Related Experiment Video
Updated: May 24, 2026
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
Mapping quantum-classical Liouville equation: projectors and trajectories.
Aaron Kelly1, Ramses van Zon, Jeremy Schofield
1Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada. atkelly@stanford.edu
The Journal of Chemical Physics
|March 3, 2012
Summary
This study introduces a mapping formalism for mixed quantum-classical systems, ensuring dynamics remain within physical space. An approximation is discussed, highlighting its limitations and potential for dynamical instabilities.
Area of Science:
- Quantum mechanics
- Chemical physics
- Theoretical chemistry
Background:
- Mixed quantum-classical systems present challenges in describing quantum-classical dynamics.
- The mapping formalism offers a phase space representation for quantum degrees of freedom.
Purpose of the Study:
- To develop and analyze a mapping formalism for mixed quantum-classical systems.
- To ensure the dynamics remain confined to the physical quantum space.
- To investigate the validity and limitations of an approximate dynamics.
Main Methods:
- Mapping discrete quantum states to oscillator states.
- Defining projection operators onto mapping states.
- Constructing trajectory-based solutions using entangled trajectories.
- Analyzing an approximation retaining only the Poisson bracket contribution.
Main Results:
- The mapping quantum-classical Liouville operator commutes with projection operators, confining dynamics to physical space.
- An approximate evolution equation allows independent trajectories but may lead the system outside physical space.
- Dynamical instabilities and the domain of validity of the approximate dynamics were discussed.
Conclusions:
- The mapping formalism provides a rigorous framework for mixed quantum-classical dynamics.
- Approximations must be carefully considered to avoid unphysical dynamics.
- Simulations on quantum systems illustrate the effects and validity of the proposed methods.
Related Concept Videos
02:45The 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...
01:12Classical Mechanics
Classical mechanics provides a mathematical description of the motion of bodies under the influence of forces. A key principle within this field is the work-energy theorem, which establishes a bridge between the net work done on an object and its kinetic energy.The work-energy theorem states that the net work done on a particle by all the forces acting on it equals the change in its kinetic energy.In simple terms, the work-energy theorem is a method to analyze the effects of forces on an...
02:32The 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...
01:21Principle of Linear Impulse and Momentum for a System of Particles
In the context of a system of particles moving relative to an inertial frame of reference, the equation of motion is a crucial tool for understanding the dynamics of the system. This equation, which accounts for external forces acting on each particle, plays a fundamental role in describing the system's behavior.
Notably, internal forces between particles, occurring in equal and opposite collinear pairs, cancel out and are not part of the equation of motion. This exclusion simplifies the...
Notably, internal forces between particles, occurring in equal and opposite collinear pairs, cancel out and are not part of the equation of motion. This exclusion simplifies the...
00:49Impulse-Momentum Theorem
The total change in the motion of an object is proportional to the total force vector acting on it and the time over which it acts. This product is called impulse, a vector quantity with the same direction as the total force acting on the object.
By writing Newton's second law of motion in terms of the momentum of an object and the external force acting on it, and simultaneously using the definition of the impulse vector, it can be shown that the total impulse on an object is equal to its net...
By writing Newton's second law of motion in terms of the momentum of an object and the external force acting on it, and simultaneously using the definition of the impulse vector, it can be shown that the total impulse on an object is equal to its net...
01:20Principle of Linear Impulse and Momentum for a Single Particle
Linear momentum is a fundamental concept in physics that describes the motion of an object. It is a vector quantity, having a magnitude equal to the product of its mass and its velocity, and direction along the object's velocity. On the other hand, linear impulse, also known as momentum impulse, is a concept in physics related to the change in the linear momentum of an object. Impulse is a vector quantity defined as the product of force and the time over which the force is applied.
Delving into...
Delving into...