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Physical Review E

Two-dimensional oscillator in time-dependent fields: Comparison of some exact and approximate calculations

Ochbadrakh Chuluunbaatar, Alexander A. Gusev, Sergue Il'ich Vinitsky, Vladimir Leonardovich Derbov, Artbazar Galtbayar, Tugal Zhanlav

Published Jul 22, 2008• 2 citations
Publisher

Abstract

Operator-difference multilayer schemes for solving the time-dependent Schrödinger equation up to sixth order of accuracy in the time step are presented. Reduced schemes for solving a set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised using expansion of a wave packet over the set of appropriate basis angular functions. Further discretization of the resulting problem is realized by means of the finite-element method. The convergence of the expansion with respect to the number of basis functions and the efficiency of the numerical schemes are demonstrated in the exactly solvable model of an electric-field-driven two-dimensional oscillator (or a charged particle in a constant uniform magnetic field), in which we explicitly observed an effect of the periodical focusing and defocusing of the probability density flux.

DOI
10.1103/physreve.78.017701

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Operator-difference multilayer schemes for solving the time-dependent Schrödinger equation up to sixth order of accuracy in the time step are presented. Reduced schemes for solving a set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised using expansion of a wave packet over the set of appropriate basis angular functions. Further discretization of the resulting problem is realized by means of the finite-element method. The convergence of the expansion with respect to the number of basis functions and the efficiency of the numerical schemes are demonstrated in the exactly solvable model of an electric-field-driven two-dimensional oscillator (or a charged particle in a constant uniform magnetic field), in which we explicitly observed an effect of the periodical focusing and defocusing of the probability density flux.
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The operator-difference multi-layer (ODML) schemes for solving the time-dependent Schrödinger equation (TDSE) till six order accuracy by a time step are presented. The reduced schemes for solving a set of the coupled TDSE’s are devised by using a set of appropriate basis angular functions and a finite element method with respect to a hyper-radial variable. Convergence by a number of the basis functions and efficiency of the numerical schemes are demonstrated in the case of an exactly solvable model of the two-dimensional oscillator in time-depended electric fields.
A symbolic algorithm to generate the multilayer operator-difference schemes for solving the evolution problem of the time-dependent Schrödinger equation is elaborated. An additional gauge transformation of operator-difference schemes to make good use of the finite-element discretization is applied. The efficiency of the generated numerical schemes until the sixth order with respect to the time step and until the seventh order with respect to the spatial step is demonstrated by calculations of some finite-dimensional quantum systems in external fields.
The use of a time-dependent basis set for solving the time-dependent Schrödinger equation is proposed. The basis set is a generalization of the harmonic oscillator functions. The proposed method is similar to the approaches formulated by Lee and Heller and by Coalson and Karplus. However, in the present method the time dependence of the basis functions is determined differently. Numerical studies on the quartic oscillator and on the Morse potential are carried out to investigate the performance of the various methods.
The time-dependent Schrödinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei–Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin–Man’ko–Trifonov and the Lewis–Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy–Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.