Толқындық пакеттер: физикалық сипаттама және қасиеттері

Short "burst" or "envelope" of restricted wave action that travels as a unit

In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an envelope. A wave packet can be analyzed into, or can be synthesized from, a potentially 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. Any signal of a limited width in time or space requires many frequency components around a center frequency within a bandwidth inversely proportional to that width; even a gaussian function is considered a wave packet because its Fourier transform is a "packet" of waves of frequencies clustered around a central frequency. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant (no dispersion) or it may change (dispersion) while propagating.

Ideas related to wave packets – modulation, carrier waves, 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 W. R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877. Erwin Schrödinger introduced the idea of wave packets just after publishing his famous wave equation. 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 oscillators, not for example to Coulomb potential characteristic of atoms. Darwin showed that at time later the position of the packet traveling at velocity would be

where is the uncertainty in the initial position. Later in 1927 Paul Ehrenfest showed that the time, for a matter wave packet of width and mass to spread by a factor of 2 was Since is so small, wave packets on the scale of macroscopic objects, with large width and mass, double only at cosmic time scales. 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 The spread is so fast that in the distance of once around an atom the wave packet is unrecognizable.

Particle interactions are called scattering in physics; wave packet mathematics play an important role in quantum scattering approaches. A monochromatic (single momentum) source produces convergence difficulties in the scattering models. 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.