Editing Matter wave (section)

=== Group velocity ===
In the de Broglie hypothesis, the velocity of a particle equals the [[group velocity]] of the matter wave.<ref name="WhittakerII" />{{rp|214}}
In isotropic media or a vacuum the [[group velocity]] of a wave is defined by:
<math display="block"> \mathbf{v_g} = \frac{\partial \omega(\mathbf{k})}{\partial \mathbf{k}} </math>
The relationship between the angular frequency and wavevector is called the [[Dispersion relation#De Broglie dispersion relations|dispersion relationship]]. For the non-relativistic case this is:
<math display="block">\omega(\mathbf{k}) \approx \frac{m_0 c^2}{\hbar} + \frac{\hbar k^2}{2m_{0} }\,.</math>
where <math>m_0</math> is the rest mass. Applying the derivative gives the (non-relativistic) '''matter wave group velocity''':
<math display="block">\mathbf{v_g} = \frac{\hbar \mathbf{k}}{m_0}</math>
For comparison, the group velocity of light, with a [[Dispersion_relation#Electromagnetic_waves_in_vacuum|dispersion]] <math>\omega(k)=ck</math>, is the [[speed of light]] <math>c</math>.

As an alternative, using the relativistic [[Dispersion relation#De Broglie dispersion relations|dispersion relationship]] for matter waves
<math display="block"> \omega(\mathbf{k}) = \sqrt{k^2c^2 + \left(\frac{m_0c^2}{\hbar}\right)^2} \,.</math>
then
<math display="block">\mathbf{v_g} = \frac{\mathbf{k}c^2}{\omega} </math>
This relativistic form relates to the phase velocity as discussed below.

For non-isotropic media we use the [[Energy–momentum relation|Energy–momentum]] form instead:
<math display="block">\begin{align}
  \mathbf{v}_\mathrm{g} &= \frac{\partial \omega}{\partial \mathbf{k}} = \frac{\partial (E/\hbar)}{\partial (\mathbf{p} /\hbar)} = \frac{\partial E}{\partial \mathbf{p}} = \frac{\partial}{\partial \mathbf{p}} \left( \sqrt{p^2c^2+m_0^2c^4} \right)\\
    &= \frac{\mathbf{p} c^2}{\sqrt{p^2c^2 + m_0^2c^4}}\\
    &= \frac{\mathbf{p} c^2}{E} .
\end{align}</math>

But (see below), since the phase velocity is <math>\mathbf{v}_\mathrm{p} = E/\mathbf{p} = c^2/\mathbf{v}</math>, then
<math display="block">\begin{align}
 \mathbf{v}_\mathrm{g} &= \frac{\mathbf{p}c^2}{E}\\
    &= \frac{c^2}{\mathbf{v}_\mathrm{p}}\\
    &= \mathbf{v} ,
\end{align}</math>
where <math>\mathbf{v}</math> is the velocity of the center of mass of the particle, identical to the group velocity.