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Article (Scientific journals)
Stein’s method and approximating the quantum harmonic oscillator
McKeague, I. W.; Peköz, E. A.; Swan, Yvik 
2019 • In Bernoulli, 25 (1), p. 89-111
 

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Keywords :
Higher energy levels; Interacting particle system; Maxwell distribution; Stein’s method
Abstract :
[en] Hall et al. [Phys. Rev. X 4 (2014) 041013] recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical “worlds.” A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall et al. [Phys. Rev. X 4 (2014) 041013] and proven by McKeague and Levin [Ann. Appl. Probab. 26 (2016) 2540–2555] using Stein’s method. In this article we show how quantum position probability densities for higher energy levels beyond the ground state may arise as distributional fixed points in a new generalization of Stein’s method. These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual “density approach” Stein solution (see Chatterjee and Shao [Ann. Appl. Probab. 21 (2011) 464–483] has a singularity. © 2019 ISI/BS
Disciplines :
Mathematics
Author, co-author :
McKeague, I. W.;  Department of Biostatistics, Columbia University, 722 West 168th Street, New York, NY 10032, United States
Peköz, E. A.;  Questrom School of Business, Boston University, 595 Commonwealth Avenue, Boston, MA 02215, United States
Swan, Yvik ;  Université de Liège - ULg
Language :
English
Title :
Stein’s method and approximating the quantum harmonic oscillator
Publication date :
2019
Journal title :
Bernoulli
ISSN :
1350-7265
Publisher :
Chapman & Hall, United Kingdom
Volume :
25
Issue :
1
Pages :
89-111

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