Reference : Stein’s method and approximating the quantum harmonic oscillator
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Stein’s method and approximating the quantum harmonic oscillator
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 mailto [Université de Liège - ULg]
Chapman & Hall
United Kingdom
[en] Higher energy levels ; Interacting particle system ; Maxwell distribution ; Stein’s method
[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

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