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

SIGMA 4 (2008), 043, 16 pages      arXiv:0805.1687      https://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.

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