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Probabilistic aspects of the SU(2)-harmonic oscillator

Part of: Groups and algebras in quantum theory Distribution theory - Probability General mathematical topics and methods in quantum theory Limit theorems

Published online by Cambridge University Press:  14 July 2016

Shunlong Luo
Shunlong Luo*
Affiliation:
Academia Sinica, Beijing
*
∗Postal address: Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing, 100080, People's Republic of China. Email address: luosl@amath4.amt.ac.cn
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Abstract

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.

Keywords

Harmonic oscillatorSU(2)Bargmann transformbinomial distributionsGaussian distributionsPoisson distributions

MSC classification

Primary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 60E10: Characteristic functions; other transforms
Secondary: 81R05: Finite-dimensional groups and algebras motivated by physics and their representations 60F05: Central limit and other weak theorems
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 306 - 314
Copyright
Copyright © 2000 by The Applied Probability Trust 

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