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


SIGMA 4 (2008), 043, 16 pages      arXiv:0805.1687      http://dx.doi.org/10.3842/SIGMA.2008.043
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems

Dieter Schuch
Institut für Theoretische Physik, J.W. Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany

Received December 28, 2007, in final form May 07, 2008; Published online May 12, 2008

Abstract
The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system. The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem. The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation. The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation. It can be expressed in terms of a complex variable that linearizes the Riccati equation. This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem. From there, also the relation between the classical and quantum dynamics of the systems can be obtained. Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown. Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations. This provides the link to a nonlinear formulation of time-independent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law. Comparison with SUSY and the time-dependent problems concludes our analysis.

Key words: Riccati equation; Ermakov invariant; wave packet dynamics; nonlinear quantum mechanics.

pdf (281 kb)   ps (189 kb)   tex (20 kb)

References

  1. Schrödinger E., Quantisierung als Eigenwertproblem, Ann. Phys. 79 (1926), 361-376.
  2. Reinisch G., Nonlinear quantum mechanics, Phys. A 206 (1994), 229-252.
    Reinisch G., Classical position probability distribution in stationary and separable quantum systems, Phys. Rev. A 56 (1997), 3409-3416.
  3. Ermakov V.P., Second-order differential equations, Conditions of complete integrability, Univ. Izv. Kiev 20 (1880), no. 9, 1-25.
  4. Milne W.E., The numerical determination of characteristic numbers, Phys. Rev. 35 (1930), 863-867.
    Pinney E., The nonlinear differential equation y" + p(x)y + cy-3 = 0, Proc. Amer. Math. Soc. 1 (1950), 681.
    Lewis H.R., Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians, Phys. Rev. Lett. 18 (1967), 510-512.
  5. Lutzky M., Noether's theorem and the time-dependent harmonic oscillator, Phys. Lett. A 68 (1979), 3-4.
    Ray J.R., Reid J.L., More exact invariants for the time-dependent harmonic oscillator, Phys. Lett. A 71 (1979), 317-318.
  6. Malkin I.A., Man'ko V.I., Trifonov D.A., Linear adiabatic invariants and coherent states, J. Math. Phys. 14 (1973), 576-582.
    Markov M.A. (Editor), Invariants and evolution of nonstationary quantum systems, Proceedings of the Lebedev Physical Institute, Vol. 183, Nova Science, New York, 1989.
  7. Feynman R.P., Hibbs A.R., Quantum mechanics and path integrals, McGraw-Hill, New York, 1965.
  8. Schleich W.P., Quantum optics in phase space, Wiley-VCh, Berlin, Chapter 17, 2001.
  9. Schuch D., Moshinsky M., Connection between quantum-mechanical and classical time evolution via a dynamical invariant, Phys. Rev. A 73 (2006), 062111, 10 pages.
    Schuch D., Connection between quantum-mechanical and classical time evolution of certain dissipative systems via a dynamical invariant, J. Math. Phys. 48 (2007), 122701, 19 pages.
  10. Wigner E.P., On the quantum correction for thermodynamical equilibrium, Phys. Rev. 40 (1932), 749-759.
    Hillery M. O'Connell R.F., Scully M.O., Wigner E.P., Distribution functions in physics: fundamentals, Phys. Rep. 106 (1984), 121-167.
  11. Schuch D., On the relation between the Wigner function and an exact dynamical invariant, Phys. Lett. A 338 (2005), 225-231.
  12. Lewis H.R., Riesenfeld W.B., An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, J. Math. Phys. 10 (1969), 1458-1473.
    Hartley J.G., Ray J.R., Ermakov systems and quantum-mechanical superposition law, Phys. Rev. A 24 (1981), 2873-2876.
  13. Cooper F., Khare A., Sukhatme U., Supersymmetry in Quantum Mechanics, World Scientific, Singapore, 2001.
    Kalka H., Soff G., Supersymmetrie, Teubner, Stuttgart, 1997.
  14. Madelung E., Quantentheorie in hydrodynamischer Form, Z. Phys. 40 (1926), 322-326.
  15. Lee R.A., Quantum ray equations, J. Phys. A: Math. Gen. 15 (1982), 2761-2774.
  16. Kaushal R.S., Quantum analogue of Ermakov systems and the phase of the quantum wave function, Intern. J. Theoret. Phys. 40 (2001), 835-847.
  17. Korsch H.J., Laurent H., Milne's differential equation and numerical solutions of the Schrödinger equation. I. Bound-state energies for single- and double minimum potentials, J. Phys. B: At. Mol. Phys. 14 (1981), 4213-4230.
    Korsch H.J., Laurent H. and Mohlenkamp, Milne's differential equation and numerical solutions of the Schrödinger equation. II. Complex energy resonance states, J. Phys. B: At. Mol. Phys. 15 (1982), 1-15.
  18. Schuch D., Relations between wave and particle aspects for motion in a magnetic field, in New Challenges in Computational Quantum Chemistry, Editors R. Broer, P.J.C. Aerts and P.S. Bagus, University of Groningen, 1994, 255-269.
    Maamache M. Bounames A., Ferkous N., Comment on "Wave function of a time-dependent harmonic oscillator in a static magnetic field", Phys. Rev. A 73 (2006), 016101, 3 pages.
  19. Ray J.R., Time-dependent invariants with applications in physics, Lett. Nuovo Cim. 27 (1980), 424-428.
    Sarlet W., Class of Hamiltonians with one degree-of-freedom allowing applications of Kruskal's asymptotic theory in closed form. II, Ann. Phys. (N.Y.) 92 (1975), 248-261.
  20. Lewis H.R., Leach P.G.L., Exact invariants for a class of time-dependent nonlinear Hamiltonian systems, J. Math. Phys. 23 (1982), 165-175.
  21. Sebawa Abdalla M., Leach P.G.L., Linear and quadratic invariants for the transformed Tavis-Cummings model, J. Phys. A: Math. Gen. 36 (2003), 12205-12221.
    Sebawa Abdalla M., Leach P.G.L., Wigner functions for time-dependent coupled linear oscillators via linear and quadratic invariant processes, J. Phys. A: Math. Gen. 38 (2005), 881-893.
  22. Kaushal R.S., Classical and quantum mechanics of noncentral potentials. A survey of 2D systems, Springer, Heidelberg, 1998.

Previous article   Next article   Contents of Volume 4 (2008)