Algebraic solution for the classical harmonic oscillator
- URL: http://arxiv.org/abs/2306.13488v1
- Date: Fri, 23 Jun 2023 13:12:14 GMT
- Title: Algebraic solution for the classical harmonic oscillator
- Authors: Murilo B. Alves
- Abstract summary: The harmonic oscillator is one of the most studied systems in Physics.
One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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