| Wigner-ville Distribution |
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The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study Quantum corrections to classical statistical mechanics. The goal was to supplant the Wavefunction that appears in Schrödinger's Equation with a probability distribution in Phase Space . It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction ψ(x), and corresponds to the quantum Density Matrix in the map between real phase-space functions and Hermitean operators introduced by Hermann Weyl in 1931, in Representation Theory in mathematics (cf. Weyl Quantization in physics). It was later rederived by J. Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal. It is sometimes known as the "Wigner function" or the "Wigner-Ville distribution". In 1949, José Enrique Moyal , who had also rederived it independently, recognized it as the quantum moment-generating functional, and thus the basis of an elegant encoding of all quantum expectation values and hence quantum mechanics in phase space (cf Weyl Quantization ). It has applications in Statistical Mechanics , Quantum Chemistry , Quantum Optics , classical Optics and signal analysis in diverse fields such as Electrical Engineering , Seismology , Biology , speech processing, and engine design. A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection ( Ensemble ) of particles, the probability of finding a particle at a certain position in phase space is specified by a probability distribution, the Liouville density. This fails for a quantum particle, due to the Uncertainty Principle . Instead, the above quasi-probability Wigner distribution plays an analogous role, but does not satisfy all the properties of a conventional probability distribution; and, conversely, satisfies boundedness properties unavailable to classical distributions. For instance, the Wigner distribution can and normally does go negative for states which have no classical model---and is a convenient indicator of quantum mechanical interference. Smoothing the Wigner distribution through a filter of size larger than (e.g., convolving with a phase-space Gaussian to yield the Hussimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one. Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions larger than a few , and hence disappear in the classical limit. They are shielded by the Uncertainty Principle , which does not allow precise location within phase-space regions smaller than , and thus renders such "negative probabilities" less paradoxical. The Wigner distribution ''P''(''x'', ''p'') is defined as:
where ψ is the wavefunction and x and p are position and momentum but could be any conjugate variable pair. (ie. real and imaginary parts of the electric field or frequency and time of a signal). It is symmetric in x and p:
where φ is the Fourier Transform of ψ. In the general case, which includes mixed states, it is the Wigner transform of the Density Matrix : | ||
| <math>\int {-\infty}^{\infty}dp\,P(x,p) | \langle x\hat{
ho}x
angle</math> If the system can be described by a Pure State , one gets <math>\int_{-\infty}^{\infty}dp\,P(x,p)= \psi(x)^2</math> |
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| <math>\int {-\infty}^{\infty}dx\,P(x,p) | \langle p\hat{
ho}p
angle</math> If the system can be described by a Pure State , one gets <math>\int_{-\infty}^{\infty}dx\,P(x,p)=\phi(p)^2</math> |
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| <math>\langle \psi Heta Angle^2 | 2\pi\hbar\int_{-\infty}^{\infty}dx\,\int_{-\infty}^{\infty}dp\,P_{\psi}(x,p)P_{ heta}(x,p)</math> |
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| <math>\langle \psi\hat{G}\psi Angle | Tr(\hat{
ho}\hat{G})=\int_{-\infty}^{\infty}dx\, \int_{-\infty}^{\infty}dp P(x,p)g(x,p)~ |
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| In The Modelling Of Optical Systems Such As Telescopes Or Fibre Telecommunications Devices, The Wigner Function Is Used To Bridge The Gap Between Simple | "http://wwwinformationdelightinfo/information/entry/ray_tracing" class="copylinks">Ray Tracing and the full wave analysis of the system Here <math>p / \hbar</math> is replaced with ''k''=''k''sin&theta&asymp''k''&theta in the small angle (paraxial) approximation In this context, the Wigner function is the closest one can get to describing the system in terms of rays at position ''x'' and angle &theta while still including the effects of interference If it becomes negative at any point then simple ray-tracing will not suffice to model the system |
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