Editing Free particle (section)

===Spread of the wave packet===
The notion of group velocity is based on a linear approximation to the dispersion relation <math>\omega(k)</math> near a particular value of <math>k</math>.<ref>{{harvnb|Hall|2013}} Equation 4.24</ref> In this approximation, the amplitude of the wave packet moves at a velocity equal to the group velocity ''without changing shape''. This result is an approximation that fails to capture certain interesting aspects of the evolution a free quantum particle. Notably, the width of the wave packet, as measured by the uncertainty in the position, grows linearly in time for large times. This phenomenon is called the [[Wave_packet#Gaussian_wave_packets_in_quantum_mechanics |spread of the wave packet]] for a free particle.

Specifically, it is not difficult to compute an exact formula for the uncertainty <math>\Delta_{\psi(t)}X</math> as a function of time, where <math>X</math> is the position operator. Working in one spatial dimension for simplicity, we have:<ref>{{harvnb|Hall|2013}} Proposition 4.10</ref>
<math display="block">(\Delta_{\psi(t)}X)^2 = \frac{t^2}{m^2}(\Delta_{\psi_0}P)^2+\frac{2t}{m}\left(\left\langle \tfrac{1}{2}({XP+PX})\right\rangle_{\psi_0} - \left\langle X\right\rangle_{\psi_0} \left\langle P\right\rangle_{\psi_0} \right)+(\Delta_{\psi_0}X)^2,</math>
where <math>\psi_0</math> is the time-zero wave function. The expression in parentheses in the second term on the right-hand side is the quantum covariance of <math>X</math> and <math>P</math>.

Thus, for large positive times, the uncertainty in <math>X</math> grows linearly, with the coefficient of <math>t</math> equal to <math>(\Delta_{\psi_0}P)/m</math>. If the momentum of the initial wave function <math>\psi_0</math> is highly localized, the wave packet will spread slowly and the group-velocity approximation will remain good for a long time. Intuitively, this result says that if the initial wave function has a very sharply defined momentum, then the particle has a sharply defined velocity and will (to good approximation) propagate at this velocity for a long time.