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arXiv:1203.2526

The Dirac factorization method and the harmonic oscillator

D. Babusci, G. Dattoli

Published Mar 12, 2012• 0 citations
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Abstract

We apply the Dirac factorization method to the nonrelativistic harmonic oscillator and, more in general, to Hamiltonians with a generic potential. It is shown that this procedure naturally leads to a supersymmetric formulation of the problems under study. It is also speculated on the physical meaning underlying this method and it is suggested that the vacuum field fluctuations can be viewed as the spontaneous emission of the associated two-level system, whose quantization is due to the noncommuting nature of the harmonic oscillator canonical variables.

DOI
10.48550/arxiv.1203.2526

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Factorization of quantum mechanical Hamiltonians has been a useful technique for some time. This procedure has been given an elegant description by supersymmetric quantum mechanics, and the subject has become well developed. We demonstrate that the existence of raising and lowering operators for the harmonic oscillator (and many other potentials) can be extended to their supersymmetric partners. The double supersymmetry (or a factorization chain) is used to obtain non-singular isospectral potentials, and the explicit expressions for the ladder operators, wavefunctions and probability densities are provided. This application avoids the technical complexities of the most general approaches, and requires relatively modest methods from supersymmetric quantum mechanics.
Review paper is devoted to the relativistic configuration space (RCS) concept, a version of the relativistic Quantum Mechanics in RCS, the generalization of the Dirac-Infeld-Hall factorization method in the framework of the noncommutative differential calculus natural for RCS, different versions of the deformed oscillators, emerging as the generalization of the harmonic oscillator for RCS. In the formulation of the Newton-Wigner postulates for the relativistic localized states the hypothesis of commutativity of the position operator components is silently accepted as an evident fact. In the present work it is shown that commutativity is not necessary condition and the alternative (noncommutative) approach to the relativistic position operator and localization concept can be realized in a framework of the physically as well as mathematically comprehensive scheme. The different generalizations of the Dirac-Infeld-Hall factorization method for this case are constructed. This method enables us to find out all possible generalizations of the most important nonrelativistic integrable case—the harmonic oscillator. It is shown also that the relativistic oscillator = q —oscillator.
We review SUSY in non-relativistic quantum mechanics as a powerful tool for studying one dimensional potentials and discuss some applications. The Dirac operatorial factorization method for the harmonic oscillator in QM is generalized to a broader scope. A Schr¨odinger second order operator can be factorized into two first-order ones by solving a Riccati equation and determining the corresponding superpotential. We apply the technique to square well, P¨oschl-Teller and finite barrier potentials, analyzing the resulting energy spectra and the transmission and reflection coefficients. Both discrete and continuous energy spectrum cases are discussed.
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.
A new scheme of field quantization is proposed. Instead of associating with different frequencies different oscillators we begin with a single oscillator that can exist in a superposition of different frequencies. The idea is applied to the electromagnetic radiation field. Using the standard Dirac-type mode-quantization of the electromagnetic field we obtain several standard properties such as coherent states or spontaneous and stimulated emission. As opposed to the standard approach the vacuum energy is finite and does not have to be removed by any ad hoc procedure.