- Volume 25, Number 1 (2019), 89-111.
Stein’s method and approximating the quantum harmonic oscillator
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.
Bernoulli, Volume 25, Number 1 (2019), 89-111.
Received: November 2016
Revised: March 2017
First available in Project Euclid: 12 December 2018
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McKeague, Ian W.; Peköz, Erol A.; Swan, Yvik. Stein’s method and approximating the quantum harmonic oscillator. Bernoulli 25 (2019), no. 1, 89--111. doi:10.3150/17-BEJ960. https://projecteuclid.org/euclid.bj/1544605240