The wave packet is one of the most widely misunderstood and misused concepts in physics. In general, a wave packet is an envelope or packet containing an arbitrary number of wave forms. This is also true in quantum mechanics however in this case the wave packet is ascribed a special significance. In quantum mechanics the wave packet is interpreted to be a "probability wave" describing the probability that a particular state will have a given position and momentum. Fig. ...

By applying the Schrodinger 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. In its most fundamental form the wave packet represents a mathematical solution to the Schrodinger equation. Then the square of the area under the wave form solution is interpreted to be the probability of finding the particle in a region of consideration. In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... In physics, Hamiltonian has distinct but closely related meanings. ... In physics, Classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ...

A graphical representation of a wave packet. The horizontal axes is position. The momentum of the wavepacket is related to the width of the wave packet.

An important consequence of this interpretation is that the width of the packet in the coordinate representation of the wave (such as the Cartesian coordinate system) corresponds to the momentum of the wave. Moreover, the position of the wave is given by the position of the packet. Upon inspection of the image it is possible to see that the position is not particularly clear. In fact, the larger the spread of the wave packet, and hence the greater the knowledge of the specific momentum of the particle, the less clear the actual location of the particle is. This tradeoff is known as the Heisenberg uncertainty principle. A Wave Packet File links The following pages link to this file: Wave packet ... A Wave Packet File links The following pages link to this file: Wave packet ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Werner Heisenberg Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ...

Background

In the early 1900's 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 lead scientists to favor a wave description of electromagnetism. It wasn't until the 1930's that the particle nature of light really began to be widely accepted in Physics. The development of quantum mechanics was at the foundation of this acceptance. Sir Isaac Newton in Knellers 1689 portrait Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemist who wrote... The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ...

One of the most important concepts in the formulation of quantum mechanics is the idea that the spectra of atoms is discrete. Therefore, the energy of light packets is a discrete function of frequency:

E = nhf

The energy, E, is given as an integer, n, multiple of Planck's constant, h, and frequency, f. This interpretation resolved a significant problem in classical physics. Max Planck This article is about Planck, the German physicist. ... The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of 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 and interacting with discrete particles. However in quantum mechanics 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. At the same time, the wave-like phenomena could be characterized as consequences of the wave packet nature of particles. (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the... The Stanford Linear Accelerator Center (SLAC) is a U.S. national laboratory operated by Stanford University for the U.S. Department of Energy. ...

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. In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...

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.

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.

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. Electron diffraction is a technique used to examine solids by firing a beam of electrons at a sample and observing their deflection. ... Quantum superposition is the application of superposition principle to quantum mechanics. ...

The Collapse of the Wave Packet

The superposition principle of quantum mechanics allows any solution to the Schrodinger equation to be composed of a linear combination of any number of possible states in a complete set of commuting observables. Nevertheless, the act of measurement typically collapses these superimposed states into a single outcome. So while the state of an electron passing through a diffraction grating is most correctly described as a combination of each of the possible individual paths it might take. The observation or measurement is most correctly described by only a single resulting path, i.e., the one it actually is observed to take. Quantum superposition is the application of superposition principle to quantum mechanics. ...

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 conclude, as the early physcists, including Albert Einstein, eroneously did, that the particle was in the observed state all along. This is the classical interpretation and does not agree with experimental evidence. Consequently, many physicists conclude that quantum mechanics simply does not answer this question. Quantum mechanics provides no way of knowing how or why the wave packet collapsed, and what if any significance the event holds. Portrait of Albert Einstein taken by Yousuf Karsh on February 11, 1948 Albert Einstein (March 14, 1879 – April 18, 1955) was a German theoretical physicist who is widely regarded as the greatest scientist of the 20th century. ...

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 abject success of those predictions, culminating in nearly overwhelming experimental confirmation of the theory. Although, it may leave some with a feeling of hopelessness, 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.

Metaphysical Claims

Many discussions about the collapse of the wave packet and quantum superposition occur in metaphysical circles. Arguments have been made across a whole spectrum of metaphysical topics including proof of the existence of god, proof of human super-consciousness, and other similar statements. At fault for some claims are scientists who in their fervor to introduce lay people to modern physics often introduce vague analogies and incomplete analysis to induce interest in potential audiences. Unfortunately, no reasonable interpretation of the principles of quantum mechanics or the experimental results demonstrating the validity of quantum physics affirms any of the metaphysical speculation.

Having dispelled any rational basis for such claims it is fun to explore a few examples of meta-quantum physics on a contingency basis. The speculation about metaphysical interpretations of the collapse of the wave packet is essentially as boundless as the body of metaphysical thought itself, however a few ideas are informally interesting enough to be presented here.

The collapse of the wave packet is a highly enigmatic process. In one instant of time evolution a quantum state might exist as a linear superposition of a complete set of commuting observables. In another instant, concomitant with measurement, the wave packet takes on the nature of a single observed state. Many interpreters of quantum mechanics say that as we are intelligent, and we observe the resulting state, so must the wave packet possess intelligence to 'know' how to confine itself to the observed state. Some philosophers reason that this intelligence is the endowment of a quantum mechanical creator, essentially proof of the existence of god born in the laws of Physics. However interesting, such speculations are futile from the standpoint of science. The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ...

In another instance of metaphysical speculation, the information carried by the wave packet is considered along with human intuition. If a particle is observed to travel a certain distance, as in ordinary particle motion, special relativity establishes that the information carried by that particle can travel no faster than light itself. However, one might frame an argument in which an entangled particle, after traveling for some time, undergoes observation which therefore causes collapse of the wavefunction. If this collapse implies that any information has been transferred, then this might also nefariously imply so called "instantaneous action at a distance" or unbounded communication of information. If observed, such results might be consistent with faster than light travel, or human super-consiousness. In reality, no such observation has ever been made and there is no quantum mechanical basis to such claims. Although quantum mechanical entanglement allows for quantum correlation across unbounded distances, there is no way to exploit these correlations to transmit information across these distances. A simple introduction to this subject is provided in Special relativity for beginners Special relativity(SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ...

Another alternative theory concerning the collapse of the wave packet is the so called Many-worlds interpretation. This interpretation is often supported or at least supposed by science writers. The Many-worlds interpretation supposes that each superimposed state of the wave packet represents a probabilistic alternative reality. As such, measurement does not so much collapse the wave packet as "choose" the reality followed by our experiment. It scarcely need be pointed out that such speculation, although fantastical, is relatively indefensable from the standpoint of the postulates of quantum mechanics. This article may be too technical for most readers to understand. ...

Although the ideas of quantum mechanics are deep and sometimes lead to mysterious physical predictions, they do not have to be intangible or supernormal. Indeed the science and the study of quantum mechanics is at its heart about physical prediction and modeling. We can explore probability waves and consider superposition principles without mystery, without vagueness, and without outlandish extrapolation into metaphysical realms.

Though they seem strange to us, quantum principals are as mundane to the electron as any other physics.