Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-04T21:17:13.813Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arecchi, F. T., Couptens, E., Gilmore, R., and Thomas, H. (1972). Atomic coherent states in quantum optics. Phys. Rev. A6, 2211–2237.Google Scholar
Bach, A. (1988). Quantum and coherent states. Lett. Math. Phys. 15, 75–79.Google Scholar
Bargmann, V. (1961). On a Hilbert space of holomorphic functions and an associated integral transform. Commun. Pure Appl. Math. 14, 187–214.Google Scholar
Berezin, F. A. (1975). General concept of quantization. Commun. Math. Phys. 40, 153–174.Google Scholar
Feller, W. (1966). An Introduction to Probability and its Applications. John Wiley, New York.Google Scholar
Lieb, E. H. (1973). The classical limit of quantum spin systems. Commun. Math. Phys. 31, 327–340.Google Scholar
Meyer, P. A. (1994). Quantum Probability for Probabilists. Springer, Berlin.Google Scholar
Parthasarathy, K. R. (1992). An Introduction to Quantum Stochastic Calculus. Birkhäuser, Basel.Google Scholar
Perelomov, A. (1986). Generalized Coherent States and their Applications. Springer, Berlin.Google Scholar
Radcliff, J. M. (1971). Some properties of coherent spin states. J. Phys. A: Gen. Phys. 4, 313–323.Google Scholar