Construction of Exactly Solvable Ring-Shaped Potentials

Affiliation(s)

Bajali College, Pathsala-781325, Assam, India.

Department of Physics, Gauhati University, Guwahati-781014, Assam, India.

Bajali College, Pathsala-781325, Assam, India.

Department of Physics, Gauhati University, Guwahati-781014, Assam, India.

ABSTRACT

We propose a method for construction of exactly solvable ring-shaped potentials where the linear homogeneous second-order differential equation satisfied by special function is subjected to the extended transformation comprising a coordinate transformation and a functional transformation to retrieve the standard Schr?dinger polar angle equation form in non-relativistic quantum mechanics. By invoking plausible ansatze, exactly solvable ring-shaped potentials and corresponding angular wave functions are constructed. The method is illustrated using Jacobi and hypergeometric polynomials and the wave functions for the constructed ring-shaped potentials are normalized.

Cite this paper

A. Bharali and N. Singh, "Construction of Exactly Solvable Ring-Shaped Potentials,"*Journal of Modern Physics*, Vol. 4 No. 4, 2013, pp. 463-467. doi: 10.4236/jmp.2013.44065.

A. Bharali and N. Singh, "Construction of Exactly Solvable Ring-Shaped Potentials,"

References

[1] J. Sadeghi and B. Pourhassan, “Exact Solution of the Non-Central Modified Kratzer Plus a Ring-Shaped Like Potential by the Factorization Method,” Electronic Journal of Theoretical Physics, Vol. 5, No. 17, 2008, pp. 193-202.

[2] M. Zhang, B. An and H. Guo-Qing, “Exact Solutions of a New Coulomb Ring-Shaped Potential,” Journal of Mathematical Chemistry, Vol. 48, No. 4, 2010, pp. 876-882. doi:10.1007/s10910-010-9715-1

[3] L. Chetouani, L. Guechi and T. F. Hammann, “Exact Path Integral for the Ring Potential,” Physics Letters A, Vol. 125, No. 6-7, 1987, pp. 277-281.

[4] M. Zhang, G. Sun and S. Dong, “Exactly Complete Solutions of the Schrodinger Equation with a Spherically Harmonic Oscillatory Ring-Shaped Potential,” Physics Letters A, Vol. 374, No. 5, 2010, pp. 704-708. doi:10.1016/j.physleta.2009.11.072

[5] S. Ikhdair, “Exact Solutions of the D-Dimensional Schrodinger Equation for a Pseudo-Coulomb Potential Plus Ring-Shaped Potential,” Chinese Journal of Physics, Vol. 46, No. 3, 2008, pp. 291-306.

[6] A. Antia, N. Ikot and L. Akpatio, “Exact Solutions of the Schrodinger Equation with Manning-Rosen Potential Plus a Ring-Shaped Like Potential by Nikiforov-Uvarov Method,” European Journal of Scientific Research, Vol. 45, No. 1, 2010 pp. 107-118.

[7] B. Gonul and I. Zorba, “Supersymmetric Solutions of Non- Central Potentials,” Physics Letters A, Vol. 269, No. 2-3, 2000, pp. 83-88. doi:10.1016/S0375-9601(00)00252-8

[8] S. A. S. Ahmed, “A Transformation Method of Generating Exact Analytic Solutions of the Schrodinger Equation,” International Journal of Theoretical Physics, Vol. 36, No. 8, 1997, pp. 1893-1905. doi:10.1007/BF02435851

[9] C. Chen, C. Liu and F. Lu, “Exact Solutions of Schrodinger Equation for the Makarov Potential,” Physics Letters A, Vol. 374, No. 11-12, 2010, pp. 1346-1349. doi:10.1016/j.physleta.2010.01.018

[10] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” Dover Publications, New York, 1978.

[11] D. E. Alvarez-Castillo and M. Kirchbach, “Exact Spectrum and Wave Functions of the Hyperbolic Scarf Potential in Terms of Finite Romanovski Polynomials,” Revista Mexicana de Fisica, Vol. E53, No. 2, 2007, pp. 143-154.

[1] J. Sadeghi and B. Pourhassan, “Exact Solution of the Non-Central Modified Kratzer Plus a Ring-Shaped Like Potential by the Factorization Method,” Electronic Journal of Theoretical Physics, Vol. 5, No. 17, 2008, pp. 193-202.

[2] M. Zhang, B. An and H. Guo-Qing, “Exact Solutions of a New Coulomb Ring-Shaped Potential,” Journal of Mathematical Chemistry, Vol. 48, No. 4, 2010, pp. 876-882. doi:10.1007/s10910-010-9715-1

[3] L. Chetouani, L. Guechi and T. F. Hammann, “Exact Path Integral for the Ring Potential,” Physics Letters A, Vol. 125, No. 6-7, 1987, pp. 277-281.

[4] M. Zhang, G. Sun and S. Dong, “Exactly Complete Solutions of the Schrodinger Equation with a Spherically Harmonic Oscillatory Ring-Shaped Potential,” Physics Letters A, Vol. 374, No. 5, 2010, pp. 704-708. doi:10.1016/j.physleta.2009.11.072

[5] S. Ikhdair, “Exact Solutions of the D-Dimensional Schrodinger Equation for a Pseudo-Coulomb Potential Plus Ring-Shaped Potential,” Chinese Journal of Physics, Vol. 46, No. 3, 2008, pp. 291-306.

[6] A. Antia, N. Ikot and L. Akpatio, “Exact Solutions of the Schrodinger Equation with Manning-Rosen Potential Plus a Ring-Shaped Like Potential by Nikiforov-Uvarov Method,” European Journal of Scientific Research, Vol. 45, No. 1, 2010 pp. 107-118.

[7] B. Gonul and I. Zorba, “Supersymmetric Solutions of Non- Central Potentials,” Physics Letters A, Vol. 269, No. 2-3, 2000, pp. 83-88. doi:10.1016/S0375-9601(00)00252-8

[8] S. A. S. Ahmed, “A Transformation Method of Generating Exact Analytic Solutions of the Schrodinger Equation,” International Journal of Theoretical Physics, Vol. 36, No. 8, 1997, pp. 1893-1905. doi:10.1007/BF02435851

[9] C. Chen, C. Liu and F. Lu, “Exact Solutions of Schrodinger Equation for the Makarov Potential,” Physics Letters A, Vol. 374, No. 11-12, 2010, pp. 1346-1349. doi:10.1016/j.physleta.2010.01.018

[10] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” Dover Publications, New York, 1978.

[11] D. E. Alvarez-Castillo and M. Kirchbach, “Exact Spectrum and Wave Functions of the Hyperbolic Scarf Potential in Terms of Finite Romanovski Polynomials,” Revista Mexicana de Fisica, Vol. E53, No. 2, 2007, pp. 143-154.