Variational Approximation to Electron Trapping by Soliton-Like Localized Excitations in One-Dimensional Anharmonic Latti
Electron trapping by soliton-like (traveling) localized excitations in one-dimensional anharmonic lattices is discussed with particular emphasis on the case of an initially completely delocalized electron.
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Abstract Electron trapping by soliton-like (traveling) localized excitations in onedimensional anharmonic lattices is discussed with particular emphasis on the case of an initially completely delocalized electron.
1 Introduction Let us consider a one-dimensional (1d) lattice with units interacting with anharmonic interactions. Then for appropriate anharmonic interactions it can be shown that the lattice can exhibit traveling solitary waves or periodic nonlinear waves, with soliton features as defined by Zabusky and Kruskal [1]. One particular such lattice is the Toda lattice where interactions are of exponential form for the repulsive component [2]. Its soliton solutions are known exactly as the system is integrable. For our purpose here we shall consider Morse interactions [3] which are not significantly different from Toda’s in their repulsive component while offering a physically justified attractive component. The latter is physically meaningless in the Toda case. Moreover, though the Morse lattice is non-integrable its computer L.A. Cisneros-Ake Department of Mathematics, ESFM, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Edificio 9, México DF-07738, México e-mail: [email protected] A.A. Minzoni FENOMEC, Department of Mathematics and Mechanics, IIMAS, Universidad Nacional Autónoma de México, Apdo. 20-726, México, DF-01000, México e-mail: [email protected] M.G. Velarde () Instituto Pluridisciplinar, Universidad Complutense, Paseo Juan XXIII, 1, Madrid-28040, Spain Fundación Universidad Alfonso X El Sabio, Villanueva de la Cañada-28691, Madrid, Spain e-mail: [email protected]; [email protected] R. Carretero-González et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, DOI 10.1007/978-3-319-02057-0__11, © Springer International Publishing Switzerland 2014
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solutions differ little from the solutions of the Toda lattice [4–6] and hence we shall benefit from such significantly useful approximation. Little has been established about the quantization of the Toda or other anharmonic lattices and, moreover, for the purpose of our present communication it is not needed here. The next item to be introduced in our mathematical model is an electron. As a quantum mechanical object, its (space and time) evolution obeys the corresponding (linear) Schrödinger equation. We shall consider the electron evolution on a lattice using the tight binding approximation (TBA) . If we disregard the lattice dynamics and hence consider a free evolving electron on the discrete lattice space, if we place it at a given site with maximum probability density, then as time proceeds the electron probability density spreads “uniformly” all over the lattice sites though the probability remains normalized to unity. Such delocalization of the electron can be considered as a form of “dust”, tiny spots on the lattice sites. The electron-phonon interaction or, in more general terms the electron-lattice dynamics interaction, is non
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