
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 TimeDependent and TimeIndependent Quantum Systems
Dieter Schuch
Institut für Theoretische Physik, J.W. GoetheUniversität, MaxvonLaueStr. 1, D60438 Frankfurt am Main, Germany
Received December 28, 2007, in final form May 07,
2008; Published online May 12, 2008
Abstract
The timeevolution of the maximum and the width of exact analytic
wave packet (WP) solutions of the timedependent 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 timedependent 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
timeindependent 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 timedependent problems concludes our analysis.
Key words:
Riccati equation; Ermakov invariant; wave packet dynamics; nonlinear quantum mechanics.
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