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arXiv:quant-ph/0311128

About evolution of particle transitions from one well to another in double-well potential

V. S. Olkhovsky, Sergei P. Maydanyuk

Published Nov 19, 2003• 1 citation
arXivHTMLPDF

Abstract

The Poincare's period of particle oscillations between wells is obtained in double-well potential. The dependencies of oscillation period on transmission coefficient on distance between levels are obtained. The cases of squared potentials and some potentials having rounded off forms are considered.
DOI
10.48550/arxiv.quant-ph/0311128

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Methods previously developed for the case of a symmetric double-well potential are here extended to study tunneling in an asymmetric, piecewise harmonic, double-well potential. The approach involves the decomposition of the real stationary states into complex right- and left-moving states and the determination of exact transmission coefficients for the latter. Numerical solutions for a selection of representative parameter values are obtained for the time evolution of states which correspond initially to a Gaussian wave packet localized in the upper well. Both resonant and nonresonant cases are studied and the significance of the transmission coefficient is established for each.
In this paper, particle tunneling through a 1-dimensional double square potential barrier is considered under fluctuations of an observer’s frame of reference (OFR). The concept of duration density—the proportion of the duration of a measurement that a particle has a particular energy—is adopted to formulate the average transmission probability of a particle for two types of fluctuations in the OFR. For single-period square-wave fluctuations, the average transmission probability is found to oscillate with a period of π or 2π as a function of the energy of the particle; the period of the oscillations rapidly switches from one value to another as a result of extremely small changes in the wavenumber configuration. For single-period sawtooth-wave fluctuations, the period of the oscillations in the transmission probability gradually increases from π to 2π as the wavenumber configuration changes. It is concluded, therefore that the average transmission probability has the potential to provide information about the nature of the fluctuations in an OFR.
For a model 1d asymmetric double-well potential we calculated so-called survival probability (i.e. the probability for a particle initially localised in one well to remain there). We use a semiclassical (WKB) solution of Schroedinger equation. It is shown that behaviour essentially depends on transition probability, and on dimensionless parameter which is a ratio of characteristic frequencies for low energy non-linear in-well oscillations and inter wells tunnelling. For the potential describing a finite motion (double-well) one has always a regular behaviour. For the small value of the parameter there is well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However for the large value of the parameter no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case one may not restrict oneself to only resonance pair levels. The number of perturbed by tunnelling levels grows proportionally to the value of this parameter (by other words instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of the parameter one has a crossover between both limiting cases, namely the exponential decay with subsequent long period recurrent behaviour.