A Class of Stochastic and Distributions-Free Quantum Mechanical Evolution Equations

Gregorio Jose Costanza

doi:10.12743/quanta.v10i1.159

Abstract

A procedure allowing to construct rigorously discrete as well as continuum deterministic evolution equations from stochastic evolution equations is developed using Dirac's bra–ket notation. This procedure is an extension of an approach previously used by the author coined Discrete Stochastic Evolution Equations. Definitions and examples of discrete as well as continuum one-dimensional lattices are developed in detail in order to show the basic tools that allow to construct Schrödinger-like equations. Extension to multi-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 allow the construction of other evolution equations. Also, using a limiting procedure, it is possible to derive the Schrödinger equation from the Schrödinger-like equations. Another possible approach is given in the appendix.Quanta 2021; 10: 22–33.

Highlights

Tion equation is derived from a set of stochastic quantum mechanical evolution equations using Dirac’s bra–ket notation, which can be considered an extension of an approach coined Discrete Stochastic Evolution Equations (DSEE) [1]
The possibility of obtaining deterministic evolution equations from stochastic evolution equations is based on the assumption that the Hamiltonian and the wave functions are statistically independent
The question that will be answered is: Is it possible sion to d-dimensional lattices is considered and a conto find a Hamiltonian, independent of distributions, that tinuum Schrodinger-like equation is obtained. The oneafter introducing it in the general evolution equations dimensional case is obtained as a special case, as expected

Summary

Costanza

Definitions and examples of discrete as well as continuum one-dimensional lattices are developed in detail in order to show the basic tools that allow to construct Schrodinger-like equations. This apderivative of the delta function δ(x − x′), V(x) is the po- proach allows to prove that the usual proposed evolution tential, m is the mass of the particle, and ħ is the reduced equations can be obtained rigorously from first princi-. If it is needed to obtain a Schrodingerlike equation, the Hamiltonian must be split in a sum like

The continuum one-dimensional lattice

Extension to a system of many particles

The Schrodinger equation as a limiting case of the Schrodinger-like equation

Conclusions