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. 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, see figure) or it may change (dispersion) while propagating. ψ t ( x ) = ∫ ψ 0 ( y ) 1 2 π i t e i ( x − y ) 2 / 2 t d y . {displaystyle psi _{t}(x)=int psi _{0}(y){1 over {sqrt {2pi it}}}e^{i(x-y)^{2}/2t}dy,.} 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. 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, 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 amplitude, its norm squared describing the probability density that a particle or particles in a particular state will be measured to have a given position or momentum. The wave equation is in this case the Schrödinger equation. It is possible to deduce the time evolution of a quantum mechanical system, similar to the process of the Hamiltonian formalism in classical mechanics. 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 position of the physical object's localized probability is specified by the position of the packet solution. Moreover, 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,and will be illustrated below. In the early 1900s, it became apparent that classical mechanics had some major failings. Isaac Newton originally proposed the idea that light came in discrete packets, which he called corpuscles, but the wave-like behavior of many light phenomena quickly led scientists to favor a wave description of electromagnetism. It wasn't until the 1930s that the particle nature of light really began to be widely accepted in physics. The development of quantum mechanics – and its success at explaining confusing experimental results – was at the root of this acceptance. Thus, one of the basic concepts in the formulation of quantum mechanics is that of light coming in discrete bundles called photons. The energy of a photon is a function of its frequency, The photon's energy is equal to Planck's constant, h, multiplied by its frequency, ν. This resolved a problem in classical physics, called the ultraviolet catastrophe.