A vectorial form of the Schr\"odinger equation

TL;DR

This paper reformulates the time-dependent Schrödinger equation using only vector algebra, eliminating complex numbers, and demonstrates its equivalence to traditional formulations for key quantum systems, also relating it to Maxwell's equations.

Contribution

It introduces a real vectorial form of the Schrödinger equation, simplifying its mathematical structure and connecting it to classical electromagnetism.

Findings

Equivalent predictions for hydrogen atom and harmonic oscillator

Reformulation avoids complex numbers

Connection to Maxwell-Ampère equation

Abstract

We rewrite the time dependent Schr\"odinger equation by using only three dimensional vector algebra and by avoiding to introduce any complex numbers. We show that this equation leads to the same conclusions than the "complex version" concerning the hydrogen atom and the harmonic oscillator. We show also that this equation can be written as a Maxwell-Amp\`ere equation.