Wave Dispersion

In physics, dispersion typically refers to a frequency-dependent effect relating to the propagation of a wave. In the presence of dispersion, the wave velocity can be described in terms of a phase and group velocity – see next discussion for further details. By way of an initial example, dispersion might be characterised in terms of the separation of white light into its colour components, when passing through a prism.

1

However, the effect of dispersion depends on the interrelation of wave properties, such as wavelength [λ], frequency [f],  velocity [v].  Dispersion may also be caused by geometric boundary conditions, e.g. the refractive index [n] of the medium through which the wave is propagating. However, in the context of the present discussion, elementary particles are being considered in terms of matter waves, which appear to have a dispersion effect even in the absence of any geometric boundaries, i.e. when passing through a vacuum. Before considering the specific effect of dispersion on matter waves, there may be some value in reviewing some of the general effects of dispersion.

Dispersive and Non-dispersive Media:

Sound waves travelling through air can be described as non-dispersive, as all the different frequencies making up a sound arrive in synchronization at any point. If this were not the case, a musical concert comprising of a ‘spectrum’ of sound waves would seem discordant when heard from the back of the hall, i.e. if different notes arrived at different times. In contrast, surface waves travelling across the ocean can be described as dispersive, as water waves of longer wavelength, as found in tsunami, can travel much faster than shorter wavelengths:

Plane waves in vacuum:

So, as a generalization, dispersion occurs when pure plane waves of different wavelengths have a different propagation velocity [v], such that a superposition of waves of mixed wavelengths would tend to spread out in space. The use of [ω] and [k] can be linked to the description of both the phase velocity [ω/k] and the group velocity [dω/dk] and the basic formulation of a sine wave as follows:

[1]      1

Electromagnetic waves:

In the case of electromagnetic waves, in vacuum, we have a clear example of the relationship between energy [E] and frequency [f] and a defined propagation velocity [v=c], which then defines the wavelength [λ]:

[2]      2

This is a linear or non-dispersive relationship. As such, the phase and group velocity are said to be both equal to [v=c], although physically, the group wave does not exist in this case.

[3]      3

de Broglie matter waves:

These waves are being characterized by the deBroglie relationship between the wavelength [λ] and momentum [p] of the particle:

[4]      4

However, the deBroglie relationship also extends to the energy [E] definition of the matter wave, which we might relate to the frequency [f], when described in wave-like form and the kinetic energy, when in particle-like form:

[5]      5

While there are some ambiguities in [5] regarding the scope of the total energy versus the kinetic energy, if we simply equate the two definitions in [5] and substitute for momentum [p], as defined in [6], we appear to have a dispersive relationship for a matter wave.

[6]      6

At this point, we shall simply defer discussing the implication of the dispersive nature of matter waves until after we have introduced a few other topics, which will eventually lead to Schrodinger’s wave theory and the time evolution of matter waves.

Note: Aspects of this discussion will eventually be expanded into a speculative discussion of the 'Wave Structure of Everything (WSE), where the issue of matter  wave dispersion will also be discussed further in 'A Matter of Perspective.'