Particle and Wave: Developing the Quantum Wave Accompanying a Classical Particle

TL;DR

This paper constructs a wave function associated with a classical particle using relativity and periodicity, deriving a wave description that aligns with quantum mechanics and reproduces the Schrödinger equation.

Contribution

It introduces a novel method to derive quantum wave behavior from classical particles by combining relativity and intrinsic periodicity, leading to a Schrödinger-like differential equation.

Findings

Wave functions for classical particles are constructed using special relativity.

A standing wave pattern emerges from bidirectional motion analysis.

Derived equations match the form of the Schrödinger equation.

Abstract

The relationship between classical and quantum mechanics is explored in an intuitive manner by the exercise of constructing a wave in association with a classical particle. Using special relativity, the time coordinate in the frame of reference of a moving particle is expressed in terms of the coordinates in the laboratory frame of reference in order to provide an initial spatiotemporal function to work from in initiating the development of a quantum wave. When temporal periodicity is ascribed to the particle, a provisional spatiotemporal function for a particle travelling at constant velocity manifests itself as an running wave characterized by parameters associated with the moving particle. A wave description for bidirectional motion is generated based on an average time coordinate for a combination of oppositely directed elementary running waves, and the resulting spatiotemporal function exhibits wave behavior characteristic of a standing wave. Ascribing directional orientation to the intrinsic periodicity of the particle introduces directional sub-states; variations in the relative number of sub-states as a function of angle in combined states lead to spatially varying magnitudes for the associated waves. Further analysis leads to full mathematical expression for all waves representing free particle motion. A generalization for particles subject to force fields enables us to develop a governing differential equation identical in form to the Schroedinger equation.