Coupling Interactions and Trapped Effects for a Triple-Well Potential

Sezgin Aydın,Mehmet Şimşek

doi:10.4236/jmp.2011.28106

Abstract

Weak and strong coupling interactions and trapped effects have always played a significant role in understanding physical and chemical properties of materials. Triple-well anharmonic potential may be modeled for interpretation of energy spectra from the nuclear to macro molecular systems, and also crystalline systems. Exact periods of a trapped particle in each well of the potential are explicitly derived. For the extended Duffing system, it is predicted that infinite series of both frequency and spatial trajectory approach to exact results in the limit of weak-coupling cases (g→0).

Highlights

The physics of nonlinear systems is one of the important research interests in both quantum and classical mechanics, and its oscillatory representation [1,2,3,4,5,6,7] occupies special place for dynamical systems
For the extended Duffing system, it is predicted that infinite series of both frequency and spatial trajectory approach to exact results in the limit of weak-coupling cases (g→0)
Note that if the parameters in the equation are chosen as c = w02, g = –1/6 w02 and a = –1/20, it turns to fifth order non-linear equation of elementary (1+1) dimensionally pendulum, and to the Duffing equation [26,27,28] for a = 0

Summary

Introduction

The physics of nonlinear systems is one of the important research interests in both quantum and classical mechanics, and its oscillatory representation [1,2,3,4,5,6,7] occupies special place for dynamical systems. Using the perturbation approximations [21], a model of nonlinear problem, other words the extended Duffing equation, reduces to approximately solvable case. We called it here after the extended Duffing oscillator. The arbitrary parameters (i.e., g and a) stand for the perturbation parameter and coupling constants for weak-coupling systems, respectively It may be modeled for coherent tunneling via adiabatic passage in a triplewell system and coherent transport of electrons between quantum dots or atoms in micro-magnetic traps [18]. Which has different shapes, as a modeling of systems, for given potential parameters w0, a and g (see Figure 1).

Weak-Coupling Interactions

Exact Solutions of a Particle in the Triple-Well Potential

Strong-Coupling Limit

Conclusions