Algebraic solution for the classical harmonic oscillator

TL;DR

This paper introduces an algebraic method for solving the classical harmonic oscillator using Hamiltonian formalism, highlighting parallels with quantum operators and aiding in quantization procedures.

Contribution

It presents a novel algebraic approach to the classical harmonic oscillator, emphasizing the connection with quantum operator methods and facilitating quantization.

Findings

Algebraic solution parallels quantum operator methods

Provides a straightforward quantization procedure

Highlights similarities between classical and quantum formalisms

Abstract

The harmonic oscillator is one of the most studied systems in Physics with a myriad of applications. One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator with analytic and algebraic approaches. In the algebraic solution, creation and annihilation operators are introduced to factorize the Hamiltonian. This work presents an algebraic solution for the simple harmonic oscillator in the context of Classical Mechanics, exploring the Hamiltonian formalism. In this solution, similarities between the canonical coordinates in a convenient basis for the classical problem and the corresponding operators in Quantum Mechanics are highlighted. Moreover, the presented algebraic solution provides a straightforward procedure for the quantization of the classical harmonic oscillator, motivating and justifying some operator definitions commonly used to solve the correspondent problem in Quantum Mechanics.