Wave packet

# Wave packet

Discussion

Encyclopedia
In physics, a wave packet (or wave train) 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 wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...

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 mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

ascribes a special significance to the wave packet: it is interpreted to be a "probability wave" describing the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

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 equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it 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 evolution
Time evolution
Time 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 Hamiltonian
Hamiltonian mechanics
Hamiltonian 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 mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

. The wave packet is a mathematical solution to the Schrödinger equation. The area under the square of 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 rule
Born rule
The 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. The Born rule is one of the key principles of quantum mechanics...

.

In the coordinate representation of the wave (such as the Cartesian coordinate system
Cartesian coordinate system
A 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 momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

of the wave. This trade-off between spread in position and spread in momentum is one example of the Heisenberg uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

.

## Background

In the early 1900s it became apparent that classical mechanics had some major failings. Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

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
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

. It wasn't until the 1930s that the particle nature of light really began to be widely accepted in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

. 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 a positive integer, n, multiple of Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

's constant, h, and frequency, ν. This resolved a significant problem in classical physics, called the ultraviolet catastrophe
Ultraviolet catastrophe
The 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 amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air 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 equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like 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 transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented 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 equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

(with m and h-bar set equal to one)

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 velocity
Group velocity
The 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 width increasing according to .