Editing Wave packet (section)

== The Airy wave train ==

In contrast to the above Gaussian wave packet, which moves at constant group velocity, and always disperses, there exists a wave function based on [[Airy function]]s, that propagates freely without envelope dispersion, maintaining its shape, and accelerates in free space:<ref>{{Citation |last1=Berry |first1=M. V. |title=Nonspreading wave packets |journal=Am J Phys |volume=47 |issue=3 |pages=264–267 |year=1979 |bibcode=1979AmJPh..47..264B |doi=10.1119/1.11855 |last2=Balazs |first2=N. L.}}</ref>
<math display="block">\psi = \operatorname{Ai}\left[\frac{B}{\hbar^{2/3}}\left(x-\frac{B^3t^2}{4m^2}\right)\right]e^{(iB^3t/2m\hbar)[x-(B^3t^2/6m^2)]},</math>
where, for simplicity (and [[nondimensionalization]]), choosing {{math|1=''ħ'' = 1}}, {{math|1=''m'' = 1/2}}, and ''B'' an arbitrary constant results in
<math display="block">\psi = \operatorname{Ai}[B(x-B^3t^2)] \, e^{iB^3t (x-\tfrac{2}{3}B^3t^2)} \, .</math>

There is no dissonance with [[Ehrenfest's theorem]] in this force-free situation, because the state is both non-normalizable and has an undefined (infinite) {{math|⟨''x''⟩}} for all times. (To the extent that it could be defined, {{math|1=⟨''p''⟩ = 0}} for all times, despite the apparent acceleration of the front.)

The Airy wave train is the only dispersionless wave in one dimensional free space.<ref>{{Cite journal |last=Unnikrishnan |first=K. |last2=Rau |first2=A. R. P. |date=1996-08-01 |title=Uniqueness of the Airy packet in quantum mechanics |url=https://pubs.aip.org/ajp/article/64/8/1034/530435/Uniqueness-of-the-Airy-packet-in-quantum-mechanics |journal=American Journal of Physics |language=en |volume=64 |issue=8 |pages=1034–1035 |doi=10.1119/1.18322 |issn=0002-9505|url-access=subscription }}</ref> In higher dimensions, other dispersionless waves are possible.<ref>{{Cite journal |last=Efremidis |first=Nikolaos K. |last2=Chen |first2=Zhigang |last3=Segev |first3=Mordechai |last4=Christodoulides |first4=Demetrios N. |date=2019-05-20 |title=Airy beams and accelerating waves: an overview of recent advances |url=https://opg.optica.org/abstract.cfm?URI=optica-6-5-686 |journal=Optica |language=en |volume=6 |issue=5 |pages=686 |doi=10.1364/OPTICA.6.000686 |issn=2334-2536|arxiv=1904.02933 }}</ref>

[[File:Airy wave train.webm|thumb|The Airy wave train in phase space. Its shape is a series of parabolas with the same axis, but oscillating according to the Airy function. Its time-evolution is a shearing along the <math>x</math>-direction. Each parabola retains its shape under this shearing, and its apex performs a translation along another parabola. Thus, the Airy wave train does not disperse, and the group motion of the wave train undergoes constant acceleration.]]
In [[Phase space quantum mechanics|phase space]], this is evident in the [[pure state]] [[Wigner quasiprobability distribution]] of this wavetrain, whose shape in ''x'' and ''p'' is invariant as time progresses, but whose features accelerate to the right, in accelerating parabolas. The Wigner function satisfies<math display="block">\begin{aligned}
W(x,p;t) 
&= W(x-B^3 t^2, p-B^3 t ;0) \\
&= \frac{1}{2^{1/3} \pi B} \, \mathrm{Ai} \left(2^{2/3} \left(B(x-B^3 t^2\right)+\left(p / B-t B^2)^2 \right)\right) \\
&= W(x - 2pt, p; 0).
\end{aligned} </math>The three equalities demonstrate three facts:

# Time-evolution is equivalent to a translation in phase-space by <math>(B^3 t^2 , B^3 t)</math>.
# The contour lines of the Wigner function are parabolas of form <math display="inline">B\left(x-B^3 t^2\right)+\left(p / B-t B^2\right)^2 = C </math>.
# Time-evolution is equivalent to a shearing in phase space along the <math>x</math>-direction at speed <math>p/m = 2p</math>.

Note the momentum distribution obtained by integrating over all {{mvar|x}} is constant. Since this is the [[Wigner quasiprobability distribution#Mathematical properties|probability density in momentum space]], it is evident that the wave function itself is not normalizable.