Editing Wave packet (section)

== Historical background ==
Ideas related to wave packets – [[modulation]], [[carrier wave]]s, [[phase velocity]], and [[group velocity]] – date from the mid-1800s. The idea of a group velocity distinct from a wave's phase velocity was first proposed by [[William Rowan Hamilton|W.R. Hamilton]] in 1839, and the first full treatment was by [[John Strutt, 3rd Baron Rayleigh|Rayleigh]] in his "Theory of Sound" in 1877.<ref>{{Citation |last=Brillouin |first=Léon |title=Wave Propagation and Group Velocity |publisher=Academic Press Inc. |location=New York |year=1960 |oclc=537250}}</ref>

[[Erwin Schrödinger]] introduced the idea of wave packets just after publishing his famous [[Schrodinger equation|wave equation]].<ref name="Kragh">{{Cite book |last=Kragh |first=Helge |title=Compendium of Quantum Physics |chapter-url=https://link.springer.com/10.1007/978-3-540-70626-7_232 |chapter=Wave Packet |date=2009 |publisher=Springer Berlin Heidelberg |isbn=978-3-540-70622-9 |editor-last=Greenberger |editor-first=Daniel |location=Berlin, Heidelberg |pages=828–830 |language=en |doi=10.1007/978-3-540-70626-7_232 |editor-last2=Hentschel |editor-first2=Klaus |editor-last3=Weinert |editor-first3=Friedel}}</ref> He solved his wave equation for a [[quantum harmonic oscillator]], introduced the [[superposition principle]], and used it to show that a compact state could persist. While this work did result in the important concept of [[coherent states]], the wave packet concept did not endure.  The year after Schrödinger's paper, [[Werner Heisenberg]] published his paper on the [[uncertainty principle]], showing in the process, that Schrödinger's results only applied to [[quantum harmonic oscillator]]s, not for example to [[Electric potential|Coulomb potential]] characteristic of atoms.<ref name="Kragh"/>{{rp|829}}

The following year, 1927, [[Charles Galton Darwin]] explored [[Schrödinger equation|Schrödinger's equation]] for an unbound electron in free space, assuming an initial [[Wave packet#Gaussian wave packets in quantum mechanics|Gaussian wave packet]].<ref>Darwin, Charles Galton. "Free motion in the wave mechanics." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117.776 (1927): 258-293.</ref> Darwin showed that at time <math>t</math> later the position <math>x</math> of the packet traveling at velocity <math>v</math> would be

<math display=block>x_0 + vt \pm \sqrt{\sigma^2 + (ht/2\pi\sigma m)^2}</math>

where <math>\sigma</math> is the uncertainty in the initial position.

Later in 1927 [[Paul Ehrenfest]] showed that the time, <math>T</math> for a [[matter wave]] packet of width <math>\Delta x</math> and mass <math>m</math> to spread by a factor of 2 was <math display=inline>T\approx m {\Delta x}^2/\hbar </math>. Since <math>\hbar</math> is so small, wave packets on the scale of macroscopic objects, with large width and mass, double only at cosmic time scales.<ref>{{Cite journal |last=Kragh |first=Helge |last2=Carazza |first2=Bruno |date=2000 |title=Classical Behavior of Macroscopic Bodies from Quantum Principles: Early Discussions |url=https://www.jstor.org/stable/41134097 |journal=Archive for History of Exact Sciences |volume=55 |issue=1 |pages=43–56 |issn=0003-9519}}</ref>{{rp|49}}