Related Experiment Video
Updated: Mar 13, 2026
09:23
Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
Published on: May 30, 2014
15.1K
Semiclassical treatment of quantum propagation with nonlinear classical dynamics: A third-order thawed Gaussian
1Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA.
Physical Review. E
|October 15, 2016
Summary
This study introduces a novel third-order expansion for semiclassical dynamics, offering a closed-form solution to treat nonlinearity in quantum systems. The method accurately captures anharmonicity, extending wave-packet semiclassical techniques to complex systems.
Area of Science:
- Quantum Mechanics
- Physical Chemistry
- Chemical Physics
Background:
- Semiclassical methods are crucial for describing quantum systems, especially in regimes far from classical behavior.
- Treating nonlinearity in semiclassical dynamics has been a significant challenge, limiting applications to highly anharmonic systems.
- Existing approximations often fail to accurately capture the complex dynamics arising from significant system curvature.
Purpose of the Study:
- To develop a novel, third-order time-dependent WKB approximation for coherent states.
- To enable the accurate treatment of nonlinearity in the semiclassical dynamics of physical systems.
- To extend the applicability of real-trajectory semiclassical methods to highly anharmonic systems.
Main Methods:
- Expansion of the time-dependent WKB approximation for coherent states to third order around a guiding real trajectory.
- Derivation of a closed-form solution involving Airy functions and their derivatives.
- Analysis of autocorrelation functions in anharmonic systems to validate the method's ability to capture nonlinearity.
Main Results:
- A closed-form solution was obtained, expressed as a linear combination of Airy functions and their derivatives, multiplied by an exponential.
- The method successfully demonstrated its ability to capture nonlinearity by examining coherent states in anharmonic systems.
- The third-order expansion showed superior accuracy compared to quadratic expansions in regimes with significant curvature (small ℏ).
Conclusions:
- The derived expression provides a robust method for uniformization over coalescing saddle points, indicative of significant curvature.
- This work extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time.
- The study establishes the validity and applicability of this advanced semiclassical approach for complex quantum dynamics.
Related Concept Videos
01:21Linear Approximation in Time Domain
388
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
388
01:12Classical Mechanics
56
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...
56
02:32The de Broglie Wavelength
34.2K
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...
34.2K
01:17Differential Form of Maxwell's Equations
1.4K
James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and...
1.4K
02:45The Quantum-Mechanical Model of an Atom
60.9K
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.
60.9K
01:18Second Order systems II
449
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
449