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Stein’s method and approximating the multidimensional quantum harmonic oscillator

Part of: General mathematical topics and methods in quantum theory Limit theorems

Published online by Cambridge University Press:  11 April 2023

Ian W. McKeague  and
Yvik Swan Open the ORCID record for Yvik Swan [Opens in a new window]
Ian W. McKeague*
Affiliation:
Columbia University
Yvik Swan*
Affiliation:
Université libre de Bruxelles
*
*Postal address: Department of Biostatistics, Columbia University, 722 West 168th Street, New York, NY 10032 U.S.A. Email: im2131@cumc.columbia.edu
**Postal address: Université libre de Bruxelles, Département de Mathématique – CP 210, Boulevard du Triomphe, 1050 Bruxelles, Belgium. Email: yvik.swan@ulb.be
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Abstract

Stein’s method is used to study discrete representations of multidimensional distributions that arise as approximations of states of quantum harmonic oscillators. These representations model how quantum effects result from the interaction of finitely many classical ‘worlds’, with the role of sample size played by the number of worlds. Each approximation arises as the ground state of a Hamiltonian involving a particular interworld potential function. Our approach, framed in terms of spherical coordinates, provides the rate of convergence of the discrete approximation to the ground state in terms of Wasserstein distance. Applying a novel Stein’s method technique to the radial component of the ground state solution, the fastest rate of convergence to the ground state is found to occur in three dimensions.

Keywords

CouplingHamiltonianpoint configurationsWasserstein distance

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 81Q65: Alternative quantum mechanics
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 855 - 873
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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