Table of Contents Author Guidelines Submit a Manuscript
Advances in High Energy Physics
Volume 2016, Article ID 8710604, 12 pages
http://dx.doi.org/10.1155/2016/8710604
Research Article

Unified Treatment of a Class of Spherically Symmetric Potentials: Quasi-Exact Solution

Department of Physics, University of Guilan, Rasht 41635-1914, Iran

Received 12 May 2016; Accepted 19 June 2016

Academic Editor: Shi-Hai Dong

Copyright © 2016 H. Panahi and M. Baradaran. 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.

Linked References

  1. G. Lèvai, “On some exactly solvable potentials derived from supersymmetric quantum mechanics,” Journal of Physics A: Mathematical and General, vol. 25, no. 9, pp. L521–L524, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  2. F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Physics Reports, vol. 251, no. 5-6, pp. 267–385, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. S.-H. Dong, Factorization Method in Quantum Mechanics, vol. 150 of Fundamental Theories of Physics, Springer, Dordrecht, The Netherlands, 2007. View at MathSciNet
  4. M. Azizi, N. Salehi, and A. A. Rajabi, “Exact solution of the dirac equation for the yukawa potential with scalar and vector potentials and tensor interaction,” ISRN High Energy Physics, vol. 2013, Article ID 310392, 6 pages, 2013. View at Publisher · View at Google Scholar
  5. H. Panahi and L. Jahangiri, “Exact solution of the curved Dirac equation in polar coordinates: master function approach,” Advances in High Energy Physics, vol. 2015, Article ID 612757, 8 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  6. L. D. Landau and E. M. Lifshitz, Quantum Mechanics-Nonrelativistic Theory, Pergamon, New York, NY, USA, 1977.
  7. L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, “Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations,” Physical Review D, vol. 65, no. 12, Article ID 125027, 8 pages, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  8. G. Pöschl and E. Teller, “Bemerkungen zur Quantenmechanik des anharmonischen Oszillators,” Zeitschrift für Physik, vol. 83, no. 3-4, pp. 143–151, 1933. View at Publisher · View at Google Scholar · View at Scopus
  9. M. M. Nieto, “Exact wave-function normalization constants for the B0tanhz-U0cosh-2z and Pöschl-Teller potentials,” Physical Review A, vol. 17, no. 4, pp. 1273–1283, 1978. View at Publisher · View at Google Scholar
  10. D. Griffiths, Introduction to Quantum Mechanics, Pearson, Hoboken, NJ, USA, 1995.
  11. S. M. Al-Jaber, “Hydrogen atom in N dimensions,” International Journal of Theoretical Physics, vol. 37, no. 4, pp. 1289–1298, 1998. View at Publisher · View at Google Scholar
  12. S. H. Dong, Wave Equations in Higher Dimensions, Springer, New York, NY, USA, 2011.
  13. P. M. Morse, “Diatomic molecules according to the wave mechanics. II. Vibrational levels,” Physical Review, vol. 34, no. 1, pp. 57–64, 1929. View at Publisher · View at Google Scholar
  14. G. Chen, “The exact solutions of the Schrödinger equation with the Morse potential via Laplace transforms,” Physics Letters A, vol. 326, no. 1-2, pp. 55–57, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  15. N. Rosen and P. M. Morse, “On the vibrations of polyatomic molecules,” Physical Review, vol. 42, no. 2, pp. 210–217, 1932. View at Publisher · View at Google Scholar · View at Scopus
  16. M. F. Manning and N. Rosen, “A potential function for the vibrations of diatomic molecules,” Physical Review, vol. 44, p. 953, 1933. View at Google Scholar
  17. S.-H. Dong and J. García-Ravelo, “Exact solutions of the s-wave Schrödinger equation with Manning–Rosen potential,” Physica Scripta, vol. 75, no. 3, pp. 307–309, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  18. T. Tietz, “Potentialenergy function for diatomic molecules,” Journal of Chemical Physics, vol. 38, no. 12, pp. 3036–3037, 1963. View at Publisher · View at Google Scholar
  19. C. Eckart, “The penetration of a potential barrier by electrons,” Physical Review, vol. 35, no. 11, pp. 1303–1309, 1930. View at Publisher · View at Google Scholar · View at Scopus
  20. F. L. Scarf, “New soluble energy band problem,” Physical Review, vol. 112, pp. 1137–1140, 1958. View at Google Scholar · View at MathSciNet
  21. A. K. Bose, “A class of solvable potentials,” Il Nuovo Cimento, vol. 32, no. 3, pp. 679–688, 1964. View at Publisher · View at Google Scholar
  22. J. N. Ginocchio, “A class of exactly solvable potentials. I. One-dimensional Schrödinger equation,” Annals of Physics, vol. 152, no. 1, pp. 203–219, 1984. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. M. A. Shifman, “New findings in quantum mechanics (partial algebraization of the spectral problem),” International Journal of Modern Physics A, vol. 4, no. 12, pp. 2897–2952, 1989. View at Publisher · View at Google Scholar
