A Fundamental Form of the Schrodinger Equation

TL;DR

This paper introduces a new first-order equation from which the Schrödinger equation can be derived, providing solutions for electron scattering and extending to three dimensions, offering a novel foundational approach.

Contribution

It presents a fundamental first-order equation that generalizes the Schrödinger equation and demonstrates its application in 1D and 3D electron scattering problems.

Findings

Sum of spin coefficients matches quantum results

Derived 1D and 3D Schrödinger equations from the new formulation

Validated the approach with electron scattering scenarios

Abstract

We propose a first order equation from which the Schrodinger equation can be derived. Matrices that obey certain properties are introduced for this purpose. We start by constructing the solutions of this equation in 1D and solve the problem of electron scattering from a step potential. We show that the sum of the spin up and down, reflection and transmission coefficients, is equal to the quantum mechanical results for this problem. Furthermore, we present a 3D version of the equation which can be used to derive the Schrodinger equation in 3D.