Wave Packet

In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Depending on the evolution equation, the wave packet's envelope may remain constant (no dispersion, see figure) or it may change (dispersion) while propagating.

Quantum mechanics ascribes a special significance to the wave packet: it is interpreted as a "probability wave", describing the probability that a particle or particles in a particular state will be measured to have a given position and momentum. It is in this way related to the wave function. Through application of the Schrödinger equation in quantum mechanics, it is possible to deduce the time evolution of a system, similar to the process of the Hamiltonian formalism in classical mechanics. The wave packet is thus a mathematical solution to the Schrödinger equation. The area under the absolute square of the wave packet solution is interpreted as the probability density of finding the particle in a region. The dispersive character of solutions of the Schrödinger equation has played an important role in rejecting Schrödinger's original interpretation, and accepting the Born rule.

In the coordinate representation of the wave (such as the Cartesian coordinate system), the localized position of the physical object's probability is given by the position of the packet. Moreover, the narrower the spatial wave packet, and therefore the better defined 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 one example of the Heisenberg uncertainty principle.