Quantum revivals of Morse oscillators and Farey-Ford geometry

Abstract

Analytical eigensolutions for Morse oscillators are used to investigate quantum resonance and revivals and show how Morse anharmonicity affects revival times. A minimum semi-classical Morse revival time T min−rev found by Heller is related to a complete quantum revival time T rev using a quantum deviation δ N parameter that in turn relates T rev to the maximum quantum beat period T max−beat . Also, number theory of Farey and Thales-circle geometry of Ford is shown to elegantly analyze and display fractional revivals. Such quantum dynamical analysis may have applications for spectroscopy or quantum information processing and computing.

Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time in terms of Husimi distributions

“…The distinguish of rational and irrational times (in the units of T ) in the semiclassical limit reveals the relation of quantum mechanics to number-theoretic issues discovered in some other models [19,27,36].…”

“…An interesting problem would be the calculation of semiclassical measures for more general potentials, for example, the Morse potential, as well as various multi-dimensional bounded domains and compact manifolds. Coherent states for the Morse potential were constructed in [5], the structure of revivals was studied in [36,55]. One can also consider quantum optimal control problems (with both coherent and incoherent controls [46,47]) in semiclassical long-time limit.…”

“…Coherent states for the Morse potential were constructed in [5], the structure of revivals was studied in [36,55]. One can also consider quantum optimal control problems (with both coherent and incoherent controls [46,47]) in semiclassical long-time limit.…”

Implementation of Morse potential for approximation of vibrational terms of diatomic molecules

“…It is also often given in textbooks and monographs [2][3][4][5][6][7]. Sometimes the authors give both options, mentioning their equivalence, for example in [14,15], although the expression (4) is laden with a systematic error, the nature of which requires mentioning, since it inevitably leads to two possible values of the parameter " a".…”

Возможности Формулы Морза При Аппроксимации Энергии Колебательных Уровней Двухатомной Молекулы

“…Он также чаще приводится и в учебниках и монографиях [2][3][4][5][6][7]. Иногда авторы приводят оба варианта, упоминая об их эквивалентности, например в [14,15], хотя выражение (4) отягощено систематической ошибкой, природа которой заслуживает упоминания, поскольку она неизбежно ведет к двум возможным значениям параметра " a". Первая систематическая ошибка заложена в дефиниции глубины потенциальной ямы D e , которую Морз принимает равной положению последнего колебательного уровня v max , и таким образом, величина D e оказывается заниженной на величину, несколько меньшую последнего колебательного кванта.…”