Stein's method and approximating the multidimensional quantum harmonic oscillator

TL;DR

This paper applies Stein's method to analyze how well discrete models of multidimensional quantum harmonic oscillators approximate the true ground state, revealing convergence rates and dimensional effects.

Contribution

It extends Stein's method to multidimensional quantum harmonic oscillators, providing convergence rates in Wasserstein distance for the first time.

Findings

Fastest convergence occurs in three dimensions.

Convergence rate depends on the dimension.

Radial component analysis is key to the approach.

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. Such approximations have previously been studied for one-dimensional quantum harmonic oscillators, but the multidimensional case has remained unresolved. 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. The fastest rate of convergence to the ground state is found to occur in three dimensions. This result is obtained using a discrete density approach to Stein's method applied to the radial component of the ground state solution.