A New Description of Quantum Behaviors for a Simple Harmonic Oscillator

TL;DR

This paper proposes a novel approach to quantum behaviors of a harmonic oscillator by integrating classical mechanics with a discrete time assumption, deriving energy quantization from phase space conditions.

Contribution

It introduces a new description linking classical phase space dynamics with quantum energy levels through a symplectic algorithm and discrete time assumption.

Findings

Derives discrete energy levels satisfying $E_n au_n = h$

Successfully integrates classical and quantum mechanics in a unified framework

Shows that imposing phase point conditions yields quantum behaviors from classical mechanics

Abstract

We describe quantum behaviors of a simple harmonic oscillator, starting from the classical mechanics. By imposing two conditions on the phase points generated from a symplectic algorithm, we obtain discrete energy levels, satisfying $E_n \tau_n = h$ where $E_n$ is the total energy of the oscillator and $\tau_n$ is the time step for the closed orbit of $n$-polygon in phase space. We can thus successfully integrate classical and quantum mechanics into a single frame, if we assume that time is discrete.