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This is usually written in the form |
In flat Minkowski space (special relativity), with coordinates |
the metric is, depending on choice of metric signature, |
For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a timelike curve, the length formula gives the proper time along the curve. |
In this case, the spacetime interval is written as |
The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. With coordinates |
where (inside the matrix) is the gravitational constant and represents the total mass-energy content of the central object. |
Classical fluids are systems of particles which retain a definite volume, and are at sufficiently high temperatures (compared to their Fermi energy) that quantum effects can be neglected. A system of hard spheres, interacting only by hard collisions (e.g., billiards, marbles), is a model classical fluid. Such a system is well described by the Percus–Yevik equation. Common liquids, e.g., liquid air, gasoline etc., are essentially mixtures of classical fluids. Electrolytes, molten salts, salts dissolved in water, are classical charged fluids. A classical fluid when cooled undergoes a freezing transition. On heating it undergoes an evaporation transition and becomes a classical gas that obeys Boltzmann statistics. |
A system of charged classical particles moving in a uniform positive neutralizing background is known as a one-component plasma (OCP). This is well described by the Hyper-netted chain equation (see CHNC). |
An essentially very accurate way of determining the properties of classical fluids is provided by the method of molecular dynamics. |
An electron gas confined in a metal is "not" a classical fluid, whereas a very high-temperature plasma of electrons could behave as a classical fluid. Such non-classical Fermi systems, i.e., quantum fluids, can be studied using quantum Monte Carlo methods, Feynman path integral equation methods, and approximately via CHNC integral-equation methods. |
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution after Eugene Wigner and ) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space. |
It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction . |
Thus, it maps on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927, in a context related to representation theory in mathematics (cf. Weyl quantization in physics). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It was later rederived by Jean Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal, effectively a spectrogram. |
In 1949, José Enrique Moyal, who had derived it independently, recognized it as the quantum moment-generating functional, and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (cf. phase space formulation). It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields such as electrical engineering, seismology, time–frequency analysis for music signals, spectrograms in biology and 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 strict interpretation fails |
for a quantum particle, due to the uncertainty principle. Instead, the above quasiprobability 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 take on negative values for states which have no classical model—and is a convenient indicator of quantum mechanical interference. (See below for a characterization of pure states whose Wigner functions are non-negative.) |
Smoothing the Wigner distribution through a filter of size larger than (e.g., convolving with a |
phase-space Gaussian, a Weierstrass transform, to yield the Husimi 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 of a pure state is defined as: |
where is the wavefunction and and are position and momentum but could be any conjugate variable pair (e.g. real and imaginary parts of the electric field or frequency and time of a signal). Note that it may have support in even in regions where has no support in ("beats"). |
where is the normalized momentum-space wave function, proportional to the Fourier transform of . |
In the general case, which includes mixed states, it is the Wigner transform of the density matrix, |
where ⟨"x"|"ψ"⟩ = . This Wigner transformation (or map) is the inverse of the Weyl transform, which maps phase-space functions to Hilbert-space operators, in Weyl quantization. |
Thus, the Wigner function is the cornerstone of quantum mechanics in phase space. |
how the Wigner function provides the integration measure (analogous |
to a probability density function) in phase space, to yield expectation values from phase-space c-number functions uniquely associated to suitably ordered operators through Weyl's transform (cf. Wigner–Weyl transform and property 7 below), in a manner evocative of classical probability theory. |
Specifically, an operator's expectation value is a "phase-space average" of the Wigner transform of that operator, |
1. "W"("x", "p") is a real valued function. |
2. The "x" and "p" probability distributions are given by the marginals: |
3. "W"("x", "p") has the following reflection symmetries: |
5. The equation of motion for each point in the phase space is classical in the absence of forces: |
7. Operator expectation values (averages) are calculated as phase-space averages of the respective Wigner transforms: |
8. In order that "W"("x", "p") represent physical (positive) density matrices: |
9. By virtue of the Cauchy–Schwarz inequality, for a pure state, it is constrained to be bounded, |
10. The Wigner transformation is simply the Fourier transform of the antidiagonals of the density matrix, when that matrix is expressed in a position basis. |
Let formula_19 be the formula_20-th Fock state of a quantum harmonic oscillator. Groenewold (1946) discovered its associated Wigner function, in dimensionless variables, isformula_21 |
where formula_22 denotes the formula_20-th Laguerre polynomial. |
This may follow from the expression for the static eigenstate wavefunctions, formula_24, |
where formula_25 is the formula_20-th Hermite polynomial. From the above definition of the Wigner function, |
The expression then follows from the integral relation between Hermite and Laguerre polynomials. |
The Wigner transformation is a general invertible transformation of an operator on a Hilbert space to a function "g(x,p)" on phase space, and is given by |
Hermitian operators map to real functions. The inverse of this transformation, |
so from phase space to Hilbert space, is called the Weyl transformation, |
(not to be confused with the distinct Weyl transformation in differential geometry). |
The Wigner function "W"("x,p") discussed here is thus seen to be the Wigner transform of the density matrix operator "ρ̂". Thus, the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap of "g"("x", "p") with the Wigner function. |
The Wigner transform of the von Neumann evolution equation of the density matrix in the Schrödinger picture is |
where H(x,p) is Hamiltonian and { {•, •} } is the Moyal bracket. In the classical limit "ħ" → 0, the Moyal bracket reduces to the Poisson bracket, while this evolution equation reduces to the Liouville equation of classical statistical mechanics. |
