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Wave packet

From Academic Kids

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.

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.

Missing image
Wavepacket.png
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.

Contents

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.

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:

<math> E = nhf <math>

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.

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.

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.

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.

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 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.

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. In fact, no experimental results demonstrating the validity of quantum physics affirms any of the metaphysical speculation.

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 outside of the realm of science.

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 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. Although quantum mechanical entanglement allows for quantum correlation across unbounded distances, there is no known way to exploit these correlations to transmit information across these distances.

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-fiction. 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.

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.de:Wellenpaket

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