Infinitesimal and Infinite Numbers as an Approach to Quantum Mechanics

TL;DR

This paper explores the use of non-Archimedean mathematics and ultrafunctions to provide a richer framework for quantum mechanics, especially for solving the Schrödinger equation with delta potential.

Contribution

It introduces ultrafunctions within non-Archimedean fields as a novel mathematical tool for quantum mechanics applications.

Findings

Ultrafunctions offer a more comprehensive description of quantum systems.

Application to Schrödinger equation with delta potential demonstrates advantages.

Non-Archimedean approach simplifies handling of singular potentials.

Abstract

Non-Archimedean mathematics is an approach based on fields which contain infinitesimal and infinite elements. Within this approach, we construct a space of a particular class of generalized functions, ultrafunctions. The space of ultrafunctions can be used as a richer framework for a description of a physical system in quantum mechanics. In this paper, we provide a discussion of the space of ultrafunctions and its advantages in the applications of quantum mechanics, particularly for the Schr\"{o}dinger equation for a Hamiltonian with the delta function potential.