Editing Wave packet (section)

== Significance in quantum mechanics ==

[[Quantum mechanics]] describes the nature of atomic and subatomic systems using [[Schrödinger equation|Schrödinger's wave equation]].  The classical limit of quantum mechanics and many formulations of quantum scattering use wave packets formed from various solutions to this equation.
Quantum wave packet profiles change while propagating; they show dispersion. Physicists have concluded that "wave packets would not do as representations of subatomic particles".<ref name="Kragh"/>{{rp|829}}

=== Wave packets and the classical limit ===
Schrodinger developed wave packets in hopes of interpreting quantum wave solutions as locally compact wave groups.<ref name="Kragh"/> Such packets tradeoff position localization for spreading momentum. In the coordinate representation of the wave (such as the [[Cartesian coordinate system]]), the position of the particle's localized probability is specified by the position of the packet solution.  The narrower the spatial wave packet, and therefore the better localized the position of the wave packet, the larger the spread in the [[momentum]] of the wave. This trade-off between spread in position and spread in momentum is a characteristic feature of the Heisenberg uncertainty principle.

One kind of optimal tradeoff minimizes the product of position uncertainty <math>\Delta x</math> and momentum uncertainty <math>\Delta p_x</math>.<ref name="Schiff">{{Cite book |last=Schiff |first=Leonard I. |title=Quantum mechanics |date=1995 |publisher=McGraw-Hill |isbn=978-0-07-055287-6 |edition=3. ed., 29. print |series=International series in pure and applied physics |location=New York}}</ref>{{rp|60}} If we place such a packet at rest it stays at rest: the average value of the position and momentum match a classical particle. However it spreads out in all directions with a velocity given by the optimal momentum uncertainty <math>\Delta p_x</math>. The spread is so fast that in the distance of once around an atom the wave packet is unrecognizable.

=== Wave packets and quantum scattering ===
Particle interactions are called [[scattering]] in physics; the wave packet concept plays an important role in [[Lippmann–Schwinger equation#Creating wavepackets|quantum scattering approaches]]. A monochromatic (single momentum) source produces convergence difficulties in the scattering models.<ref name="Newton">{{Cite book |last=Newton |first=Roger G. |title=Scattering theory of waves and particles |date=1982 |publisher=Springer |isbn=978-0-387-10950-3 |edition=2|series=Texts and monographs in physics |location=New York, Heidelberg, Berlin}}</ref>{{rp|150}} Scattering problems also have classical limits. Whenever the scattering target (for example an atom) has a size much smaller than wave packet, the center of the wave packet follows scattering classical trajectories. In other cases, the wave packet distorts and scatters as it interacts with the target.<ref name="Susskind-Friedman">{{Cite book |last1=Susskind |first1=Leonard |title=Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics] |last2=Friedman |first2=Art |last3=Susskind |first3=Leonard |date=2014 |publisher=Basic Books |isbn=978-0-465-08061-8 |series=The theoretical minimum / Leonard Susskind and George Hrabovsky |location=New York, NY}}</ref>{{rp|295}}