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Journal of Physics A Mathematical and General

Exact quantum motion of a particle trapped by oscillating fields

I. A. Pedrosa, Alexandre Rosas, I. Guedes

Published Aug 16, 2005• 34 citations
Publisher

Abstract

The exact wavefunctions for a particle trapped by oscillating fields are obtained in terms of Mathieu functions with the help of linear invariants and the dynamical invariant method. In addition, we construct Gaussian wave packet solutions and calculate the quantum fluctuations in the coordinate and momentum as well as the quantum correlations between coordinate and momentum.

DOI
10.1088/0305-4470/38/35/009

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The exact wavefunctions for a particle trapped by oscillating fields are obtained in terms of Mathieu functions with the help of linear invariants and the dynamical invariant method. In addition, we construct Gaussian wave packet solutions and calculate the quantum fluctuations in the coordinate and momentum as well as the quantum correlations between coordinate and momentum.
In this work we investigate the quantum dynamics of a particle trapped by oscillating fields. With the help of quadratic invariants and of the invariant method we solve analytically and exactly the Schrödinger equation for this system and write the corresponding wave functions in terms of solutions of the Milne–Pinney equation. We also construct coherent and squeezed states for this system and evaluate the quantum fluctuations in coordinate and momentum as well as the uncertainty product between canonical coordinate and momentum.
We derive quantum solutions of a generalized inverted or repulsive harmonic oscillator with arbitrary time-dependent mass and frequency using the quantum invariant method and linear invariants, and write its wave functions in terms of solutions of a second-order ordinary differential equation that describes the amplitude of the damped classical inverted oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum, the associated uncertainty relation, and the quantum correlations between coordinate and momentum. As a particular case, we apply our general development to the generalized inverted Caldirola–Kanai oscillator.
The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.
We use the Lewis and Riesenfeld invariant method [J. Math. Phys. 10, 1458 (1969)]10.1063/1.1664991 and a unitary transformation to obtain the exact Schrödinger wave functions for time-dependent harmonic oscillators exhibiting log-periodic-type behavior. For each oscillator we calculate the quantum fluctuations in the coordinate and momentum as well as the quantum correlations between the coordinate and momentum. We observe that the oscillator with m = m0t/t0 and ω = ω0t0/t, which exhibits an exact log-periodic oscillation, behaves as the harmonic oscillator with m and ω constant.
In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the  Schrodinger equation for the Caldirola-Kanai time-dependent forced harmonic oscillator in terms of the  solutions of a second order ordinary differential equation that describes the amplitude of the classical  unforced damped oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the  quantum fluctuations in coordinate and momentum as well as the uncertainty product. It is also shown  that the quantum fluctuations and the width of the Gaussian packet do not depend on the external force.
Abstract The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined by solving a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied by linearizing the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. The effect of a dissipative, linear velocity dependent friction force on these systems is discussed.