• 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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

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.

Export citation


  • [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.