Advertisement

The European Physical Journal D

, Volume 46, Issue 1, pp 41–50 | Cite as

Energy levels of a particle confined in a super-circular box

  • N. Bera
  • J. K. Bhattacharjee
  • S. Mitra
  • S. P. Khastgir
Molecular Physics and Chemical Physics

Abstract.

We find the energy levels of a free particle confined in a two dimensional infinite potential well having super-circular boundary (|x|n+|y|n=an where n is a rational number and a is a positive real number) by perturbing about the equivalent circle (n=2). The ground state energies are very accurate over a wide range of n and can be improved further by introducing a phenomenological constant determined from the knowledge of exact results available for diamond (n=1). For excited states, we find that the shape effect can cause parametric resonance which can lead to singlet-triplet crossing.

PACS.

03.65.-W Quantum mechanics 31.15.Md Perturbation theory 03.65.Ge Solutions of wave equations: bound states 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.R. Krishnamurthy, H.S. Mani, H.C. Verma, J. Phys. A: Math. Gen. 15, 2131 (1982) CrossRefADSMathSciNetGoogle Scholar
  2. J.K. Bhattacharjee, K. Banerjee, J. Phys. A: Math. Gen. 20, L759 (1987) Google Scholar
  3. K. Lis, S. Bednarek, B. Szafran, J. Adamowski, Physica E 17, 494 (2003) CrossRefADSGoogle Scholar
  4. P.S. Drouvelis, P. Schmelcher, F.K. Diakonos, Phys. Rev. B 69, 155312 (2004) CrossRefADSGoogle Scholar
  5. I. Magnúsdóttir, V. Gudmundsson, Phys. Rev. B 60, 16590 (1999) CrossRefGoogle Scholar
  6. H. Ichikawa, K. Sakata, Intern. J. Quant. Chem. 87, 135 (2002) and referrences therein CrossRefGoogle Scholar
  7. S. Sakai, J. Phys. Chem. A 110, 6339 (2006) CrossRefGoogle Scholar
  8. M. Gardner, “Piet Hein's Superellipse”, in Mathematical Carnival: A new Round-Up of Tantalizers and Puzzles from Scientific American (Vintage, New York, 1977), Ch. 18, pp. 240–254 Google Scholar
  9. G. Lamé, Examen des différentes méthodes employées pour résoudre les problémes de geometrie (Oxford University, 1818) Google Scholar
  10. N.T. Gridgeman, Math. Gaz. 54, 31 (1970) MATHCrossRefGoogle Scholar
  11. J.A. Gielis, Am. J. Bot. 90, 333 (2003) Google Scholar
  12. I. Peterson, Science News 163, 18 (2003) Google Scholar
  13. J. Whitfield, Nature Science Update (April 2, 2003) Google Scholar
  14. Y. Shimizu, Chaos, Solitons Fractals 5, 1337 (1995) MATHCrossRefMathSciNetGoogle Scholar
  15. M.V. Berry, Eur. J. Phys. 2, 91 (1981) CrossRefMathSciNetGoogle Scholar
  16. I.M. Erham, H. Taseli, J. Comput. Appl. Math. 194, 227 (2006) CrossRefADSMathSciNetGoogle Scholar
  17. Handbook of Mathematical functions with Formulas, Graphs and Mathematical Tables, edited by M. Abramowitz, I. Stegun, National Bureau of Standards, Washington, D.C. (1964) Google Scholar
  18. K.F. Riley, M.P. Hobson, S.C. Bence, Mathematical methods for physics and engeneering (Cambridge University Press, 2002), Ch. 19 Google Scholar
  19. M.R. Spiegel, Vector Analysis (Schaum's Outline Series, New York, 1959) Google Scholar
  20. G. Arfken, Mathematical Methods for Physicists (Academy Press, 1995) Google Scholar
  21. D.J. Griffiths, Introduction to Quantum Mechanics, 2nd edn. (Prentice Hall, 2004) Google Scholar
  22. F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, 2001) Google Scholar
  23. W.D. Carl, Chemistry Education Material, University of Connecticut (2006) Google Scholar
  24. S. Agmon, Lectures on Elliptic Boundary Value Problems (Van Nostrand, Princeton, N.J., 1965) Google Scholar
  25. I. Babuŝka, U. Banerjee, J.E. Osborn, Survey of meshless and generalized finite element methods: A unified approach (Acta Numerica, 2003), pp. 1–125 Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  • N. Bera
    • 1
  • J. K. Bhattacharjee
    • 1
  • S. Mitra
    • 2
  • S. P. Khastgir
    • 2
  1. 1.Department of Theoretical PhysicsKolkataIndia
  2. 2.Department of Physics and MeteorologyIndian Institute of TechnologyKharagpurIndia

Personalised recommendations