  24. A. G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics, IOP, Bristol, UK, 1994. View at MathSciNet
  25. A. V. Turbiner, “Lie-algebras and linear operators with invariant subspaces,” AMS eBooks: Contemporary Mathematics, vol. 160, p. 263, 1994. View at Publisher · View at Google Scholar
  26. A. González-López, N. Kamran, and P. J. Olver, “Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators,” Communications in Mathematical Physics, vol. 153, no. 1, pp. 117–146, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  27. S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, and V. S. Varma, “Eigenvalues of λx2m anharmonic oscillators,” Journal of Mathematical Physics, vol. 14, no. 9, pp. 1190–1195, 1973. View at Publisher · View at Google Scholar
  28. A. Salam and J. Strathdee, “Momentum-space behavior of integrals in nonpolynomial lagrangian theories,” Physical Review D, vol. 1, no. 12, pp. 3296–3312, 1970. View at Publisher · View at Google Scholar · View at Scopus
  29. A. K. Mitra, “On the interaction of the type λx2/1+gx2,” Journal of Mathematical Physics, vol. 19, article 2018, 1978. View at Publisher · View at Google Scholar
  30. G. P. Flessas, “On the Schrödinger equation for the x2 + λx2(1 + gx2) interaction,” Physics Letters A, vol. 83, no. 3, pp. 121–122, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  31. H. Risken and H. D. Vollmer, “The influence of higher order contributions to the correlation function of the intensity fluctuation in a Laser near threshold,” Zeitschrift für Physik, vol. 201, no. 3, pp. 323–330, 1967. View at Publisher · View at Google Scholar · View at Scopus
  32. H. Haken, Laser Theory, Springer, Berlin, Germany, 1970.
  33. M. Znojil, “Potential r2+λr2/(1+gr2) and the analytic continued fractions,” Journal of Physics A: Mathematical and General, vol. 16, no. 2, p. 293, 1983. View at Publisher · View at Google Scholar
  34. S. K. Bose and N. Varma, “Exact solution of the Schrödinger equation for the central nonpolynomial potential Vr=r2+λr2/1+gr2 in two and three dimensions,” Physics Letters A, vol. 141, no. 3-4, pp. 141–146, 1989. View at Publisher · View at Google Scholar
  35. B. Roy, R. Roychoudhury, and P. Roy, “Shifted 1/N expansion approach to the interaction V(r)=r2+λr2/(1+gr2),” Journal of Physics A: Mathematical and General, vol. 21, no. 7, p. 1579, 1988. View at Publisher · View at Google Scholar
  36. D. Agboola and Y.-Z. Zhang, “Unified derivation of exact solutions for a class of quasi-exactly solvable models,” Journal of Mathematical Physics, vol. 53, no. 4, 042101, 13 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. M. Znojil, “Analytic green function and bound states for the screened coulomb potential V(r) = F/r2 + G/r + H/(r + z2),” Physics Letters A, vol. 94, no. 3-4, pp. 120–124, 1983. View at Publisher · View at Google Scholar
  38. Y.-Z. Zhang, “Exact polynomial solutions of second order differential equations and their applications,” Journal of Physics A: Mathematical and Theoretical, vol. 45, no. 6, Article ID 065206, 20 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  39. M. E. Portnoi and I. Galbraith, “Variable-phase method and Levinson's theorem in two dimensions: application to a screened Coulomb potential,” Solid State Communications, vol. 103, no. 6, pp. 325–329, 1997. View at Publisher · View at Google Scholar · View at Scopus
  40. G. C. Maitland, M. Rigby, E. B. Smith, and W. A. Wakeham, Intermolecular Forces, Oxford University Press, Oxford, UK, 1987.
  41. R. J. LeRoy and W. Lam, “Near-dissociation expansions in the spectroscopic determination of diatom dissociation energies: method, and application to BeAr+,” Chemical Physics Letters, vol. 71, no. 3, pp. 544–548, 1970. View at Google Scholar
  42. E. Vogt and G. H. Wannier, “Scattering of ions by polarization forces,” Physical Review, vol. 95, no. 5, pp. 1190–1198, 1954. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  43. L. D. Laudau and E. M. Lifshitz, Quantum Mechanics, Pergamon, Oxford, UK, 1977.
  44. S. H. Dong, “Schrödinger equation with the potential Vr=Ar-4+Br-3+Cr-2+Dr-1,” Physica Scripta, vol. 64, no. 4, pp. 273–276, 2001. View at Publisher · View at Google Scholar
  45. S.-H. Dong, Z.-Q. Ma, and G. Esposito, “Exact solutions of the Schrödinger equation with inverse-power potential,” Foundations of Physics Letters, vol. 12, no. 5, pp. 465–474, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  46. S.-H. Dong, “The Ansatz method for analyzing Schrödinger's equation with three anharmonic potentials in D dimensions,” Foundations of Physics Letters, vol. 15, no. 4, pp. 385–395, 2002. View at Publisher · View at Google Scholar · View at MathSciNet