# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1901.10945