In physics, a
wave packet is a short "burst" or "envelope" of 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
wavenumberWavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters . Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of wavelengths per unit distance, or more commonly times that, or the number...
s, 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.
In physics, a
wave packet is a short "burst" or "envelope" of 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
wavenumberWavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters . Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of wavelengths per unit distance, or more commonly times that, or the number...
s, 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 mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
ascribes a special significance to the wave packet: it is interpreted to be a "probability wave" describing the
probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
that a particle or particles in a particular state will be measured to have a given position and momentum. It is in this way similar to the wave function.
By applying the
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
in quantum mechanics it is possible to deduce the
time evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...
of a system, similar to the process of the
HamiltonianHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
formalism in
classical mechanicsIn the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain...
. The wave packet is a mathematical solution to the Schrödinger equation.The square of the area under the wave packet solution is interpreted to be 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 ruleThe Born rule is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born...
.
In the coordinate representation of the wave (such as the
Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length....
) the position of the wave 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
momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
of the wave. This trade-off between spread in position and spread in momentum is one example of the Heisenberg
uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known...
.
Background
In the early 1900s it became apparent that classical mechanics had some major failings.
Isaac NewtonSir Isaac Newton FRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is perceived and considered by a substantial number of scholars and the general public as one of the most influential men in history...
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
electromagnetismElectromagnetism is the physics of the electromagnetic field, a field that exerts a force on particles with the property of electric charge and is reciprocally affected by the presence and motion of such particles....
. It wasn't until the 1930s that the particle nature of light really began to be widely accepted in
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
. The development of quantum mechanics — and its success at explaining confusing experimental results — was at the foundation of this acceptance.
One of the most important concepts in the formulation of quantum mechanics is the idea that light comes in discrete bundles called photons. The energy of light is a discrete function of frequency:
The energy is an integer,
n, multiple of
PlanckMax Planck was a German physicist. He is considered to be the founder of the quantum theory, and thus one of the most important physicists of the twentieth century. Planck was awarded the Nobel Prize in Physics in 1918.-Biography:Planck came from a traditional, intellectual family...
's constant,
h, and frequency,
f. This resolved a significant problem in classical physics, called the
ultraviolet catastropheThe ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power....
.
The ideas of quantum mechanics continued to be developed throughout the 20th century. The picture that was developed was of a particulate world, with all phenomena and matter made of and interacting with discrete particles; however, these particles were described by a probability wave. The interactions, locations, and all of physics would be reduced to the calculations of these probability
amplitudeAmplitude is the magnitude of change in the oscillating variable, with each oscillation, within an oscillating system. For instance, sound waves are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
waves. The particle-like nature of the world was significantly confirmed by experiment, while the wave-like phenomena could be characterized as consequences of the wave packet nature of particles.
Mathematics of wave packets
As an example of propagation without dispersion, consider wave solutions to the following
wave equationThe wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics...
:
where c is the speed of the wave's propagation in a given medium.
Using the physics time convention, , the wave equation has plane-wave solutions
where
This relation between and
should be valid so that the plane wave is a solution to the wave equation. It is called a
dispersion relation.
To simplify, consider only waves propagating in one dimension (extension to three dimensions is possible). Then the general solution is
in which we may take The first term represents a wave propagating in the positive x-direction since it is a function of only; the second term, being a function of , represents a wave propagating in the negative x-direction.
A wave packet is a localized disturbance that results from the sum of many different wave forms. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region. From the basic solutions in one dimension, a general form of a wave packet can be expressed as
.
Like in the plane-wave case the wave packet travels to the right for (since then ) and to the left for
(since then ).
The factor comes from
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
conventions. The amplitude contains the coefficients of the
linear superposition of the plane wave solutions. These coefficients can in turn be expressed as a function of evaluated at by inverting the Fourier transform relation above:
.
For instance, choosing
we obtain
and
The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.
As an example of propagation
with dispersion consider
solutions to the
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
yielding as dispersion relation
Once again restricting ourselves to one dimension the solution to the Schrödinger equation satisfying the initial condition is found according to
An impression of the dispersive behaviour of this wave packet is obtained by looking at
(note that is not a solution of the Schrödinger equation). It is seen that the dispersive wave packet, while moving with constant
group velocityThe group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....
, has a time-dependent
widthIn mathematics, a Gaussian function is a function of the form:for some real constants a > 0, b, c > 0, and e ≈ 2.718281828 ....
increasing according to .