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 Quantum-Mechanical Model of an Atom02:45

The Quantum-Mechanical Model of an Atom

60.2K
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.
60.2K
Electron Orbital Model01:18

Electron Orbital Model

74.3K
Orbitals are the areas outside of the atomic nucleus where electrons are most likely to reside. They are characterized by different energy levels, shapes, and three-dimensional orientations. The location of electrons is described most generally by a shell or principal energy level, then by a subshell within each shell, and finally, by individual orbitals found within the subshells.
The first shell is closest to the nucleus, and it has only one subshell with a single spherical orbital called the...
74.3K
The de Broglie Wavelength02:32

The de Broglie Wavelength

33.9K
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...
33.9K
The Uncertainty Principle04:08

The Uncertainty Principle

33.6K
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...
33.6K
The Bohr Model02:18

The Bohr Model

81.8K
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...
81.8K
Molecular Orbital Theory I02:35

Molecular Orbital Theory I

48.1K
Overview of Molecular Orbital Theory
48.1K

You might also read

Related Articles

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

Sort by
Same author

Generalization of Bandlimited Functions and Applications to Quantum Probability Distributions.

Entropy (Basel, Switzerland)·2026
Same author

Conditional Values in Quantum Mechanics.

Entropy (Basel, Switzerland)·2024
Same author

The eigenvalue problem in phase space.

Journal of computational chemistry·2017
Same journal

The Anionic States of Ubiquinone Characterized by Second-Order Approximate Coupled-Cluster Theory.

Journal of computational chemistry·2026
Same journal

Hydrogen Bond Energy Estimation in Large Molecular Clusters via the Method of Synergistic Cyclic Cooperativity: A Software Update H-BEE 2.0.

Journal of computational chemistry·2026
Same journal

The Intricate Mechanism of Nitric Oxide Synthase.

Journal of computational chemistry·2026
Same journal

A Molecular "Thermometer" for Measuring Effective Non-Local Exchange.

Journal of computational chemistry·2026
Same journal

Insights to Orientation Dependence of Molecular Conduction Modeled by High-Level Quantum Embedding.

Journal of computational chemistry·2026
Same journal

AutoSTOP-RT-TDDFT: Adaptive and Selected Real-Time Time-Dependent Density Functional Theory for Simulation of X-Ray Absorptions.

Journal of computational chemistry·2026
See all related articles

Related Experiment Video

Updated: Feb 25, 2026

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

9.8K

Contracted Schrödinger equation in quantum phase-space.

Carol Frishberg1, Leon Cohen2

  • 1Department of Chemistry, Ramapo College of New Jersey, 505 Ramapo Valley Rd, Mahwah, New Jersey, 07430.

Journal of Computational Chemistry
|July 28, 2017
PubMed
Summary
This summary is machine-generated.

This study derives quantum hierarchy equations using phase space quantum mechanics. The Wigner distribution and Moyal formulation reveal similar equation structures across different representations.

Keywords:
Wigner distributioncontracted Schrödinger equationquantum hierarchy equations

More Related Videos

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
09:04

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

10.0K
Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method
05:51

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method

Published on: July 19, 2019

6.7K

Related Experiment Videos

Last Updated: Feb 25, 2026

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

9.8K
Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
09:04

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

10.0K
Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method
05:51

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method

Published on: July 19, 2019

6.7K

Area of Science:

  • Quantum mechanics
  • Statistical mechanics
  • Quantum information

Background:

  • The phase space formulation offers an alternative to standard quantum mechanics.
  • It allows for calculations analogous to classical statistical mechanics.

Purpose of the Study:

  • To derive quantum hierarchy equations in the phase space representation.
  • To investigate the structure of these equations using specific formulations.

Main Methods:

  • Utilizing the phase space formulation of quantum mechanics.
  • Employing the Wigner distribution for phase space functions.
  • Applying the Moyal phase space eigenvalue formulation.

Main Results:

  • Successfully derived the quantum hierarchy equations (contracted Schrödinger equation).
  • Demonstrated structural similarities in the hierarchy equations across position, momentum, and position-momentum representations.

Conclusions:

  • The phase space formulation provides an equivalent framework to standard quantum mechanics.
  • The derived hierarchy equations exhibit consistent structural properties regardless of the representation.