Editing Matter wave (section)

=== Phase velocity ===
The [[phase velocity]] in isotropic media is defined as:
<math display="block">\mathbf{v_p} = \frac{\omega}{\mathbf{k}}</math>
Using the relativistic group velocity above:<ref name="WhittakerII"/>{{rp|p=215}}
<math display="block">\mathbf{v_p} = \frac{c^2 }{\mathbf{v_g}}</math>
This shows that <math>\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2</math> as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952. Electromagnetic waves also obey  <math>\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2</math>, as both <math>|\mathbf{v_p}|=c</math> and <math>|\mathbf{v_g}|=c</math>. Since for matter waves, <math>|\mathbf{v_g}| < c</math>, it follows that <math>|\mathbf{v_p}| > c</math>, but only the group velocity carries information. The [[Faster-than-light|superluminal]] phase velocity therefore does not violate special relativity, as it does not carry information.

For non-isotropic media, then
<math display="block">\mathbf{v}_\mathrm{p} = \frac{\omega}{\mathbf{k}} = \frac{E/\hbar}{\mathbf{p}/\hbar} = \frac{E}{\mathbf{p}}. </math>

Using the [[special relativity|relativistic]] relations for energy and momentum yields
<math display="block">\mathbf{v}_\mathrm{p} = \frac{E}{\mathbf{p}} = \frac{m c^2}{m \mathbf{v}} = \frac{\gamma m_0 c^2}{\gamma m_0 \mathbf{v}} = \frac{c^2}{\mathbf{v}}.</math>
The variable <math>\mathbf{v}</math> can either be interpreted as the speed of the particle or the group velocity of the corresponding matter wave—the two are the same. Since the particle speed <math>|\mathbf{v}| < c </math> for any particle that has nonzero mass (according to [[special relativity]]), the phase velocity of matter waves always exceeds ''c'', i.e.,
<math display="block">| \mathbf{v}_\mathrm{p} | > c ,</math>
which approaches ''c'' when the particle speed is relativistic. The [[Faster-than-light|superluminal]] phase velocity does not violate special relativity, similar to the case above for non-isotropic media. See the article on ''[[Dispersion (optics)#Group velocity dispersion|Dispersion (optics)]]'' for further details.