Editing Free particle (section)

===Group velocity and phase velocity===
[[File:Wave_packet_propagation.png|thumb|right|Propagation of a wave packet, with the motion of a single peak shaded in purple. The peaks move at the phase velocity while the overall packet moves at the group velocity.]]
The [[phase velocity]] is defined to be the speed at which a plane wave solution propagates, namely

<math display="block"> v_p=\frac{\omega}{k}=\frac{\hbar k}{2m} = \frac{p}{2m}. </math>

Note that <math>\frac{p}{2m}</math> is ''not'' the speed of a classical particle with momentum <math>p</math>; rather, it is half of the classical velocity.

Meanwhile, suppose that the initial wave function <math>\psi_0</math> is a [[wave packet]] whose Fourier transform <math>\hat\psi_0</math> is concentrated near a particular wave vector <math>\mathbf k</math>. Then the [[group velocity]] of the plane wave is defined as
<math display="block"> v_g= \nabla\omega(\mathbf k)=\frac{\hbar\mathbf k}{m}=\frac{\mathbf p}{m},</math>

which agrees with the formula for the classical velocity of the particle. The group velocity is the (approximate) speed at which the whole wave packet propagates, while the phase velocity is the speed at which the individual peaks in the wave packet move.<ref>{{harvnb|Hall|2013}} Sections 4.3 and 4.4</ref> The figure illustrates this phenomenon, with the individual peaks within the wave packet propagating at half the speed of the overall packet.