Stein’s method and approximating the quantum harmonic oscillator

Abstract

Hall et al. (2014) 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. (2014) and proven by McKeague and Levin (2016) 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 (2011)) has a singularity.

Some recent advances for limit theorems

“…Since then, the method has been developed for numerous probability distributions both in univariate and multivariate contexts producing quantitative bounds for limit theorems (1) with corresponding asymptotic distributions. Let us mention the following series of papers (together with their associated target limiting laws): for exponential and Laplace approximations [27,55,93,98], for gamma and chi-squared approximations [43,64,85,97], for Compound Poisson approximation [6], for negative binomial approximation [22], for beta approximation [42,67], for semicircular approximation [69], for variance-gamma approximation [58,62], for two-sided Maxwell approximation [86] and for symmetric α-stable approximation [122].…”

The day the universes interacted: quantum cosmology without a wave function

“…A note before we move on... In a recent paper by I. McKeague, E. Peköz and Y. Swan [20] a more general case has been studied, in which the interworld potential has the form…”

Stein’s method and approximating the multidimensional quantum harmonic oscillator

“…An excited state of a quantum system is any state that has a higher energy than the ground state. The first excited state of the one-dimensional quantum harmonic oscillator can be represented in the MIW setting via the extension of the interworld potential introduced in [14], and leads to the two-sided Maxwell distribution as the limit (agreeing with quantum theory). Nonlocality was studied in [7] by introducing other extensions of the MIW interworld potential for the first excited state using higher-order smoothing methods.…”