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arXiv:1909.01824

A class of stochastic and distributions-free quantum mechanics evolution equations: The Schrödinger-like equations

G. Costanza

Published Sep 2, 2019• 0 citations
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Abstract

A procedure allowing to construct rigorously discrete as well as continuum deterministic evolution equations from stochastic evolution equations is developed using a Dirac's bra and ket notation. This procedure is an extension of an approach previously used by the authors coined Discrete Stochastic Evolution Equations. Definitions and examples of discretes as well as continuum one-dimensional lattices are developed in detail in order to show the basic tools that allows to construct Schrödinger-like equations. Extension to d-dimensional lattices are studied in order to provide a wider exposition and the one-dimensional cases are derived as special cases, as expected. Some variants of the procedure allows to construct other evolution equations. Also, using a limiting procedure, it is possible to derive the Schrödinger equations from the Schrödinger like equations.

DOI
10.48550/arxiv.1909.01824

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Abstract This chapter begins by studying the one-dimensional time-dependent Schrödinger equation. In order to do this the chapter discretizes the spatial coordinates, and thus the differential equation becomes a set of coupled difference equations first order in time. Dealing with a finite number or a numerable set of states makes it easier to understand the concepts of Hilbert space, its measure, and the overlap between states. Furthermore, the chapter only has to deal with the Kronecker symbol as opposed to the Dirac delta function. Later, in Chapter 3, it will consider the continuum limit where we will naturally recover the Schrödinger differential equation, the state normalization condition as an integration condition, and the delta function.
In this article, the following results are obtained: the process of a randomly wandering particle having a size and a continuous trajectory of motion is considered; (b) based on the study of this probabilistic process, a derivation of the Schrodinger equation is proposed; the Planck constant h is expressed in terms of the characteristics of the stationary stochastic process under consideration; the probability distribution density of the n-order derivative for a n-times differentiable stationary stochastic process is determined; it is shown that the obtained generalized Schrodinger equation is applicable not only to describe the possible states of microcosm objects, but also for discrete states of a number of macrocosm stochastic systems
The Schrödinger equation is a beautiful wave equation that leads us directly to the widely used Hamiltonians in quantum mechanics. The Schrödinger equation was also there at the very beginning of quantum mechanics. Therefore it should not be surprising that when most people refer to quantum mechanics they do so from the vantage point of the Schrödinger equation. However, there are also several other versions of quantum mechanics. Most notable is the version of quantum mechanics introduced by Heisenberg that was transformed into a matrix formalism via a collaborative effort of Born, Heisenberg, and Jordan. Then around came our favorite, Dirac's bra ket notation. Dirac also inspired Feynman to introduce his path integrals. But there are also others. A brief description follows.
The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.