Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

The Bohr Model02:18

The Bohr Model

Following the work of Ernest Rutherford and his colleagues in the early twentieth century, the picture of atoms consisting of tiny dense nuclei surrounded by lighter and even tinier electrons continually moving about the nucleus was well established. This picture was called the planetary model since it pictured the atom as a miniature “solar system” with the electrons orbiting the nucleus like planets orbiting the sun. The simplest atom is hydrogen, consisting of a single proton as the nucleus...
The de Broglie Wavelength02:32

The de Broglie Wavelength

In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...
The Uncertainty Principle04:08

The Uncertainty Principle

Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He mathematically...
The Quantum-Mechanical Model of an Atom02:45

The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. Schrödinger...
Molecular Orbital Theory I02:35

Molecular Orbital Theory I

Overview of Molecular Orbital Theory
Reaction Mechanisms: The Steady-State Approximation01:26

Reaction Mechanisms: The Steady-State Approximation

The steady-state approximation, also referred to as the quasi-steady-state approximation to differentiate it from a true steady state, is a widely used method for simplifying calculations in complex reaction mechanisms. This approach is particularly useful when dealing with multi-step reactions that involve reverse reactions or several steps, which can significantly increase mathematical complexity and make the reactions nearly unsolvable analytically.The steady-state approximation operates on...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Authors reply to letter to the editor regarding "Selective serotonin reuptake inhibitor use during pregnancy and maternal depression: a nationwide birth cohort study on risks to the mother and the newborn".

American journal of obstetrics & gynecology MFM·2026
Same author

Selective serotonin reuptake inhibitor use during pregnancy and maternal depression-a nationwide birth cohort study on risks to the mother and the newborn.

American journal of obstetrics & gynecology MFM·2026
Same author

Assessing a Mitochondrial Disease Treatment via a Novel Statistical Technique for Accelerometer Data.

Annals of clinical and translational neurology·2025
Same author

Post-selection inference for high-dimensional mediation analysis with survival outcomes.

Scandinavian journal of statistics, theory and applications·2025
Same author

Association between maternal c-reactive protein levels in pregnancy and growth trajectories of head circumference during the first year of postnatal life - A secondary data analysis of a case-control study on autism.

Pediatrics and neonatology·2025
Same author

Maternal vitamin B12 during pregnancy and schizophrenia in offspring.

Psychiatry research·2024
Same journal

Data fusion methods for the heterogeneity of treatment effect and confounding function.

Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability·2026
Same journal

Semiparametric regression of panel count data with informative terminal event.

Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability·2025
Same journal

Nonparametric inference for reversed mean models with panel count data.

Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability·2025
Same journal

Universal sieve-based strategies for efficient estimation using machine learning tools.

Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability·2021
Same journal

Bayesian graph selection consistency under model misspecification.

Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability·2021
Same journal

Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields.

Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability·2019
See all related articles

Related Experiment Video

Updated: Jun 29, 2026

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
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
PubMed
Summary
This summary is machine-generated.

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

Keywords:
Higher energy levelsInteracting particle systemMaxwell distributionStein’s method

More Related Videos

Measurement of Quantum Interference in a Silicon Ring Resonator Photon Source
12:19

Measurement of Quantum Interference in a Silicon Ring Resonator Photon Source

Published on: April 4, 2017

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Related Experiment Videos

Last Updated: Jun 29, 2026

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
09:23

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Published on: May 30, 2014

Measurement of Quantum Interference in a Silicon Ring Resonator Photon Source
12:19

Measurement of Quantum Interference in a Silicon Ring Resonator Photon Source

Published on: April 4, 2017

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

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.