Related Experiment Video
Updated: Apr 10, 2026
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
9.8K
Stochastic approach to the generalized Schrödinger equation: A method of eigenfunction expansion
Satoshi Tsuchida1, Hiroshi Kuratsuji1
1Department of Physics, Ritsumeikan University-BKC, Noji, Kusatsu City, 525-8577 Shiga, Japan.
Summary
This study develops a stochastic equation for the generalized Schrödinger equation with random fluctuations. The derived Fokker-Planck equation offers insights into quantum systems with noise, crucial for understanding quantum dynamics.
Area of Science:
- Quantum Mechanics
- Stochastic Processes
- Mathematical Physics
Background:
- The generalized Schrödinger equation describes quantum systems.
- Random fluctuations introduce stochasticity, complicating analysis.
- Eigenfunction expansion is a common technique for solving differential equations.
Purpose of the Study:
- To develop a stochastic equation for the generalized Schrödinger equation with random fluctuations.
- To derive the corresponding Fokker-Planck equation for the system's dynamics.
- To analyze the derived Fokker-Planck equation using approximate schemes.
Main Methods:
- Eigenfunction expansion of the wave field.
- Derivation of the Langevin equation for expansion coefficients.
- Conversion to a Fokker-Planck equation using functional integrals and Gaussian white noise assumption.
- Analysis of the Fokker-Planck equation via approximate schemes.
Main Results:
- A stochastic equation for the generalized Schrödinger equation with random fluctuations was successfully developed.
- The Langevin equation for expansion coefficients was derived and converted to a Fokker-Planck equation.
- The crucial role of the functional Jacobian in determining the Fokker-Planck equation's form was highlighted.
Conclusions:
- The eigenfunction expansion method provides a robust framework for analyzing stochastic quantum systems.
- The derived Fokker-Planck equation is a valuable tool for studying the dynamics of quantum systems under random fluctuations.
- Approximate schemes are effective for analyzing the complex Fokker-Planck equation.
More Related Videos
Related Concept Videos
02:45The Quantum-Mechanical Model of an Atom
61.9K
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.
61.9K
01:18Electron Orbital Model
77.3K
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...
The first shell is closest to the nucleus, and it has only one subshell with a single spherical orbital called the...
77.3K
02:32The de Broglie Wavelength
34.8K
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...
34.8K
02:44Atomic Orbitals
47.5K
An atomic orbital represents the three-dimensional regions in an atom where an electron has the highest probability to reside. The radial distribution function indicates the total probability of finding an electron within the thin shell at a distance r from the nucleus. The atomic orbitals have distinct shapes which are determined by l, the angular momentum quantum number. The orbitals are often drawn with a boundary surface, enclosing densest regions of the cloud.
47.5K
04:08The Uncertainty Principle
34.8K
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...
34.8K
02:35Molecular Orbital Theory I
49.5K
Overview of Molecular Orbital Theory
49.5K