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Journal of Physics A Mathematical and General

A harmonic oscillator on the Poincaré disc and hypercontractivity

Shunlong Luo

Published Jul 21, 1997• 1 citation
Publisher

Abstract

Based on the geometry of a Poincaré disc, we construct a relativistic analogue of quantum mechanical harmonic oscillator with a hyperbolic phase space. The Hamiltonian is closely related to the ultraspherical operator and enjoys hypercontractivity. In the large-radius and large-spin limit, we recover the ordinary harmonic oscillator.
DOI
10.1088/0305-4470/30/14/022

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For the Dirac equation in one space dimension, we examine the possibility of harmonic oscillator (HO) potentials. A HO potential is such that it has only bound states and the energy levels are equally spaced. Regarding their Lorentz transformation properties, there are three types of potentials for the Dirac equation: vector, scalar, and pseudoscalar. HO potentials are possible for the scalar and pseudoscalar types, but not for the vector type. We show how HO potentials can be constructed by means of the ``inverse scattering method.'' We also examine the behavior of a wave packet in the HO potentials so constructed. The behavior is, in most of the cases, very similar to that of Schr\"odinger's coherent wave packet of the nonrelativistic harmonic oscillator.
In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.