The effective potential for a charged particle induced by a rapidly oscillating field is in general difficult to determine. An analytical solution for such a problem is available for the case of Paul trap, i.e., when the sinusoidally oscillating potential is a quadratic function of position. In this paper, we describe time-domain simulation of the time-dependent Schrödinger equation to determine the effective two-dimensional lattice potential induced by an oscillating standing wave. Such a scheme can be generalized to determine the effective potential induced by any rapidly oscillating field.The effective potential for a charged particle induced by a rapidly oscillating field is in general difficult to determine. An analytical solution for such a problem is available for the case of Paul trap, i.e., when the sinusoidally oscillating potential is a quadratic function of position. In this paper, we describe time-domain simulation of the time-dependent Schrödinger equation to determine the effective two-dimensional lattice potential induced by an oscillating standing wave. Such a scheme can be generalized to determine the effective potential induced by any rapidly oscillating field.

Journal of Applied PhysicsAug 15, 20060 citations

Some Solutions of the Steady State Schroedinger Equation

Walter Kauzmann

Elsevier eBooksJan 1, 19570 citations

On the time-dependent solutions of the Schrödinger equation

Alejandro Palma, I. Pedraza

Progress in theoretical chemistry and physicsJan 1, 20073 citations

Inference of Schrödinger's Equation from Classical-Mechanical Solution

J. X. Zheng-Johansson, Johansson, P-I.

We set up the classical wave equation for a particle formed of an oscillatory zero-rest-mass charge together with its resulting electromagnetic waves, traveling in a potential field $V$ in a susceptible vacuum. The waves are Doppler-displaced upon the source motion, and superpose into a total, traveling- and in turn a standing- beat wave, or de Broglie phase wave, described by a corresponding total classical wave equation. By back-substitution of the explicitly known total, standing beat wave function and upon appropriate reductions at classic-velocity limit, we separate out from the total a component wave equation describing the kinetic motion of particle, which is equivalent to the Schrödinger equation. The Schrödinger wave function follows to be the envelope function of the standing beat wave at classic-velocity limit.We set up the classical wave equation for a particle formed of an oscillatory zero-rest-mass charge together with its resulting electromagnetic waves, traveling in a potential field $V$ in a susceptible vacuum. The waves are Doppler-displaced upon the source motion, and superpose into a total, traveling- and in turn a standing- beat wave, or de Broglie phase wave, described by a corresponding total classical wave equation. By back-substitution of the explicitly known total, standing beat wave function and upon appropriate reductions at classic-velocity limit, we separate out from the total a component wave equation describing the kinetic motion of particle, which is equivalent to the Schrödinger equation. The Schrödinger wave function follows to be the envelope function of the standing beat wave at classic-velocity limit.

arXiv:physics/0411134Nov 15, 20042 citations

Bound State Solutions of the Schrödinger Equation

Ajoy Kumar Ghatak, S. Lokanathan

Springer eBooksJan 1, 20040 citations

Bose-Einstein Condensates in Standing Waves: The Cubic Nonlinear Schrödinger Equation with a Periodic Potential

Jared C. Bronski, Lincoln D. Carr, Bernard Deconinck +1 more

We present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with an elliptic function potential. In the limit of a sinusoidal potential our solutions model a quasi-one-dimensional dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided that the ratio of the height of the variations of the condensate to its dc offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Recent developments allow for experimental investigation of these predictions.We present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with an elliptic function potential. In the limit of a sinusoidal potential our solutions model a quasi-one-dimensional dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided that the ratio of the height of the variations of the condensate to its dc offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Recent developments allow for experimental investigation of these predictions.

arXiv:cond-mat/0007174Physical Review LettersFeb 19, 2001331 citations

We set up the classical wave equation for a particle formed of an oscillatory zero-rest-mass charge together with its resulting electromagnetic waves, traveling in a potential field

V

in a susceptible vacuum. The waves are Doppler-displaced upon the source motion, and superpose into a total, traveling- and in turn a standing- beat wave, or de Broglie phase wave, described by a corresponding total classical wave equation. By back-substitution of the explicitly known total, standing beat wave function and upon appropriate reductions at classic-velocity limit, we separate out from the total a component wave equation describing the kinetic motion of particle, which is equivalent to the Schrödinger equation. The Schrödinger wave function follows to be the envelope function of the standing beat wave at classic-velocity limit.