Wave packet

Wave packet
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Wave_packet".

A wave packet without dispersion.

In physics, a wave packet is an envelope or packet containing an arbitrary number of wave forms. In quantum mechanics the wave packet is ascribed a special significance: it is interpreted to be 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.

By applying 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 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.

In the coordinate representation of the wave (such as the Cartesian coordinate system) 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 of the wave. This trade-off between spread in position and spread in momentum is one example of the Heisenberg uncertainty principle.

content

1 Background
2 Mathematics of wave packets
3 See also
4 References

Background

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 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:

E = n h f

The energy is an integer, n, multiple of Planck's constant, h, and frequency, f. This resolved a significant problem in classical physics, called the ultraviolet catastrophe.

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 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, consider wave solutions to the following wave equation:

${ \partial^2 u \over \partial t^2 } = c^2 { \nabla^2 u }$

where c is the speed of the wave's propagation in a given medium. Using the physics time convention, $e - i ω t$ , the wave equation has plane-wave solutions

$u(\bold{x},t) = e^{i{(\bold{k\cdot x}}-\omega t)}$

where $|\bold{k}|=\frac{\omega}{c}$ .

To simplify, consider only waves propagating in one dimension. Then the general solution is

$u(x,t)= A e^{i(kx-\omega t)} + B e^{-i(kx+\omega t)} \,$ ,

where the first term represents a wave propagating in the positive x-direction and the second term 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 the one dimension, a general form of a wave packet can be expressed as

$f(x,t) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{-\infty} A(k) ~ e^{i(kx-\omega(k)t)} \,dk$ .

The factor $1/\sqrt{2\pi}$ comes from Fourier transform conventions. The amplitude $A (k)$ contains the coefficients of the linear superposition of the plane wave solutions. These coefficients can in turn be expressed as a function of $f (x, t)$ evaluated at $t = 0$ by inverting the Fourier transform relation above:

$A(k) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{-\infty} f(x,0) ~ e^{-ikx}\,dx$ .

This differential equation has a simple and useful solution in agreement with the Maxwell distribution:

$A(k) = A_{0} ~ \exp(-\frac{1}{\sqrt{2\pi}}|k - k_{0}|)$

where $A o$ and $k o$ are constants.

References

Jackson, J.D. (1975). Classical Electrodynamics (2nd Ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-43132-X

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Contents

Background

Mathematics of wave packets

See also

References