ANALYTICAL SOLUTION OF THE POSITION DEPENDENT
MASS SCHRÖDINGER EQUATION WITH A HYPERBOLIC
TANGENT POTENTIAL
A. Kharab1, H. Eleuch2
1Department of Applied Sciences and Mathematics
Abu Dhabi University, Abu Dhabi 59911, UAE
2Institute for Quantum Science and Engineering
Texas A&M University, College Station
Texas 77843, USA
2Department of Applied Sciences and Mathematics
Abu Dhabi University, Abu Dhabi, UAE

Abstract


Abstract. An analytical solution of the position dependent mass Schrödinger equation with a hyperbolic tangent potential is presented. The state energy and the corresponding wave function are obtained using the Nikiforov-Uvarov method. The energy eigenvalues and eigenfunctions are discussed and results are presented for some values of potential parameters.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 32
Issue: 2
Year: 2019

DOI: 10.12732/ijam.v32i2.14

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References

  1. [1] R.N. Costa Filho, M.P. Almeida, G.A. Farias and J.S. Andrade, Jr., Displacement operator for quantum systems with position-dependent mass, Phys. Rev. A, 84 (2011), 050102(R).
  2. [2] V.D. Dinh, Well-posedness of nonlinear fractional Schr¨odinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31, No 4 (2018), 483–525; DOI: 10.12732/ijam.v31i4.1.
  3. [3] H. Eleuch, Y.V. Rostovtsev and M.S. Abdalla, Analytical solution for one-dimensional scattering problem, Optics Communications, 284 (2011), 5457.
  4. [4] H. Eleuch, Y.V. Rostovtsev and M.O. Scully, New analytical solution of Schr¨odinger equation, Europhys. EPL, 89 (2010), 50004.
  5. [5] N. Froman and P.O. Froman, JWKB Approximation, AbeBooks, NorthHolland, Amsterdam (1965).
  6. [6] T.K. Jana and P. Roy, Potential algebra approach to position-dependent mass Schr¨odinger equations, Europhys. Lett., 87, No 3 (2009), 30003.
  7. [7] P.K. Jha, H. Eleuch and Y.V. Rostovtsev, Analytical solution to position dependent mass Schr¨odinger equation, J. Mod. Optic., 58 (2011), 652–656.
  8. [8] H.A. Kramers, Wellenmechanik und halbzahlige quantisierung, Z. Physik, 39 (1926), 828–840.
  9. [9] H.W. Lee and M.O. Scully, TheWigner phase-space description of collision processes, Found. Phys., 13, No 1 (1983), 61–72.
  10. [10] S.H. Mazharimousavi, Revisiting the displacement operator for quantum systems with position-dependent mass, Phys. Rev. A, 85 (2011), 034102(R).
  11. [11] S. Meyur, Position dependent mass for the Hulth´en plus hyperbolic cotangent potential, Bulg. J. Phys., 38 (2011), 357–363.
  12. [12] O. Mustafa and S.H. Mazharimousavi, D-dimensional generalization of the point canonical transformation for a quantum particle with positiondependent mass, J. Phys. A, 39 (2006), 10537–10547.
  13. [13] A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathematical Physics, Birkhauser, Basel (1988).
  14. [14] E. Pozdeeva and A. Schulze-Halberg, Trace formula for Green’s functions of effective mass Schr¨odinger equations and N-th order Darboux transformations, Int. J. Mod. Phys. A, 23 (2008), 2635–2647.
  15. [15] R. Renan, M.H. Pacheco and A.C.S. Almeida, Treating some solid state problems with the Dirac equation, J. Phys. A, 33, No 50 (2000), L509.
  16. [16] R.W. Robinet, Quantum Mechanics, Oxford Univ. Press, Oxford (2006).
  17. [17] E.C. Romao, M.D. de Campos and L.F.M. de Moura, Solutions for problems in the finite amplitude wave propagations using the method of characteristics, Int. J. Appl. Math., 24, No 3 (2011), 339–347.
  18. [18] B. Sangar´e, A moving mesh method for the nonlinear Schr¨odinger equation, Int. J. Appl. Math., 29, No 1 (2016), 105-126; DOI:10.12732/ijam.v29i1.9.
  19. [19] A. Schmidt, Wave-packet revival for the Schr¨odinger equation with position-dependent mass, Phys. Lett. A, 353, No 6 (2006), 459–462.
  20. [20] E. Schr¨odinger, Quantisierung als eigenwertproblem, Ann. Phys. (Leipzig), 384, No 4 (1926), 361–376. ANALYTICAL SOLUTION OF THE POSITION DEPENDENT... 367
  21. [21] R. Sever, C. Tezcan, ¨O. Yesiltas and M. Bucurgat, Exact solution of effective mass Schr¨odinger equation for the Hulthen potential, Int. J. Theor. Phys., 47 (2008), 2243–2248.
  22. [22] E.L. Shishkina, Inversion of the mixed Riesz hyperbolic B-potentials, Int. J. Appl. Math., 30, No 6 (2017), 487–500, DOI: 10.12732/ijam.v30i6.3.
  23. [23] A. Sinha, Scattering states of a particle with position-dependent mass in a double heterojunction, Europhys. Lett., 96, No 2 (2011), 20008.
  24. [24] A. de Souza Dutra, Ordering ambiguity versus representation, J. Phys. A, 39, No 1 (2005), 203-210.
  25. [25] A. de Souza Dutra and C.A.S. Almeida, Exact solvability of potentials with spatially dependent effective masses, Phys. Lett. A, 275 (2000), 25–30.
  26. [26] C. Tezcan and R. Sever, A general approach for the exact solution of the Schr¨odinger equation, Int. J. Theor. Phys., 48 (2009), 337–357.
  27. [27] G. Wentzel, Eine verallgemeinerung der quantenbedingungen f¨ur die Zwecke der wellenmechanik, Z. Physik, 38 (1926), 518–529.