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Related Concept Videos

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...
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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...
Equations of Wave Motion01:02

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Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a period T. The wave on the string is sinusoidal and is translated in the positive x-direction as time progresses. Sine is a function of the angle θ, oscillating between +A and −A and repeating every 2π radians. To construct a wave model, the ratio of the angle θ and the position x is considered.
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A household microwave and lasers are examples of standing electromagnetic waves in a cavity. When two conducting metal plates are placed parallel at the nodal planes, it creates a cavity where standing waves are formed. The cavity between the two planes is analogous to a stretched string held at the points x = 0 and x = L. Here, the distance 'L' between the two planes must be an integer multiple of half of the wavelength. The wavelengths that satisfy this condition are given by:
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For a tube open at one end and closed at the other filled with air, the modes are such that there is always an antinode at the open end and a node at the closed end.

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Related Experiment Video

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

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Published on: June 8, 2018

Semiclassical wave functions for open quantum billiards.

Fabian Lackner1, Iva Březinová, Joachim Burgdörfer

  • 1Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstraße 8-10/136, 1040 Vienna, Austria, EU.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 17, 2013
PubMed
Summary
This summary is machine-generated.

We developed a new semiclassical method to accurately approximate quantum scattering wave functions in open billiards. This approach links accuracy to path length and dwell time, offering insights into quantum systems.

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Area of Science:

  • Quantum mechanics
  • Mathematical physics
  • Computational physics

Background:

  • Open quantum systems present challenges for traditional quantum mechanical descriptions.
  • Semiclassical approximations offer a bridge between classical and quantum mechanics.

Purpose of the Study:

  • To present a novel semiclassical approximation for the scattering wave function in open quantum billiards.
  • To assess the numerical accuracy and convergence properties of this approximation.

Main Methods:

  • Reconstruction of the Feynman path integral.
  • Numerical implementation for an open rectangular billiard.
  • Analysis of convergence based on path length and dwell time.

Main Results:

  • Demonstrated remarkable numerical accuracy for the open rectangular billiard.
  • Established that convergence is controlled by mean path length or dwell time.
  • Showed energy resolution is inversely related to maximum dwell time (ΔE~τ(max)(-1)).

Conclusions:

  • The presented semiclassical approximation is a powerful tool for studying open quantum billiards.
  • The findings provide a quantitative link between classical path properties and quantum state fidelity.
  • Potential applications include leaky billiards and systems exhibiting decoherence.