Editing Uncertainty principle (section)

===Constant momentum===
{{Main article|Wave packet}}
[[File:Guassian Dispersion.gif|360 px|thumb|right|Position space probability density of an initially Gaussian state moving at minimally uncertain, constant momentum in free space]]
Assume a particle initially has a [[momentum space]] wave function described by a normal distribution around some constant momentum ''p''<sub>0</sub> according to
<math display="block">\varphi(p) = \left(\frac{x_0}{\hbar \sqrt{\pi}} \right)^{1/2} \exp\left(\frac{-x_0^2 (p-p_0)^2}{2\hbar^2}\right),</math>
where we have introduced a reference scale <math display="inline">x_0=\sqrt{\hbar/m\omega_0}</math>, with <math>\omega_0>0</math> describing the width of the distribution—cf. [[nondimensionalization]]. If the state is allowed to evolve in free space, then the time-dependent momentum and position space wave functions are
<math display="block">\Phi(p,t) = \left(\frac{x_0}{\hbar \sqrt{\pi}} \right)^{1/2} \exp\left(\frac{-x_0^2 (p-p_0)^2}{2\hbar^2}-\frac{ip^2 t}{2m\hbar}\right),</math>
<math display="block">\Psi(x,t) = \left(\frac{1}{x_0 \sqrt{\pi}} \right)^{1/2} \frac{e^{-x_0^2 p_0^2 /2\hbar^2}}{\sqrt{1+i\omega_0 t}} \, \exp\left(-\frac{(x-ix_0^2 p_0/\hbar)^2}{2x_0^2 (1+i\omega_0 t)}\right).</math>

Since <math> \langle p(t) \rangle = p_0</math> and <math>\sigma_p(t) = \hbar /(\sqrt{2}x_0)</math>, this can be interpreted as a particle moving along with constant momentum at arbitrarily high precision. On the other hand, the standard deviation of the position is
<math display="block">\sigma_x = \frac{x_0}{\sqrt{2}} \sqrt{1+\omega_0^2 t^2}</math>
such that the uncertainty product can only increase with time as
<math display="block">\sigma_x(t) \sigma_p(t) = \frac{\hbar}{2} \sqrt{1+\omega_0^2 t^2}</math>