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arXiv:math-ph/0612038

Topological quantization of the harmonic oscillator

Francisco Nettel, Hernando Quevedo

Published Dec 13, 2006• 5 citations
arXivHTMLPDF

Abstract

We present a derivation of the energy spectrum of the harmonic oscillator by using the alternative approach of topological quantization. The spectrum is derived from the topological invariants of a particular principal fiber bundle which can be assigned to any configuration of classical mechanics, when formulated according to Maupertuis formalism.

DOI
10.48550/arxiv.math-ph/0612038

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•Dec 14, 2007•1 citation
We present a derivation of the energy spectrum of the harmonic oscillator by using the alternative approach of topological quantization. The spectrum is derived from the topological invariants of a particular principal fiber bundle which can be assigned to any configuration of classical mechanics, when formulated according to Maupertuis formalism.
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.
Two methods for obtaining the ladder spectrum characteristic of the simple harmonic oscillator are widely used in elementary quantum mechanics. In the first method, that of series solution, quantization is shown to follow from the requirement that the wave function be normalizable. In the second, it is common to derive quantisation in an operator formalism based on factorization of the wave equation. In this latter method the role of normalizability is not customarily discussed. We show explicitly that the same boundary condition implicit in the operator manipulations leads to quantization in both methods.
Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold ℳ as its configuration space is based on a quantization method, which, in general, reflects global properties of ℳ, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on ℳ. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold ℳ, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on ℳ, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self-adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self-adjoint extensions of the energy operator. An experimental setup to verify some of the results is suggested.
In this work we present a heuristic derivation for the energy levels of a linear harmonic oscillator that involves only algebra. This derivation offers the instructor a way to rationalize the energy level spectrum of a harmonic oscillator without having to solve the complicated differential equation prescribed by the Schrödinger equation for the harmonic oscillator. This derivation treats the harmonic oscillator as a quasiparticle in a box and yields qualitative results, these being the correct scaling of the energy with quantum number and vibrational frequency, but does not yield the exact result. By comparing the exact result with the result of the heuristic derivation students are introduced to the idea of tunneling in quantum mechanics and how tunneling can effect the energy levels of quantum mechanical systems.
Abstract. In this theoretical paper, we propose a general derivation of the adiabatic invariant of the n-degree-of-freedom harmonic oscillator, available whichever the physical nature of the oscillator and of the parametrical excitation it undergoes. This derivation is founded on the use of the classical Glauber variables and ends up with this simplest result: the oscillator’s adiabatic invariant is just the sum of all the semiclassical quanta numbers associated with its different eigenmodes. PACS numbers: 45.20.-d, 45.20.Jj, 45.05.+x 1.