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International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT)

Effective Schrodinger equation for one-dimensional systems with rapidly oscillating boundary conditions

N. P. Tretyakov, Павел Евгеньевич Голосов, Anton Kochanov

Published Nov 1, 2018• 1 citation

Abstract

The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.

DOI: 10.1109/icumt.2018.8631204

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The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.

The quantum dynamics of a hydrogen-like atom confined in one-dimensional box with oscillating walls is studied.The description of the system is reduced to a one-dimensional Schrödinger equation for Coulomb potential with time-dependent boundary conditions, which is solved numerically.Using the obtained solution, the average kinetic energy and binding energies are calculated as a function of time.It is found that both the average kinetic energy and the binding energies are periodic in time with the period depending on the wall's oscillation parameters.The probability density is also analyzed as a function of time and coordinate.

The Schrödinger equation is solved for the case of a particle confined to a small region of a box with infinite walls. If walls of the well are moved, then, due to an effective quantum nonlocal interaction with the boundary, even though the particle is nowhere near the walls, it will be affected. It is shown that this force apart from a minus sign is equal to the expectation value of the gradient of the quantum potential for vanishing at the walls boundary condition. Variation of this force with time is studied. A selection of Bohmian trajectories of the confined particle is also computed.

We show that the motion of a particle of mass m in a high-frequency time-dependent potential V(x\ensuremath{\rightarrow})=v(x\ensuremath{\rightarrow})cos(\ensuremath{\Omega}t) is governed by a Schr\"odinger equation with time-independent effective potential

{V}_{\mathrm{eff}}

(x\ensuremath{\rightarrow})=\ensuremath{\nabla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow})\ensuremath{\cdot}\ensuremath{\nabla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow}) /4m${\ensuremath{\Omega}}^{2}$. The validity of this approximation and an exact formal solution based on the Wigner function are discussed for the case in which v(x\ensuremath{\rightarrow}) is quadratic.

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.