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.

Keywords: Higher energy levels; Interacting particle system; Maxwell distribution; Stein’s method.

Figure 1.

Example with b(x) = x², N = 22, showing the piecewise constant density having mass 1/(N – 1) uniformly distributed over the intervals between successive x_n compared with the Maxwell density, where the breaks in the histogram are the successive x_n satisfying the recursion (4).

Figure 2.

Example with b(x) = He_k(x)²/k! for k = 2, N = 41, where the breaks in the histogram are the successive x_n satisfying the recursion (7) and the red curve is p_k(x).