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Journal of Physics A Mathematical and General

On the stochastic Lagrangian and a new derivation of the Schrodinger equation

Ian M Davies

Published Aug 21, 1989• 7 citations
Publisher

Abstract

The author obtains an expression for the stochastic Lagrangian as used in stochastic mechanics in a straightforward manner without obtaining the singular terms of Guerra et. at. (1983). He then presents a new derivation of the Schrodinger equation.
DOI
10.1088/0305-4470/22/16/010

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This is the fourth paper of a series devoted to mathematically derive the Schrödinger equation using stochastic constructs. We then show that Quantum Mechanics has a stochastic support by following two paths: we first present a stochastic derivation already known in the literature and show that it is equivalent to the characteristic function derivation, made by us in the first paper of this series. However, this approach does not furnish the true stochastic dynamics of the quantum mechanical system, but only an averaged one. We then present a new derivation of the Schrödinger equation based on Langevin equations that present all features of a true stochastic system. These two approaches then improve our understanding of Quantum Mechanics to encompass stochastic behavior.
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
Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.
In this note, we argue that the Schrodinger equation for one particle represents ensemble averages taken from stochastic momentum values of a single particle at different times, with time being removed. A new time, representing the evolution of the ensemble is then introduced especially for the case in which energy is being added or removed from a system (e.g. V(x,t)). The momentum stochastic features arise from the potential (or force) acting in a stochastic manner leading to a distribution of momentum values at x (at different times). If V(x,t), we argue the system adds or removes some average energy at x at t. We suggest that in reality, stochastic amounts of energy are added or removed (as long as the average is obtained). This leads to a second distribution, namely one in energy being established.. Thus, W(x,t) may be expanded in terms of exp(ipx) for a fixed t or in exp(i Ei t) for a fixed x. As a result, one may calculate average energy using either Average (p*p/2m) + V(x,t) or [Sum over Ei exp(i Ei t) Ei ] / W(x,t). Equating the two stochastic approaches yields the Schrodinger equation.