time-dependent harmonic oscillators
Nonlinear equations are of increasing interest in Physics; Riccati equation and Ermakov systems enter the formalism of quantum theory
in the study of cases where exact analytic Gaussian wave packet (WP) solutions of the time-dependent
Schrödinger equation (SE) do exist, and in particular, in the harmonic oscillator (HO) and the free motion cases.
One of the simplest examples of such nonlinear equations is the Milne-Pinney equation:
(1)
where
is a real constant with values depending on the field in which the equation is to be applied.
This equation was introduced in the nineteenth century by V.P. Ermakov, as a way of looking for a first integral for the time-dependent harmonic oscillator. He employed some of Lie's ideas for dealing with ordinary differential equations with the tools of classical geometry. Lie had previously obtained a characterization of non-autonomous systems
of first-order differential equations admitting a superposition rule:
(2).
This approach has been recently reformulated from a geometric perspective in which the role of the superposition function is played by an appropriate algebriac connection. This geometric approach allows one to consider a superposition of solutions of a given system in order to obtain solutions of another system as a kind of mathematical construction that might be generalized even further; such a superposition rule may be understood from a geometric viewpoint in some interesting cases, as in the Milne-Pinney equation (1), or in the Ermakov system and its generalisations. One recalls here that Ermakov systems are defined as systems of second-order differential equations composed by the Milne-Pinney differential equation (1) together with the corresponding time-dependent harmonic oscillator.
Ermakov systems have been also broadly studied in Physics since their introduction in the nineteenth century. They also appear in the study of the Bose-Einstein condensates, cosmological models, and the solution of time-dependent harmonic or anharmonic oscillators. Several recent reports are concerned with the use Hamiltonian
or Lagrangian structures in the study of such a system, and many generalisations or new insights from the mathematical point of view have ben thus obtained. Ermakov-Lewis invariants naturally emerge as functions
defining the foliation associated to the superposition rule.
It has been shown by Ermakov in 1880 that the system of differential equations
coupled via the possibly time-dependent frequency
, leads to a dynamical invariant that has been rediscovered by several authors in the 20th century:
(3)
It is straightforward to show that
The above Ermakov invariant
depends not only on the classical variables
and its time derivative, but also on the quantum uncertainty
related to
and its time derivative. Additional interesting insight into the relation
between variables
and
can be obtained by considering also the
Riccati equation.
- 1
-
Dieter Schuch. 2008. Riccati and Ermakov Equations in Time-Dependent
and Time-Independent Quantum Systems. Symmetry, Integrability and Geometry: Methods and Applications (SIGM), 4, 043: 16 pages.
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R. Goodall and P. G. L. Leach.
Generalised Symmetries and the Ermakov-Lewis Invariant.,
Journal of Nonlinear Mathematical Physics. Volume 12, Number 1, (2005), 15-26. (Letter)
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Kaushal R.S., Quantum analogue of Ermakov systems and the phase of the quantum wave function, Intnl. J. Theoret. Phys., 40, (2001), 835-847.
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Korsch H.J., Laurent H., Milne's differential equation and numerical solutions of the Schršodinger equation. I. Bound-state energies for single- and double- minimum potentials,J. Phys. B: At. Mol. Phys. 14 (1981), 4213-4230.
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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.
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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.
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A. Andriopoulos, Applicable Analysis and Discrete Mathematics,
2 (2008) 146-157. (http://pefmath.etf.bg.ac.yu/vol2num2/AADM-Vol2-No2-146-157.pdf)
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P. G. L. Leach, A. Karasu, M. C. Nucci, and A. Andriopoulos,
SIGMA, 1 (2005) 018
/www.emis.de/journals/SIGMA/2005/Paper018/
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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.
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