Deriving the time-dependent Schrodinger m- and p-equations from the Klein-Gordon equation

TL;DR

This paper introduces a novel derivation of the Schrödinger equation from the Klein-Gordon equation using a directional factorization scheme, and also derives a new momentum-dominated wavefunction propagation equation as a complement to the traditional mass-dominated Schrödinger equation.

Contribution

It presents a direct derivation of the Schrödinger equation from the Klein-Gordon equation and introduces a new momentum-dominated wavefunction equation.

Findings

Derived the Schrödinger equation from Klein-Gordon using directional factorization.

Established an alternative wavefunction propagation equation in the momentum limit.

Provides a complementary perspective to the traditional mass-based Schrödinger equation.

Abstract

I present an alternative and rather direct way to derive the well known Schr\"odinger equation for a quantum wavefunction, by starting with the Klein Gordon equation and applying a directional factorization scheme. And since if you have a directionally factorizing hammer, everything looks like a factorizable nail, I also derive an alternative wavefunction propagation equation in the momentum-dominated limit. This new Schr\"odinger $p$-equation therefore provides a potentially useful complement to the traditional Schr\"odinger $m$-equation's mass-dominated limit.