The European Physical Journal D

, Volume 8, Issue 1, pp 1–7 | Cite as

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

  • R. EnglmanEmail author
  • A. Yahalom
  • M. Baer


We calculate the open path phase in a two state model with a slowly (nearly adiabatically) varying time-periodic Hamiltonian and trace its continuous development during a period. We show that the topological (Berry) phase attains π or 2π depending on whether there is or is not a degeneracy in the part of the parameter space enclosed by the trajectory. Oscillations are found in the phase. As adiabaticity is approached, these become both more frequent and less pronounced and the phase jump becomes increasingly more steep. Integral relations between the phase and the amplitude modulus (having the form of Kramers-Kronig relations, but in the time domain) are used as an alternative way to calculate open path phases. These relations attest to the observable nature of the open path phase.


03.65.Bz Foundations, theory of measurement, miscellaneous theories (including Aharonov-Bohm effect, Bell inequalities, Berry’s phase) 03.65.Ge Solutions of wave equations: bound states 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M.V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).ADSCrossRefGoogle Scholar
  2. 2.
    B. Simon, Phys. Rev. Lett. 51, 2167 (1983).ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58, 1593 (1987).ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    A.K. Pati, Phys. Rev. A 52, 2576 (1995); S.R. Jain, A.K. Pati, Phys. Rev. Lett. 80, 650 (1998).ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    C.M. Cheng, P.C.W. Fung, J. Phys. A Math. Gen. 22, 3493 (1989).ADSCrossRefGoogle Scholar
  6. 6.
    D.J. Moore, G.E. Stedman, J. Phys. A Math. Gen. 23, 2049 (1990).ADSCrossRefGoogle Scholar
  7. 7.
    T. Bitter, D. Dubbens, Phys. Rev. Lett. 59, 251 (1998); D. Suter, K.T. Mueller, A. Pines, Phys. Rev. Lett. 60, 1218 (1988); H. von Busch, V. Dev, H.-A. Eckel, S. Kasahara, J. Wang, W. Demtroder, P. Sebald, W. Meyer, Phys. Rev. Lett. 81, 4584 (1998).Google Scholar
  8. 8.
    R. Englman, M. Baer, J. Phys. Cond. Matt. 11, 1059 (1999).ADSCrossRefGoogle Scholar
  9. 9.
    R. Resta, S. Sorella, Phys. Rev. Lett. 74, 4738 (1995) (especially bottom of first column on p. 4740).ADSCrossRefGoogle Scholar
  10. 10.
    L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (University Press, Cambridge, 1995), Sects. 3.1 and 10.7; L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).CrossRefGoogle Scholar
  11. 11.
    J.H. Shapiro, S.R. Shepard, Phys. Rev. A 43, 3795 (1991) (Footnote 53).ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    R. Englman, A. Yahalom, M. Baer, Phys. Lett. A 251, 223 (1999).ADSCrossRefGoogle Scholar
  13. 13.
    H.M. Nussenzweig, Causality and Dispersion Relations (Academic Press, NewYork, 1972), pp. 24, 212.Google Scholar
  14. 14.
    E.C. Titchmarsh, The Theory of Functions (Clarendon Press, Oxford, 1932), Sects. 7.8 and 8.1.zbMATHGoogle Scholar
  15. 15.
    R. Englman, A. Yahalom, Phys. Rev. A 60, 1890 (1999).CrossRefGoogle Scholar
  16. 16.
    R. Englman, The Jahn-Teller Effect in Molecules and Crystals (Wiley Interscience, London, 1972).Google Scholar
  17. 17.
    H.C. Longuet-Higgins, Adv. Spectrosc. 2, 429 (1961).ADSGoogle Scholar
  18. 18.
    M. Baer, R. Englman, Mol. Phys. 75, 293 (1992); R. Baer, D. Charutz, R. Kosloff, M. Baer, J. Chem. Phys. 105, 9141 (1996).ADSCrossRefGoogle Scholar
  19. 19.
    M. Baer, A. Yahalom, R. Englman, J. Chem. Phys. 109, 6550 (1998).ADSCrossRefGoogle Scholar
  20. 20.
    J.W. Zwanziger, E.R. Grant, J. Chem. Phys. 87, 2954 (1987).ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    M.S. Topaler, D.G. Truhlar, X.Y. Chang, P. Piecuch, J.C. Polanyi, J. Chem. Phys. 108, 5349 (1998).ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica Springer-Verlag 2000

Authors and Affiliations

  1. 1.Department of Physics and Applied MathematicsSoreq NRCYavneIsrael
  2. 2.Research InstituteCollege of Judea and SamariaArielIsrael
  3. 3.Faculty of EngineeringTel-Aviv UniversityTel-AvivIsrael

Personalised recommendations