Editing Wave packet (section)

=== Dispersive ===
[[File:Wave packet (dispersion).gif|right|thumb|300px|A wave packet with dispersion. Notice the wave spreads out and its amplitude reduces.]]
[[File:Guassian Dispersion.gif|360 px|thumb|right|Position space probability density of an initially Gaussian state moving in one dimension at minimally uncertain, constant momentum in free space.]]

By contrast, in the case of dispersion, a wave changes shape during propagation. For example, the [[Free_particle#Mathematical_description|free Schrödinger equation]] ,
<math display="block">i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^{2}}{2m} \nabla^2 \psi,</math>
has plane-wave solutions of the form:
<math display="block">\psi (\mathbf{r},t) = A e^{i{[\mathbf{k\cdot r}}-\omega(\mathbf{k}) t]},</math>
where <math>A</math> is a constant and the dispersion relation satisfies<ref>{{cite book | last=Hall | first=Brian C. | title=Quantum Theory for Mathematicians | publisher=Springer | publication-place=New York Heidelberg Dordrecht London | year=2013 | isbn=978-1-4614-7115-8 | pages=91-92}}</ref>{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|pp=13-15}}
<math display="block"> \omega (\mathbf{k}) = \frac{\hbar \mathbf{k}^2}{2m}=\frac{\hbar}{2m}(k_x^2 + k_y^2 + k_z^2), </math>
with the subscripts denoting [[Vector_notation#Unit_vector_notation|unit vector notation]]. As the dispersion relation is non-linear, the free Schrödinger equation is '''dispersive'''.

In this case, the wave packet is given by:
<math display="block">\psi(\mathbf{r},t) = \frac{1}{(2\pi)^{3/2}}\int g(\mathbf{k}) e^{i{[\mathbf{k\cdot r}}-\omega(\mathbf{k}) t]}d^3 k</math>
where once again <math>g(\mathbf{k})</math> is simply the Fourier transform of <math>\psi(\mathbf{k},0)</math>. If <math>\psi(\mathbf{k},0)</math> (and therefore <math>g(\mathbf{k})</math>) is a [[Gaussian function]], the wave packet is called a '''Gaussian wave packet'''.{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|pp=57,1511}}

For example, the solution to the one-dimensional free Schrödinger equation (with {{math|2Δ''x''}}, {{mvar|m}}, and ''ħ'' set equal to one) satisfying the initial condition 
<math display="block"> \psi(x,0)= \sqrt[4]{2/\pi} \exp\left({-x^2 + ik_0 x}\right),</math> 
representing a wave packet localized in space at the origin as a Gaussian function, is seen to be
<math display="block">\begin{align}
 \psi(x,t) &= \frac{ \sqrt[4]{2/\pi}}{\sqrt{1 + 2it}} e^{-\frac{1}{4}k_0^2} ~ e^{-\frac{1}{1 + 2it}\left(x - \frac{ik_0}{2}\right)^2}\\
           &= \frac{ \sqrt[4]{2/\pi}}{\sqrt{1 + 2it}} e^{-\frac{1}{1 + 4t^2}(x - k_0t)^2}~ e^{i \frac{1}{1 + 4t^2}\left((k_0 + 2tx)x - \frac{1}{2}tk_0^2\right)} ~.
\end{align} </math>

An impression of the dispersive behavior of this wave packet is obtained by looking at the probability density:
<math display="block">|\psi(x,t)|^2 = \frac{ \sqrt{2/\pi}}{\sqrt{1+4t^2}}~e^{-\frac{2(x-k_0t)^2}{1+4t^2}}~.</math>
It is evident that this dispersive wave packet, while moving with constant group velocity {{math|''k<sub>o</sub>''}}, is delocalizing rapidly: it has a [[Gaussian function|width]] increasing with time as {{math|{{radical| 1 + 4''t''<sup>2</sup>}} → 2''t''}}, so eventually it diffuses to an unlimited region of space.