Talk:PlanetPhysics/Time Dependent Harmonic Oscillators

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\begin{document}

 \section{Time-dependent harmonic oscillators}
Nonlinear equations are of increasing interest in Physics; Riccati equation and Ermakov systems enter the formalism of \htmladdnormallink{quantum theory}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} in the study of cases where exact analytic Gaussian \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} packet (WP) solutions of the time-dependent
Schr\"odinger equation (SE) do exist, and in particular, in the harmonic oscillator (HO) and the free \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} cases.

One of the simplest examples of such nonlinear equations is the Milne--Pinney equation:
$$d^2x/dt^2 = ~ -~ {\omega}^2(t)x +k x^3,$$ (1)
where $k$ is a real constant with values depending on the \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} in which the equation is to be applied.

\subsection{Ermakov systems}

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 \htmladdnormallink{ordinary differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} with the tools of classical geometry. Lie had previously obtained a characterization of non-autonomous \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of first-order \htmladdnormallink{differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} admitting a superposition rule:
$$dx_i/dt = Y i(t, x), i = 1, . . . , n, $$ (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 {\em 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 \htmladdnormallink{Hamiltonian}{http://planetphysics.us/encyclopedia/Hamiltonian2.html} or \htmladdnormallink{Lagrangian}{http://planetphysics.us/encyclopedia/LagrangesEquations.html} structures in the study of such a system, and many generalisations or new insights from the mathematical point of view have ben thus obtained. {\em Ermakov--Lewis invariants} naturally emerge as \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} defining the foliation associated to the superposition rule.
It has been shown by Ermakov in 1880 that the \htmladdnormallink{system of differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} coupled via the possibly time-dependent frequency $\omega$, leads to a dynamical invariant that has been rediscovered by several authors in the 20th century:
$$I_L = 0.5[(d\eta/dt) ~ \alpha~ - ~ \eta (d \alpha/dt)]^2 + (\eta~ \alpha)^2 = const.  22:34, 25 June 2015 (UTC)~~      (3) $$ (3)

It is straightforward to show that $d/dt (I_L) = 0.$
The above Ermakov invariant $I_L$ depends not only on the classical variables
$\eta (t)$ and its time derivative, but also on the \htmladdnormallink{quantum uncertainty}{http://planetphysics.us/encyclopedia/QuantumParticle.html} related to $\alpha (t)$ and its time derivative. Additional interesting insight into the \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} between variables $\eta$ and $\alpha$ can be obtained by considering also the
\htmladdnormallink{Riccati equation}{http://planetphysics.us/encyclopedia/RiccatiEquation2.html}.

\begin{thebibliography}{9}
\bibitem{DS2k8}
Dieter Schuch. 2008. Riccati and Ermakov Equations in Time--Dependent
and Time--Independent Quantum Systems. {\em Symmetry, Integrability and Geometry: Methods and Applications (SIGM)}, {\bf 4}, 043: 16 pages.

\bibitem{RG-PGLL2k5}
R. Goodall and P. G. L. Leach.
\htmladdnormallink{Generalised Symmetries and the Ermakov-Lewis Invariant.}{http://eqworld.ipmnet.ru/en/solutions/interesting/leach-2005.pdf},
{\em Journal of Nonlinear Mathematical Physics.} Volume {\bf 12}, Number 1, (2005), 15--26. (Letter)

\bibitem{GB-DB84}
Grammaticos B. and Dorizzi B., Two-dimensional time-dependent Hamiltonian systems with an exact invariant. {\em Journal of Mathematical Physics} 25 (1984) 2194--2199.

\bibitem{KRS2k1}
Kaushal R.S., Quantum analogue of Ermakov systems and the phase of the quantum wave function, {\em Intnl. J. Theoret. Phys.}, {\bf 40}, (2001), 835--847.

\bibitem{KHJ-LH81}
Korsch H.J., Laurent H., Milne's differential equation and numerical solutions of the Schr¨odinger equation. I. Bound-state energies for single-- and double-- minimum potentials,{\em J. Phys. B: At. Mol. Phys.} {\bf 14} (1981), 4213--4230.

\bibitem{KHJ-LH-M2k3}
Korsch H.J., Laurent H. and Mohlenkamp., Milne's differential equation and numerical solutions of the Schr\"odinger equation. II. Complex energy resonance states, {\em J. Phys. B: At. Mol. Phys.}, {\bf 15}, (1982), 1--15.

\bibitem{SD94}
Schuch D., Relations between wave and particle aspects for motion in a magnetic field, in {\em New Challenges in Computational Quantum Chemistry.}, Editors R. Broer, P.J.C. Aerts and P.S. Bagus, University of Groningen,(1994), 255--269.

\bibitem{MMB-FN2k6}
Maamache M. Bounames A., Ferkous N., Comment on ``Wave function of a time-dependent harmonic oscillator in a static magnetic field.'', {\em Phys. Rev. A} {\bf 73}, (2006), 016101, 3 pages.

\bibitem{RJR80}
Ray J.R., Time-dependent invariants with applications in physics, {\em Lett. Nuovo Cim.}, {\bf 27}, (1980), 424--428.

\bibitem{SW75}
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.

\bibitem{LHR-LP82}
Lewis H.R., Leach P.G.L., Exact invariants for a class of time-dependent nonlinear Hamiltonian systems, {\em J. Math. Phys. 23 (1982), 165--175.}

\bibitem{LHR-LP82}
``The Ermakov Equation: A Commentary.''  P. G. L. Leach and
A. Andriopoulos, Applicable Analysis and Discrete Mathematics,
2 (2008) 146-157. (http://pefmath.etf.bg.ac.yu/vol2num2/AADM-Vol2-No2-146-157.pdf)

\bibitem{LKNA2k5}
``Ermakov's Superintegrable Toy and Non-Local Symmetries.''
P. G. L. Leach, A. Karasu, M. C. Nucci, and A. Andriopoulos,
SIGMA, 1 (2005) 018

/www.emis.de/journals/SIGMA/2005/Paper018/

\bibitem{SAM-LP2k3}
Sebawa Abdalla M., Leach P.G.L., Linear and quadratic invariants for the transformed Tavis--Cummings model, {\em J. Phys. A: Math. Gen.}, {\bf 36} (2003), 12205--12221.

\bibitem{SAM-LP2k5}
Sebawa Abdalla M., Leach P.G.L., Wigner functions for time--dependent coupled linear oscillators via linear and quadratic invariant processes, {\em J. Phys. A: Math. Gen.}, {\bf 38}, (2005), 881--893.

\bibitem{KRS98}
Kaushal R.S., Classical and quantum mechanics of noncentral potentials. A survey of 2D systems, Springer, Heidelberg, (1998).

\bibitem{EV1880}
Ermakov V., Second-order differential equations. Conditions of complete integrability. {\em Universita Izvestia Kiev} Ser III 9 (1880) 1--25, trans Harin AO.

\end{thebibliography} 

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