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Updated: Jun 29, 2026
09:23
Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
Published on: May 30, 2014
Stein's method and approximating the quantum harmonic oscillator
Ian W McKeague1, Erol A Peköz2, Yvik Swan3
1Columbia University, Department of Biostatistics, Room R639, 722 West 168th Street, New York, NY 10032.
Bernoulli : Official Journal of the Bernoulli Society for Mathematical Statistics and Probability
|June 11, 2019
Summary
This study extends quantum theory by showing higher energy levels emerge from a deterministic framework. It proves quantum position densities arise from a generalized Stein
Area of Science:
- Quantum mechanics
- Mathematical physics
- Statistical mechanics
Background:
- Quantum theory can be interpreted as the continuum limit of a deterministic theory with a finite number of classical worlds.
- A Gaussian limit theorem for ground state particle positions was previously conjectured and proven using Stein's method.
Purpose of the Study:
- To demonstrate how quantum position probability densities for energy levels beyond the ground state arise from a generalized Stein's method.
- To derive a rate of convergence for particle positions in the first excited state to the Maxwell-Boltzmann distribution.
Main Methods:
- Generalization of Stein's method to analyze distributional fixed points for quantum systems.
- Development of novel techniques to handle singularities in the 'density approach' of Stein's method.
- Application of the generalized method to derive convergence rates for particle position distributions.
Main Results:
- Quantum position probability densities for higher energy levels are shown to emerge as distributional fixed points.
- A rate of distributional convergence is obtained for conjectured particle positions in the first excited state towards the Maxwell distribution.
- New mathematical techniques were developed to overcome limitations of existing methods for higher energy levels.
Conclusions:
- The study provides a novel framework for understanding quantum mechanics from a deterministic perspective, extending beyond the ground state.
- The findings offer new insights into the mathematical foundations of quantum theory and statistical mechanics.
- The developed methods have potential applications in analyzing complex quantum systems and their emergent classical behaviors.
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