Dispersion relation

In physics, the dispersion relation describes the relationship between the expiry of a physical process (frequency, energy ) and the properties of the variables describing him ( wave number, refractive index, speed of propagation, pulse). Mathematically, the dispersion relation of the relationship between the angular frequency and wave number of the circle. It is obtained from the linear wave equation by a Fourier transformation in space and time and has the form

In the simplest case, angular frequency and wavenumber circle are always proportional

With the constant phase velocity. In this case, there is no dispersion.

The speed of a wave packet, however, is the group velocity

A wave packet consists of waves of different frequencies, which can have different phase velocities. Therefore, a wave packet A., diverges. Wave packets that do not diverge due to nonlinear effects in spite of dispersion are called solitons.

Optics

In the dispersion relation of the optics, the (complex) refractive index emerges as a function of angular frequency:

With

• The speed of propagation of light in a medium
• The vacuum speed of light.

Particle Physics

Since the frequency is always in connection with the energy

And the wave number (or wave vector ) with the pulse

Refers to the energy-momentum relations of the particle as a dispersion relation ( or dispersion relation ), eg free electrons in the non- relativistic limit:

Where the reduced Planck constant and the mass of the particle respectively.

Solid State Physics

In solid-state physics, the dispersion is given as link between energy and angular frequency and wave number of a particle or quasi-particle. In solids the phonons ( lattice vibrations of the atomic lattice ) is on the one hand associated with a phonon dispersion relation, on the other hand, the electron- electron dispersion relation can be assigned, which is described with the help of the band structure.

• Wave
• Optics
• Particle Physics