Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 046, 9 pages      arXiv:0805.4024      http://dx.doi.org/10.3842/SIGMA.2008.046
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Hamiltonian Systems Inspired by the Schrödinger Equation

Vasyl Kovalchuk and Jan Jerzy Slawianowski
Institute of Fundamental Technological Research, Polish Academy of Sciences, 21, Swietokrzyska str., 00-049 Warsaw, Poland

Received October 30, 2007, in final form April 25, 2008; Published online May 27, 2008

Abstract
Described is n-level quantum system realized in the n-dimensional ''Hilbert'' space H with the scalar product G taken as a dynamical variable. The most general Lagrangian for the wave function and G is considered. Equations of motion and conservation laws are obtained. Special cases for the free evolution of the wave function with fixed G and the pure dynamics of G are calculated. The usual, first- and second-order modified Schrödinger equations are obtained.

Key words: Schrödinger equation; Hamiltonian systems on manifolds of scalar products; n-level quantum systems; scalar product as a dynamical variable; essential non-perturbative nonlinearity; conservation laws; GL(n,C)-invariance.

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References

  1. Arnold V.I., Mathematical methods of classical mechanics, Springer Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York, 1978.
  2. Doebner H.-D., Goldin G.A., Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations, Phys. Rev. A 54 (1996), 3764-3771.
  3. Doebner H.-D., Goldin G.A., Nattermann P., Gauge transformations in quantum mechanics and the unification of nonlinear Schrödinger equations, J. Math. Phys. 40 (1999), 49-63, quant-ph/9709036.
  4. Dvoeglazov V.V., The Barut second-order equation, dynamical invariants and interactions, J. Phys. Conf. Ser. 24 (2005), 236-240, math-ph/0503008.
  5. Goldin G.A., Gauge transformations for a family of nonlinear Schrödinger equations, J. Nonlinear Math. Phys. 4 (1997), 6-11.
  6. Hehl F.W., Kerlick G.D., von der Heyde P., General relativity with spin and torsion and its deviations from Einstein's theory, Phys. Rev. D 10 (1974), 1066-1069.
  7. Hehl F.W., Lord E.A., Ne'eman Y., Hadron dilatation, shear and spin as components of the intrinsic hypermomentum current and metric-affine theory of gravitation, Phys. Lett. B 71 (1977), 432-434.
  8. Kovalchuk V., Green function for Klein-Gordon-Dirac equation, J. Nonlinear Math. Phys. 11 (2004), suppl., 72-77.
  9. Kozlowski M., Marciak-Kozlowska J., From quarks to bulk matter, Hadronic Press, USA, 2001.
  10. Kruglov S.I., On the generalized Dirac equation for fermions with two mass states, Ann. Fond. Louis de Broglie 29 (2004), 1005-1016, quant-ph/0408056.
  11. Marciak-Kozlowska J., Kozlowski M., Schrödinger equation for nanoscience, cond-mat/0306699.
  12. Slawianowski J.J., Kovalchuk V., Klein-Gordon-Dirac equation: physical justification and quantization attempts, Rep. Math. Phys. 49 (2002), 249-257.
  13. Slawianowski J.J., Kovalchuk V., Slawianowska A., Golubowska B., Martens A., Rozko E.E., Zawistowski Z.J., Affine symmetry in mechanics of collective and internal modes. Part I. Classical models, Rep. Math. Phys. 54 (2004), 373-427, arXiv:0802.3027.
  14. Slawianowski J.J., Kovalchuk V., Slawianowska A., Golubowska B., Martens A., Rozko E.E., Zawistowski Z.J., Affine symmetry in mechanics of collective and internal modes. Part II. Quantum models, Rep. Math. Phys. 55 (2005), 1-45, arXiv:0802.3028.
  15. Svetlichny G., Informal resource letter - nonlinear quantum mechanics on arXiv up to August 2004, quant-ph/0410036.
  16. Svetlichny G., Nonlinear quantum mechanics at the Planck scale, Internat. J. Theoret. Phys. 44 (2005), 2051-2058, quant-ph/0410230.

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