The relation between the energy of a system and its corresponding momentum is known as its dispersion relation. For example, for massive particles in free space, the dispersion relation can easily be calculated from the definition of kinetic energy: In physics, momentum is the product of the mass and velocity of an object. ... Kinetic energy is energy that a body has as a result of its speed or energy of motion. ...

i.e. the dispersion relation in this case is a quadratic function. More complicated systems will have different dispersion relations. f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where is nonzero. ...

Derivation of physical properties

Many classical physical properties of systems, such as speed, can be extended to other system if they are recast in terms of the dispersion relation. In a classical mechanical system, the speed of the system can be defined as

Optics

For electromagnetic waves, the energy is proportional to the frequency of the wave and the momentum to the wavenumber. In this case, Maxwell's equations tell us that the dispersion relation is linear. Electromagnetic radiation or EM radiation is a combination (cross product) of oscillating electric and magnetic fields perpendicular to each other, moving through space as a wave, effectively transporting energy and momentum. ... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell (written by Oliver Heaviside), that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

ω = ck

By using the same reasoning, we can infer the speed of those waves:

This is the speed of light, a constant. Cherenkov effect in a swimming pool nuclear reactor. ...

The name "dispersion relation" originally comes from optics. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constant index of refraction. In this case, the waveform will spread over time, such that a narrow pulse will become an extended pulse, i.e. be dispersed. In these materials, is known as the group velocity and correspond to the speed at which the peak propagates, a value different from the phase velocity. The refractive index of a material is the factor by which electromagnetic radiation is slowed down (relative to vacuum) when it travels inside the material. ... The group velocity of a wave is the velocity with which the overall shape of the waves amplitude (known as the envelope of the wave) propagates through space. ... The phase velocity of a wave is the rate at which the phase of the wave propagates in space. ...

Solid State

In the study of solids, the study of the dispersion relation of electrons is of paramount importance. The periodicity of crystals mean that for a given momentum, many levels of energy are possible, and that some energies might not be available at any momentum. The collection of all possible momentums and energies is known as the band structure of a material. Properties of the band structure define whether the material is a insulator, semiconductor or conductor. In solid state physics, the electronic band structure, or simply band structure, refers to the dispersion relation (the relation between energy versus momentum) of electrons in a crystal. ... // Definition An Insulator is a material or object which resists the flow of electric charge. ... A semiconductor is a material with an electrical conductivity that is intermediate between that of an insulator and a conductor. ... Conductor can mean different things in different contexts: In science and engineering, a conductor is material which contains movable electric charges in which an electric current can be produced. ...

Phonons

The dispersion relation of phonons is also important and non-trivial. Most systems will show two separate bands on which phonons live. Phonons on the band that cross the origin are known as acoustic phonons, the others as optical phonons. In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ...

1. REDIRECT Group velocity