Advertisement

Energy spectrum of the long-range Lennard-Jones potential

  • Shahar Hod
Regular Article
  • 7 Downloads

Abstract.

The discrete energy spectra of composite inverse power-law binding potentials of the form \(V(r;\alpha,\beta,n) =-\alpha/r^{2}+\beta/r^{n}\) with \(n > 2\) are studied analytically. In particular, using a functional matching procedure for the eigenfunctions of the radial Schrödinger equation, we derive a remarkably compact analytical formula for the discrete spectra of binding energies \(\{E(\alpha,\beta, n;k)\}^{k=\infty}_{k=1}\) which characterize the highly excited bound-state resonances of these long-range binding potentials. Our results are of practical importance for the physics of polarized molecules, the physics of composite polymers, and also for physical models describing the quantum interactions of bosonic particles.

References

  1. 1.
    H.R. Thorsheim, J. Weiner, P.S. Julienne, Phys. Rev. Lett. 58, 2420 (1987)ADSCrossRefGoogle Scholar
  2. 2.
    J.D. Miller, R.A. Cline, D.J. Heinzen, Phys. Rev. Lett. 71, 2204 (1993)ADSCrossRefGoogle Scholar
  3. 3.
    R.A. Cline, J.D. Miller, D.J. Heinzen, Phys. Rev. Lett. 73, 632 (1994)ADSCrossRefGoogle Scholar
  4. 4.
    C.J. Williams, P.S. Julienne, J. Chem. Phys. 101, 2634 (1994)ADSCrossRefGoogle Scholar
  5. 5.
    R. Cote, A. Dalgarno, M.J. Jamieson, Phys. Rev. A 50, 399 (1994)ADSCrossRefGoogle Scholar
  6. 6.
    H. Wang, P.L. Gould, W.C. Stwalley, Phys. Rev. A 53, R1216 (1996)ADSCrossRefGoogle Scholar
  7. 7.
    B. Gao, Phys. Rev. A 58, 1728 (1998)ADSCrossRefGoogle Scholar
  8. 8.
    J. Trost, C. Eltschka, H. Friedrich, J. Phys. B: At. Mol. Opt. Phys. 31, 361 (1998)ADSCrossRefGoogle Scholar
  9. 9.
    B. Gao, Phys. Rev. A 59, 2778 (1999)ADSCrossRefGoogle Scholar
  10. 10.
    B. Gao, Phys. Rev. Lett. 83, 4225 (1999)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    L.D. Landau, E.M. Lifshitz, Quantum Mechanics. Non-relativistic Theory (Mir, Moscow, 1974)Google Scholar
  12. 12.
    J.E. Lennard-Jones, Proc. R. Soc. London A 106, 463 (1924)ADSCrossRefGoogle Scholar
  13. 13.
    Mott, Massey, Theory of Atomic Collisions (Clarendon Press, Oxford, 1949) p. 30Google Scholar
  14. 14.
    K.M. Case, Phys. Rev. 80, 797 (1950)ADSCrossRefGoogle Scholar
  15. 15.
    J.-M. Lévy-Leblond, Phys. Rev. 153, 1 (1967)ADSCrossRefGoogle Scholar
  16. 16.
    O.H. Crawford, Proc. Phys. Soc. London 91, 279 (1967)ADSCrossRefGoogle Scholar
  17. 17.
    C. Desfrancois, H. Abdoul-Carime, N. Khelifa, J.P. Schermann, Phys. Rev. Lett. 73, 2436 (1994)ADSCrossRefGoogle Scholar
  18. 18.
    H.E. Camblong, L.N. Epele, H. Fanchiotti, C.A. Garcia Canal, Phys. Rev. Lett. 85, 1590 (2000)ADSCrossRefGoogle Scholar
  19. 19.
    H.E. Camblong, L.N. Epele, H. Fanchiotti, C.A. Garcia Canal, Phys. Rev. Lett. 87, 220402 (2001)ADSCrossRefGoogle Scholar
  20. 20.
    V. Efimov, Phys. Lett. B 33, 563 (1970)ADSCrossRefGoogle Scholar
  21. 21.
    V. Efimov, Nucl. Phys. A 210, 157 (1973)ADSCrossRefGoogle Scholar
  22. 22.
    T. Kraemer et al., Nature 440, 315 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    E. Marinari, G. Parisi, Europhys. Lett. 15, 721 (1991)ADSCrossRefGoogle Scholar
  24. 24.
    C. Nisoli, A.R. Bishop, Phys. Rev. Lett. 112, 070401 (2014)ADSCrossRefGoogle Scholar
  25. 25.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970)Google Scholar
  26. 26.
    Y. Nishida, Y. Kato, C.D. Batista, Nat. Phys. 9, 93 (2013)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Ruppin Academic CenterEmeq HeferIsrael
  2. 2.The Hadassah Academic CollegeJerusalemIsrael

Personalised recommendations