potential via Asymptotic Iteration Method, EJTP, 9(26), 191-198, 2012.

Chen, G., Chen, Z.D. & Lou, Z.M., Bound State of Klein-Gordon and Dirac Equation for Scalar and Vector Pseudoharmonic Oscillator Potentials, Chin. Phys., 13(2), pp. 279-284, 2004.

Suparmi, A. & Cari, C., Solution of Dirac Equation for q-Deformed Eckart Potential with Yukawa-type Tensor Interaction for Spin and Pseudospin Symmetry Using Romanovski Polynomial, At. Ind. J., 39(3), pp. 112-123, 2013.

Cari, C., Suparmi, A., Deta, U.A. & Werdiningsih, I.S., Solution of Dirac Equation for Cotangent Potential with Coulomb-type Tensor Interaction for Spin and Pseudospin Symmetries Using Romanovski Polynomial, Makara. J. of Sci., 17(3), pp. 93-102, 2013.

Suparmi, A., Cari, C. & Deta, U.A., Exact Solution of Dirac equation for Scarf Potential with New Tensor Coupling Potential For Spin and Pseudospin Symmetries Using Romanovski Polynomials, Chin. Phys. B, 23(9), pp. 090304-1-10, 2014.

Hall, R.L. & Yesiltas, -., Supersymmetric Analysis for the Dirac Equation with Spin-Symmetric and Pseudo-Spin-Symmetric Interactions, arXiv:1006.4628v1 [math-ph], pp. 1-10, 2010.

Ikot, A.N. & Akpabio, L.E., Approximate Solution of the Schrodinger Equation with Rosen-Morse Potential Including the Centrifugal Term, App. Phys. Res., 2(2), pp. 202-208, 2010.

Agboola, D., Dirac Equation with Spin Symmetry for the Modified Pschl-Teller Potential in D Dimensions, Pramana, 76(6), pp. 875-885, 2011.

Greene, R.I. & Aldrich, C., Variational Wave Functions for a Screened Coulomb Potential, Phys. Rev. A, 14, pp. 2363, 1976.

Derezinski, J. & Wrochna, M., Exactly Solvable Schrdinger Operators, arXiv:1009.0541v2 [math-ph] , 2010.

Flugge, S., Practical Quantum Mechanics I, Springer, Berlin, 1994.

Witten, E., Dynamical Breaking of Supersymmetry, Nucl. Phys. B, 188, pp. 513-554, 1981.

Gendenshtein, L.E., Derivation of Exact Spectra of Schrodinger Equation by Means Supersymmetry, JETP Lett., 38, pp. 356-359, 1983.

Dutt, R., Khare, A. & Sukhatme, U.P., Supersymmetry, Shape Invariance and Exactly Solvable Potentials, Am. J. Phys., 56, pp. 163-168, 1988.

Dutt, R., Gangopadhyaya, A. & Sukhatme, U.P., Non-central potentials and Spherical Harmonics Using Supersymmetry and Shape Invariance, Am. J. Phys., 65(5), pp. 400-403, 1997.

Gangopadhyaya, A., Mallow, J.V., Rasinariu, C. & U.P. Sukhatme, Exact Solution of the Schrodinger Equation: Connection between Supersymmetric Quantum Mechanics and Spectrum Generating Algebras, Chin. J. Phys., 39(2), pp. 101-121, 2001.

Suparmi, A., Cari, C., Handhika, J., Yanuarief, C. & Marini, H., Approximate Solution of Schrodinger Equation for Modified Pschl-Teller plus Trigonometric Rosen-Morse Non-Central Potentials in Terms of Finite Romanovski Polynomials, IOSR-JAP, 2(2), pp. 43-51, 2012.

Cari, C. & Suparmi, A., Approximate Solution of Schrodinger Equation for Trigonometric Scarf Potential with the Pschl-Teller Non-central Potential Using NU Method, IOSR-JAP,2(3), pp. 13-23, 2012.

Cari, C. & Suparmi, A., Solution of Schrodinger Equation for Three Dimensional Harmonics Oscillator plus Rosen-Morse Non-central Potential Using NU Method and Romanovski Polynomials, J. Phys.: Conf. Series, 423, pp. 012031-1-10, 2013.

Suparmi, A., Cari, C. & Handhika, J., Approximate Solution of Schrodinger Equation For Eckart Potential Combined With Trigonormetric Pschl-Teller Non-Central Potential Using Romanovski Polynomials, J. of Phys.: Conf. Series, 423, pp. 012039-1-11, 2013.

Saregar, A., Suparmi, A., Cari, C. & Yuliani, H., Analysis of Energy Spectra and Wave Function of Trigonometric Pschl-Teller plus Rosen-Morse Non-Central Potential Using Supersymmetric Quantum Mechanics Approach, Res. Inv.: Int. J. Eng. and Sci., 2(3), pp. 14-26, 2013.

Suparmi, A., Cari, C., & Yuliani, H., Energy Spectra and Wave Function Analysis of q-Deformed Modified Pschl-Teller and Hyperbolic Scarf II Potentials Using NU Method and a Mapping Method, Adv. Phys. Theor. Appl., 16, pp. 64-74, 2013.

Ikhdair, S.M. & Sever, R., Approximate Bound States of the Dirac Equation with Some physical Quantum Potentials, arXiv:1204.2700v2 [quant-ph], pp. 1-25, 2012

Khare, A. & Bhaduri, R.K., Supersymmetry Shape Invariance and Exactly Solvable Noncentral Potentials, arXiv:hep-th/9310104v1, pp. 1-16, 1993.

Cari, Quantum Mechanics-Solutions of Non-Central Potentials Using SUSY Quantum Mechanics, Hypergeometry Method, Nikiforov-Uvarov Method, and Romanovski Polynomial, p. 240, Surakarta UNS Press 2013.