Quantum Mechanics without an Equation of Motion

TL;DR

This paper introduces a novel formulation of quantum mechanics for finite level systems with spherical symmetry, where dynamics are described without differential equations, using a matrix-based approach in function space.

Contribution

It presents a new quantum mechanics framework that replaces differential equations with a matrix representation in function space for finite systems.

Findings

Wavefunction expressed as an infinite sum in complete basis

Interaction modeled by a finite tridiagonal symmetric matrix

System dynamics encoded in the scattering matrix

Abstract

We propose a formulation of quantum mechanics in three dimensions with spherical symmetry for a finite level system whose dynamics is not governed by a differential equation of motion. The wavefunction is written as an infinite sum in a complete set of square integrable functions. Interaction in the theory is introduced in function space by a real finite tridiagonal symmetric matrix. Information about the structure and dynamics of the system is contained in the scattering matrix, which is defined in the usual way.