Advertisement

Journal of Mathematical Chemistry

, Volume 51, Issue 3, pp 954–975 | Cite as

A simple and effective technique to variationally interpret the structure of SUSY partners of mirror image potentials

  • Neetik Mukherjee
Original Paper
  • 189 Downloads

Abstract

Precise supersymmetric partner potentials can be generated for exactly solvable problems of the stationary Schrödinger equation. Construction of isospectral potential is not always possible for exactly solvable systems. This is a restriction, as most problems are not exactly solvable. Employment of mirror-image property can help to construct an exact isospectral partner of that potential. These potentials have chemical relevance to enantiomers. In this paper, we present a formulation as modelling to explore the form of SUSY pair of these potentials. Through polynomial fit, we correlate all possible basic SUSY partners and optimise it to best fit polynomial to present a typical energy value of N = 50.

Keywords

SUSY quantum mechanics Mirror image potentials Isospectrality Enantiomers 

Notes

Acknowledgments

Neetik Mukherjee is greatful to CSIR, INDIA for financial support. He also greatfully acknowledges the support of computational infrastructure of developed by Prof. S. N. Sarkar (Dept of Electronic Science, University of Calcutta) in his fibre optics laboratory as well as helpful and critical discussion with him.

References

  1. 1.
    L.F. Urrutia, E. Hernández. Phys. Rev. Lett. 51, 755 (1983) and references thereinGoogle Scholar
  2. 2.
    M.M. Nieto, Phys. Lett. B 145, 208 (1984)CrossRefGoogle Scholar
  3. 3.
    A.A. Andrianov, N.B. Borisov, M.V. Ioffe, Phys. Lett. A 105, 19 (1984)CrossRefGoogle Scholar
  4. 4.
    C.V. Sukumar, J. Phys. A 18, L57 (1985)CrossRefGoogle Scholar
  5. 5.
    C.V. Sukumar, J. Phys. A 18, 2917 (1984)CrossRefGoogle Scholar
  6. 6.
    R. Dutt, A. Khare, U.P. Sukhatme, Am. J. Phys. 56, 163 (1984)CrossRefGoogle Scholar
  7. 7.
    F. Cooper, A. Khare, U.P. Sukhatme, Phys. Rep. 251, 267 (1984)CrossRefGoogle Scholar
  8. 8.
    F. Cooper, A. Khare, U.P. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singapore, 2001)CrossRefGoogle Scholar
  9. 9.
    A.R.P. Rau, J. Phys. A 37, 10421 (2004)CrossRefGoogle Scholar
  10. 10.
    W.Y. Keung, E. Kovacs, U.P. Sukhatme, Phys. Rev. Lett. 60, 41 (1988)CrossRefGoogle Scholar
  11. 11.
    A. Gangopadhyay, P.K. Panigrahi, U.P. Sukhatme, Phys. Rev. A 47, 2720 (1984)CrossRefGoogle Scholar
  12. 12.
    F. Cooper, J. Dawson, H. Shepard, Phys. Lett. A 187, 140 (1994)CrossRefGoogle Scholar
  13. 13.
    D.J.C. Fernandez, V. Hussin, B. Mielnik, Phys. Lett. A 244, 309 (1998)CrossRefGoogle Scholar
  14. 14.
    J.J. Peña, G. Ovando, D. Morales-Guzmán, J. Morales, Int. J. Quantum Chem. 85, 244 (2004)CrossRefGoogle Scholar
  15. 15.
    A. Khare, U. Sukhatme, J. Phys. A 37, 10037 (2004)CrossRefGoogle Scholar
  16. 16.
    R. Dutt, A. Khare, U.P. Sukhatme, Am. J. Phys. 59, 723 (1991)CrossRefGoogle Scholar
  17. 17.
    G. Chen, Phys. Scr. 69, 257 (2004)CrossRefGoogle Scholar
  18. 18.
    G. Le’vai, J. Phys. A 37, 10179 (2004)CrossRefGoogle Scholar
  19. 19.
    J. Morales, J.J. Pena, G. Ovando, J.J. García-Ravelo, Mol. Struct. (Theochem) 769, 9 (2006)CrossRefGoogle Scholar
  20. 20.
    E. Gozzi, M. Reuter, W. Thacker, Phys. Lett. A 183, 29 (2003)CrossRefGoogle Scholar
  21. 21.
    E.D. Filho, R.M. Ricotta, Phys. Lett. A 320, 95 (2003)CrossRefGoogle Scholar
  22. 22.
    D.J. Kouri, T. Markovich, N. Maxwell, E.R. Bittner, J. Phys. Chem. 113, 15257 (2009)CrossRefGoogle Scholar
  23. 23.
    E.R. Bittner, J.B. Maddox, D.J. Kouri, J. Phys. Chem. 113, 15276 (2009)CrossRefGoogle Scholar
  24. 24.
    W.Y. Keung, U.P. Sukhatme, Q. Wang, T.D. Imbo, J. Phys. A Math. Gen. 22, L987 (1989)CrossRefGoogle Scholar
  25. 25.
    S.T. Epstein, The Variational Method in Quantum Chemistry, 2nd edn. (Academic Press, New York, 1974)Google Scholar
  26. 26.
    W. Yurgrau, S. Mandelstam, Variational Principle in Dynamics and Quantum Theory, 3rd edn. (Dover Publications, New York, 1979)Google Scholar
  27. 27.
    D.J. Griffith, Introduction to Quantum Mechanics, 2nd edn. (PEARSON, Education, Addison-Wesley, 2006)Google Scholar
  28. 28.
    C.R. Moon, L.S. Mattos, B.K. Foster, G. Zeltzer, W. Ko, H.C. Manoharan, Science 319, 782 (2008)CrossRefGoogle Scholar
  29. 29.
    N. Tyutyulkov, F. Dietz, G. Olbrich, Int. J. Quantum Chem. 62, 167 (1998)CrossRefGoogle Scholar
  30. 30.
    W.C. Herndon, Tetrahedron Lett. 15, 671 (1974)CrossRefGoogle Scholar
  31. 31.
    W.C. Herndon Jr, M.L. Ellzey, Tetrahedron 31, 99 (1975)CrossRefGoogle Scholar
  32. 32.
    E.L. Eliel, S.H. Wilen, Stereochemistry of Organic Compounds (Wiley Student edition, New York, 1994)Google Scholar
  33. 33.
    K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering, 1st edn. (Cambridge University Press, Cambridge, 1998)Google Scholar
  34. 34.
    J.L. Powell, B. Crasemann, Quantum Mech. (Narosa Publishing House, New Delhi, 1998)Google Scholar
  35. 35.
    N. Mukherjee, R.K. Pathak, K. Bhattacharyya, Int. J. Quantum Chem. 111, 3591 (2011)Google Scholar
  36. 36.
    N. Mukherjee, K. Bhattacharyya, Int. J. Quantum Chem. 112, 960 (2012)CrossRefGoogle Scholar
  37. 37.
    N. Mukherjee, J. Math. Chem. 50, 2303 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of ChemistryUniversity of CalcuttaKolkataIndia

Personalised recommendations