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Journal | MESA

Energetic fluctuation expansion in quantum evolvers dynamics from the new extended form of probabilistic evolution theoretical (PREVTH) perspective

Berfin Kalay

Published May 27, 2018• 0 citations
OpenAlex

Abstract

This paper is about the Quantum Evolver Dynamics which is based  on the Poisson Bracket algebra over the basic quantum operators for a univariate system whose potential function is analytic or  at most having polar singularity with a finite multiplicity in  the position operator. Paper is devoted to the formulation of  the evolver (temporally varying operator describing the evolution of a given operator) dynamics. The basic aim is to expand any given evolver to a fluctuation expansion with respect to the fluctuation in the Hamiltonian of the system. We reveal the relation of this (energetic) fluctuation expansion coefficient operators with the probabilistic evolution theoretical telescopic Kronecker power series which have been quite recently extended pretty much, for the isolated systems. Paper does not contain any illustration because of the concreteness in what we have developed here.

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Nancy Makri, William Hughes Miller

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This paper is about the Quantum Evolver Dynamics which is based  on the Poisson Bracket algebra over the basic quantum operators for a univariate system whose potential function is analytic or  at most having polar singularity with a finite multiplicity in  the position operator. Paper is devoted to the formulation of  the evolver (temporally varying operator describing the evolution of a given operator) dynamics. The basic aim is to expand any given evolver to a fluctuation expansion with respect to the fluctuation in the Hamiltonian of the system. We reveal the relation of this (energetic) fluctuation expansion coefficient operators with the probabilistic evolution theoretical telescopic Kronecker power series which have been quite recently extended pretty much, for the isolated systems. Paper does not contain any illustration because of the concreteness in what we have developed here.
The time variant power series expansion for the expectation value of a given quantum dynamical operator is well-known and well-investigated issue in quantum dynamics. However, depending on the operator and Hamiltonian singularities this expansion either may not exist or may not converge for all time instances except the beginning of the evolution. This work focuses on this issue and seeks certain cures for the negativities. We work in the extended space obtained by adding all images of the initial wave function under the system Hamiltonian’s positive integer powers. This requires the introduction of certain appropriately defined weight operators. The resulting better convergence in the temporal power series urges us to call the new defined entities “extended space expectation values” even though they are constructed over certain weight oparetors and are somehow pseudo expectation values.
The goal of this work is, the utilization of a method recently developed by the author, in quantum dynamical problems. Main idea is based on the relation between quantum and classical dynamics such that quantum dynamical problems become their classical dynamical counterparts when they are formulated via expectation values of certain operators and the probability density tends to be sharply localized. Expectation values can be expanded into a series in ascending appearence multiplicity of the complement of the projection operator which projects the space spanned by the wave function. This expansion contains fluctuation terms as unknowns besides the expectation values. We construct an infinite set of differential equations. The finite truncations of this set enables us to approximate the quantum dynamics of the system under consideration.
The variation of the expectation matrix of position and momentum operator in time can serve us to investigate the evolution of a quantum system in time. This brings the utilization of the ODEs instead of Schrödinger's equation at the expense of incapability for the calculation of the wave function. As long as we deal with the observables which can be expressed in terms of position and momentum operators this may be quite practical to know about the quantum dynamics of the system under consideration. Expectation matrix of an operator becomes a function of the expectation matrices of the position and momentum operator when the fluctuations diminish to zero. At this limit, the time‐variant ODEs for the expectation matrices of the position and momentum operator can be handled by using the matrix algebraic tools even in the case of nonlinearities in the potential function. This work presents certain details about these points.
This work is a new extension to our a very recent work whose paper will appear in the proceedings of a very recent international conference. What we have done in the previous work is the use of a weight operator to suppress the singularities causing nonexistence of some of temporal Maclaurin expansion coefficients. The weight operator has been constructed in such a way that certain number of expectation values of position operator’s first positive integer powers with and without the chosen weight operator match. Therein this match has not been considered for the momentum operator’s corresponding power expectation values and a finite linear combination of the spatial variable’s first reciprocal powers has been used in the construction of the weight operator. Here, that approach is extended to the case where matches for both position and momentum operators are considered and the weight operator involves finite linear combinations of the spatial variable’s both positive integer powers and their reciprocals.
Quantum dynamics (e.g., the Schrödinger equation) and classical dynamics (e.g., Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. The difference between both worlds is due to the presence of extra structure in the quantum case, that leads to the appearance of the probabilistic nature of the measurements and the indetermination and superposition principles. In this paper we first show that the quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well defined Poisson bracket allows to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann's distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. This work provides the basis for extending stochastic methods to Ehrenfest dynamics.
The coordinate matrix element of the time evolution operator, exp[−iĤt/ℏ], is determined by expanding (its exponent) in a power series in t. Recursion relations are obtained for the expansion coefficients which can be analytically evaluated for any number of degrees of freedom. Numerical application to the tunneling matrix element in a double well potential and to the reactive flux correlation function for a barrier potential show this approach to be a dramatic improvement over the standard short time approximation for the propagator. Its use in a Feynman path integral means that fewer ‘‘time slices’’ in the matrix product exp[(−i/ℏ)ΔtĤ]N, Δt=t/N, will be required. The first few terms in the present expansion constitute a fully quantum version of the short time propagator recently obtained by us using semiclassical methods [Chem. Phys. Lett. 151, 1 (1988)].