Reciprocal Symmetry and Unified Classico-Quantum Oscillator And Consistency between a Particle in a Potential Well and a Harmonic Oscillator

TL;DR

This paper introduces a reciprocal symmetric finite-difference approach to unify classical and quantum oscillators, deriving quantum-like energy levels without quantum postulates.

Contribution

It proposes a symmetric difference equation framework that naturally produces quantum energy levels, unifying classical and quantum oscillators without quantum assumptions.

Findings

Symmetric difference equations yield quantum energy levels.

Unified description of classical and quantum oscillators.

No quantum postulates needed for energy quantization.

Abstract

The function exp(iwt) describes an oscillating motion. Energy of the oscillator is proportional to the square of w. exp(iwt) is the solution of a differential equation. We have replaced this differential equation by the corresponding finite-time difference equation with d as the smallest span of time. We have, then, symmetrized the equation so that it remains invariant under the change d going to -d. This symmetric equation has solutions in pairs. The angular speed w is modified to w' or w". w' contains a part with an integer. w" contains a part with a half-integer. This corresponds to quantum mechanical oscillator energy levels. F= a.exp(iwt) describes oscillation between -a and +a. If we make w=0, F describes free oscillation between -a and +a. Reciprocal symmetric oscillator, thus, unifies quantum and classical harmonic oscillators on one hand, and a harmonic oscillator and a free particle in a potential well on the other hand. No quantum mechanical postulates are involved.