Wave packet

From Wikipedia, the free encyclopedia

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 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 wave packet, and therefore the better defined the position of the wave packet, the larger the uncertainty in the momentum of the wave. This tradeoff is known as the Heisenberg uncertainty principle.

1 Background
2 Mathematics of wave packets
3 Quantum mechanical waves
4 Superposition
5 The collapse of the wave packet
6 Metaphysical claims
7 References

[edit] 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.

[edit] 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. The wave equation has plane-wave solutions

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

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

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

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

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

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

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 $t = 0$ :

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

Let me add a little comment without which above-mentioned derivation would be hard to comprehend by most people myself included. Not everybody has simply patience to solve puzzles each time he or she tries to learn anything about quantum mechanics. This differential equation has a simple and useful solution in the form of something close to Maxwell distribution:

$A(k) = A_{0}\exp[-\frac{1}{\sqrt{2\pi}}\left(k - k_{0}\right)]$

where Ao and ko are constants.

[edit] Quantum mechanical waves

A quantum mechanical wave in its most salient and simple form is a solution to a differential equation. It is a bridge that provides mathematical insight into physical problems. The solutions of these mathematical models are postulated in quantum mechanics to provide the possible or observable outcomes for any experiment. Unfortunately, the wave packet combined with the probabilistic nature of measurement leads to some peculiarities.

First, the solutions do not provide answers about single experiments. These solutions can only be confirmed by measurement, and measurement is probabilistic in nature. Experimental confirmation of a prediction of quantum mechanics is only given in the outcomes of repeated similar experiments. The first mistake that many people make in thinking about quantum mechanical predictions is in thinking about only single experiments. In reality, quantum mechanics has little to say about the observations made in single experiments unless the system is prepared in an energy eigenstate.

Next, the quantum mechanical wave is a representation of a particle. The wave carries the information about particle position and momentum and also any other observable that can be derived from position and momentum. However, there is no reason to believe that a quantum wave actually is a particle. In the words of Dirac, the wave expresses information. A quantum mechanical wave can be nothing more than a mathematical model.

Nevertheless, it is reasonable to think about experiments in terms of quantum mechanical intuition. It is this blurring of the line between mathematical modeling and the quantum picture of the world that so often leads to confusion. For the purposes of the uninitiated it would be much safer to consider only the model, and leave the intuition to those who are better acquainted with the mathematical intricacies of quantum mechanics.

[edit] Superposition

One of the most common classes of problems discussed in a quantum mechanics are interference phenomena. These interference phenomena apparently arise from the self-interaction of particles and the wave-like nature of these interactions. Such self-interaction is enabled by the principle of superposition. A particle does not negotiate any single path through a diffraction grating; its probability wave actually coincidently traverses all possible paths. Ultimately it is the act of measurement that collapses the wave packet to the single observed outcome.

[edit] The collapse of the wave packet

The superposition principle of quantum mechanics allows any solution to the Schrödinger equation to be composed of a linear combination of any number of possible states in a complete set of commuting observables. The act of measurement typically collapses these superimposed states into a single outcome. For instance, the state of an electron passing through a double slit is most correctly described as a combination of each of the possible individual paths it might take, resulting in the famous double slit diffraction pattern. If, however, a measurement is made at the slits, then the double slit diffraction pattern disappears because the electron now travels through only a single slit.

Quantum mechanics places no constraints on how we interpret the collapse of the wave packet. We might say that during the act of observation the electron suddenly and probabilistically jumps into the state we measure. On the other hand we might suppose, as many early physicists, including Albert Einstein, erroneously did, that the particle was in the observed state all along. This is the classical interpretation and does not agree with experimental evidence. Because none of the modern interpretations can be falsified experimentally, quantum mechanics provides no way of knowing how or why the wave packet collapsed, and what, if any, significance the event holds.

Pointedly, Dirac intimates that this question has less significance than many people assume. The importance of quantum mechanics lies in the predictions that it makes, and the success of those predictions, culminating in nearly overwhelming experimental confirmation of the theory. The fact that we don't really know how to interpret the collapse of the wave packet at this time is irrelevant, what is important for physics is that the theory works at all.

[edit] Metaphysical claims

Many discussions about the collapse of the wave packet and quantum superposition occur in metaphysical and speculative fiction circles, primarily based on attempts to draw analogies between the language used in quantum physics with the world of the layman. Common themes are the many-worlds interpretation and the related multiverse, proof of the existence of God, teleportation, faster-than-light travel, and proof of human super-consciousness.

Generally, like most fiction, these themes sacrifice scientific rigor in favor of evocative concepts, as the physics of quantum mechanics is not easily described relative to human experience.