Quantum mechanics without potential function

TL;DR

This paper introduces an alternative quantum mechanics formulation that derives physical properties directly from wavefunctions expressed as polynomial basis expansions, bypassing the need for a potential function.

Contribution

It presents a novel approach to quantum systems that does not rely on potential functions, expanding the set of analytically solvable models.

Findings

Derives scattering phase shifts and bound states without potential functions.

Enables analysis of previously untreated quantum systems.

Provides illustrative examples for multi-parameter systems.

Abstract

In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schr\"odinger equation, which is solved for the wave function, bound states energy spectrum and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and there-parameter systems.