• Bernoulli
  • Volume 25, Number 1 (2019), 89-111.

Stein’s method and approximating the quantum harmonic oscillator

Ian W. McKeague, Erol A. Peköz, and Yvik Swan

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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.

Article information

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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higher energy levels interacting particle system Maxwell distribution Stein’s method


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.

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  • [1] Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. (2) 85 166–179.
  • [2] Cacoullos, T. and Papathanasiou, V. (1989). Characterizations of distributions by variance bounds. Statist. Probab. Lett. 7 351–356.
  • [3] Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 1–40.
  • [4] Chatterjee, S. and Shao, Q.-M. (2011). Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Ann. Appl. Probab. 21 464–483.
  • [5] Chen, L.H.Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Probability and Its Applications (New York). Heidelberg: Springer.
  • [6] Döbler, C. (2015). Stein’s method of exchangeable pairs for the beta distribution and generalizations. Electron. J. Probab. 20 no. 109, 34.
  • [7] Goldstein, L. and Reinert, G. (1997). Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935–952.
  • [8] Hall, M.J.W., Deckert, D.A. and Wiseman, H.M. (2014). Quantum phenomena modeled by interactions between many classical worlds. Phys. Rev. X 4 041013.
  • [9] Ley, C., Reinert, G. and Swan, Y. (2017). Stein’s method for comparison of univariate distributions. Probab. Surv. 14 1–52.
  • [10] McKeague, I.W. and Levin, B. (2016). Convergence of empirical distributions in an interpretation of quantum mechanics. Ann. Appl. Probab. 26 2540–2555.
  • [11] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
  • [12] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge: Cambridge Univ. Press.
  • [13] Peköz, E.A. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab. 39 587–608.
  • [14] Peköz, E.A., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 1188–1218.
  • [15] Peköz, E.A., Röllin, A. and Ross, N. (2016). Generalized gamma approximation with rates for urns, walks and trees. Ann. Probab. 44 1776–1816.
  • [16] Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv. 8 210–293.
  • [17] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes – Monograph Series 7. Hayward, CA: IMS.