Editing Wave packet (section)

== Basic behaviors ==
{{More citations needed section|date=October 2024}}
{{See also|Dispersion (water waves) | Dispersion (optics)}}

=== Non-dispersive ===
[[File:Wave packet (no dispersion).gif|right|thumb|300px|A wave packet without dispersion (real or imaginary part)]]
Without dispersion the wave packet maintains its shape as it propagates.
As an example of propagation ''without dispersion'', consider wave solutions to the following [[wave equation]] from [[classical physics]]
<math display="block">{ \partial^2 u \over \partial t^2 } = c^2 \, \nabla^2 u,</math>

where {{math|''c''}} is the speed of the wave's propagation in a given medium.

Using the physics time convention, {{math|''e''<sup>−''iωt''</sup>}}, the wave equation has [[plane-wave]] solutions
<math display="block"> u(\mathbf{x},t) = e^{i{(\mathbf{k\cdot x}}-\omega(\mathbf{k}) t)},</math>

where the relation between the [[angular frequency]] {{math|''ω''}} and [[angular wave vector]] {{math|'''k'''}} is given by the [[dispersion relation]]:
<math display="block"> \omega(\mathbf{k}) =\pm |\mathbf{k}| c = \pm \frac{2\pi c}{\lambda},</math>
such that <math> \omega^2/|\mathbf{k}|^2 = c^2</math>. This relation should be valid so that the plane wave is a solution to the wave equation. As the relation is ''linear'', the wave equation is said to be '''non-dispersive'''.

To simplify, consider the one-dimensional wave equation with  {{math|1=''ω(k) ''='' ±kc''}}. Then the general solution is
<math display="block"> u(x,t)= A e^{ik(x-c t)} + B e^{ik(x+c t)},</math>
where the first and second term represent a wave propagating in the positive respectively negative {{nowrap|{{math|''x''}}-direction}}.

A wave packet is a localized disturbance that results from the sum of many different [[wave form]]s. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region.{{sfn|Jackson|1998|pp=322-326}} From the basic one-dimensional plane-wave solutions, a general form of a wave packet can be expressed as
<math display="block"> u(x,t) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{-\infty} A(k) ~ e^{i(kx-\omega(k)t)} \, dk.</math>
where the amplitude {{math|''A''(''k'')}}, containing the coefficients of the [[Superposition_principle#Wave_superposition|wave superposition]], follows from taking the [[Fourier inversion theorem|inverse Fourier transform]] of a "[[Fourier_inversion_theorem#Conditions_on_the_function|sufficiently nice]]" 
initial wave {{math|''u''(''x'', ''t'')}} evaluated at {{math|1=''t'' = 0}}:
<math display="block"> A(k) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{-\infty} u(x,0) ~ e^{-ikx} \, dx.</math>
and  <math>1 / \sqrt{2 \pi}</math> comes from [[Fourier transform#Other conventions|Fourier transform conventions]].

For example, choosing
<math display="block"> u(x,0) = e^{-x^2 +ik_0x},</math>

we obtain
<math display="block"> A(k) = \frac{1}{\sqrt{2}} e^{-\frac{(k-k_0)^2}{4}},</math>

and finally
<math display="block">\begin{align}
 u(x,t) &= e^{-(x-ct)^2 +ik_0(x-ct)}\\ 
&= e^{-(x-ct)^2} \left[\cos\left(2\pi \frac{x-ct}{\lambda}\right)+ i\sin\left(2\pi\frac{x-ct}{\lambda}\right)\right].
\end{align} </math>

The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.

=== 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.