Editing Dispersion relation (section)

==Plane waves in vacuum==

Plane waves in vacuum are the simplest case of wave propagation: no geometric constraint, no interaction with a transmitting medium.

=== Electromagnetic waves in vacuum ===

For [[electromagnetic wave]]s in vacuum, the angular frequency is proportional to the wavenumber:
: <math>\omega = c k.</math>

This is a ''linear'' dispersion relation, in which case the waves are said to be '''non-dispersive'''.{{sfn|Ablowitz|2011|pp=19-20}} That is, the phase velocity and the group velocity are the same:
: <math> v = \frac{\omega}{k} = \frac{d\omega}{d k} = c,</math>
and thus both are equal to the [[speed of light]] in vacuum, which is frequency-independent.

===De Broglie dispersion relations===
For [[matter wave|de Broglie matter waves]] the frequency dispersion relation is non-linear:
<math display=block>\omega(k) \approx \frac{m_0 c^2}{\hbar} + \frac{\hbar k^2}{2m_{0} }\,.</math>
The equation says the matter wave frequency <math>\omega</math> in vacuum varies with wavenumber (<math>k=2\pi/\lambda</math>) in the non-relativistic approximation. The variation has two parts: a constant part due to the de Broglie frequency of the rest mass (<math>\hbar \omega_0 = m_{0}c^2</math>) and a quadratic part due to kinetic energy.

==== Derivation ====
While applications of matter waves occur at non-relativistic velocity, [[Matter wave#De Broglie hypothesis|de Broglie applied]] [[special relativity]] to derive his waves. 
Starting from the relativistic [[energy–momentum relation]]:
<math display=block>E^2 = (p \textrm c)^2 + \left(m_0 \textrm c^2\right)^2\,</math>
use the [[de Broglie relations]] for energy and momentum for [[matter wave]]s,
:<math>E = \hbar \omega \,, \quad \mathbf{p} = \hbar\mathbf{k}\,,</math>
where {{math|''ω''}} is the [[angular frequency]] and {{math|'''k'''}} is the [[wavevector]] with magnitude {{math|{{abs|'''k'''}} {{=}} ''k''}}, equal to the [[wave number]]. Divide by <math>\hbar</math> and take the square root. This gives the '''relativistic frequency dispersion relation''':
<!-- eqn used in Matter_wave#Group_velocity, please keep in sync.-->
<math display=block> \omega(k) = \sqrt{k^2c^2 + \left(\frac{m_0c^2}{\hbar}\right)^2} \,.</math>

[[matter wave#Applications of matter waves|Practical work with matter waves]] occurs at non-relativistic velocity. To approximate, we pull out the rest-mass dependent frequency:
<math display=block>\omega = \frac{m_0 c^2}{\hbar}\sqrt{1+ \left( \frac{k\hbar}{m_{0} c} \right)^2 } \,.</math>

Then we see that the <math>\hbar/c</math> factor is very small so for <math>k</math> not too large, we expand <math>\sqrt{1+x^2}\approx 1+x^2/2,</math> and multiply:
<!-- eqn used in Matter_wave#Group_velocity, please keep in sync.-->
<math display=block>\omega(k) \approx \frac{m_0 c^2}{\hbar} + \frac{\hbar k^2}{2m_{0} }\,.</math>
This gives the non-relativistic approximation discussed above. 
If we start with the non-relativistic [[Schrödinger equation]] we will end up without the first, rest mass, term.

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
! ''Animation:'' phase and group velocity of electrons
|-
| [[Image:deBroglie3.gif|frame|center]]
This animation portrays the de Broglie phase and group velocities (in slow motion) of three free electrons traveling over a field 0.4 [[ångström]]s in width. The momentum per unit mass (proper velocity) of the middle electron is lightspeed, so that its group velocity is 0.707 ''c''. The top electron has twice the momentum, while the bottom electron has half.  Note that as the momentum increases, the phase velocity decreases down to ''c'', whereas the group velocity increases up to ''c'', until the wave packet and its phase maxima move together near the speed of light, whereas the wavelength continues to decrease without bound. Both transverse and longitudinal coherence widths (packet sizes) of such high energy electrons in the lab may be orders of magnitude larger than the ones shown here.
|}