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DYNAMICS EVOLUTION FOR TIME-DEPENDENT COUPLED OSCILLATORS

Xiao Wang

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

By means of the time-dependent quantum transformation theory,we derive the exact solution for the time-dependent coupled oscillators.Based on this result,for the given initial states,the dynamics evolutions of Fock state and coherent state are discussed.

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We introduce a new class of quantum models with time-dependent Hamiltonians of a special scaling form. By using a couple of time-dependent unitary transformations, the time evolution of these models is expressed in terms of related systems with time-independent Hamiltonians. The mapping of dynamics can be performed in any dimension, for an arbitrary number of interacting particles and for any type of the scaling interaction potential. The exact solvability of a "dual" time-independent Hamiltonian automatically means the exact solvability of the original problem with model time-dependence.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.