Strictly formally, in terms of quantum characteristics, the solution of |
where formula_32 and formula_33 are solutions of |
so-called quantum Hamilton's equations, subject to initial conditions |
composition is understood for all argument functions. |
Since, however, formula_36-composition is thoroughly nonlocal (the "quantum probability fluid" diffuses, as observed by Moyal), vestiges of local trajectories |
are normally barely discernible in the evolution of the Wigner distribution function. |
In the integral representation of -products, successive operations by them have been adapted to a phase-space path-integral, to solve this evolution equation for the Wigner function (see also ). |
This non-trajectoral feature of Moyal time evolution is illustrated in the gallery below, for Hamiltonians more complex than the harmonic oscillator. |
In the special case of the quantum harmonic oscillator, however, the evolution is simple and appears identical to the classical motion: a rigid rotation in phase space with a frequency given by the oscillator frequency. This is illustrated in the gallery below. This same time evolution occurs with quantum states of light modes, which are harmonic oscillators. |
The Wigner function allows one to study the classical limit, offering a comparison of the classical and quantum dynamics in phase space. |
It has been suggested that the Wigner function approach can be viewed as a quantum analogy to the operatorial formulation of classical mechanics introduced in 1932 by Bernard Koopman and John von Neumann: the time evolution of the Wigner function approaches, in the limit "ħ" → 0, the time evolution of the Koopman–von Neumann wavefunction of a classical particle. |
As already noted, the Wigner function of quantum state typically takes some negative values. Indeed, for a pure state in one variable, if formula_38 for all formula_39 and formula_40, then the wave function must have the form |
for some complex numbers formula_42 with formula_43 (Hudson's theorem). Note that formula_44 is allowed to be complex, so that formula_45 is not necessarily a Gaussian wave packet in the usual sense. Thus, pure states with non-negative Wigner functions are not necessarily minimum uncertainty states in the sense of the Heisenberg uncertainty formula; rather, they give equality in the Schrödinger uncertainty formula, which includes an anticommutator term in addition to the commutator term. (With careful definition of the respective variances, all pure state Wigner functions lead to Heisenberg's inequality all the same.) |
In higher dimensions, the characterization of pure states with non-negative Wigner functions is similar; the wave function must have the form |
where formula_47 is a symmetric complex matrix whose real part is positive definite, formula_48 is a complex vector, and is a complex number. The Wigner function of any such state is a Gaussian distribution on phase space. |
The cited paper of Soto and Claverie gives an elegant proof of this characterization, using the Segal–Bargmann transform. The reasoning is as follows. The Husimi Q function of formula_45 may be computed as the squared magnitude of the Segal–Bargmann transform of formula_45, multiplied by a Gaussian. Meanwhile, the Husimi Q function is the convolution of the Wigner function with a Gaussian. If the Wigner function of formula_45 is non-negative everywhere on phase space, then the Husimi Q function will be strictly positive everywhere on phase space. Thus, the Segal–Bargmann transform formula_52 of formula_45 will be nowhere zero. Thus, by a standard result from complex analysis, we have |
for some holomorphic function formula_55. But in order for formula_56 to belong to the Segal–Bargmann space—that is, for formula_56 to be square-integrable with respect to a Gaussian measure—formula_55 must have at most quadratic growth at infinity. From this, elementary complex analysis can be used to show that formula_55 must actually be a quadratic polynomial. Thus, we obtain an explicit form of the Segal–Bargmann transform of any pure state whose Wigner function is non-negative. We can then invert the Segal–Bargmann transform to obtain the claimed form of the position wave function. |
There does not appear to be any simple characterization of mixed states with non-negative Wigner functions. |
The Wigner function in relation to other interpretations of quantum mechanics. |
It has been shown that the Wigner quasiprobability distribution function can be regarded as an -deformation of another phase space distribution function that describes an ensemble of de Broglie–Bohm causal trajectories. Basil Hiley has shown that the quasi-probability distribution may be understood as the density matrix re-expressed in terms of a mean position and momentum of a "cell" in phase space, and the de Broglie–Bohm interpretation allows one to describe the dynamics of the centers of such "cells". |
There is a close connection between the description of quantum states in terms of the Wigner function and a method of quantum states reconstruction in terms of mutually unbiased bases. |
The Wigner distribution was the first quasiprobability distribution to be formulated, but many more followed, formally equivalent and transformable to and from it (viz. Transformation between distributions in time–frequency analysis). As in the case of coordinate systems, on account of varying properties, several such have with various advantages for specific applications: |
Nevertheless, in some sense, the Wigner distribution holds a privileged position among all these distributions, since it is the "only one" whose requisite star product drops out (integrates out by parts to effective unity) in the evaluation of expectation values, as illustrated above, and so "can" be visualized as a quasiprobability measure analogous to the classical ones. |
As indicated, the formula for the Wigner function was independently derived several times in different contexts. In fact, apparently, Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac, albeit purely formally: these two missed its significance, and that of its negative values, as they merely considered it as an approximation to the full quantum description of a system such as the atom. (Incidentally, Dirac would later become Wigner's brother-in-law, marrying his sister Manci.) Symmetrically, in most of his legendary 18-month correspondence with Moyal in the mid-1940s, Dirac was unaware that Moyal's quantum-moment generating function was effectively the Wigner function, and it was Moyal who finally brought it to his attention. |
In physics, two wave sources are coherent if their frequency and waveform are identical. Coherence is an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference. It contains several distinct concepts, which are limiting cases that never quite occur in reality but allow an understanding of the physics of waves, and has become a very important concept in quantum physics. More generally, coherence describes all properties of the correlation between physical quantities of a single wave, or between several waves or wave packets. |
Spatial coherence describes the correlation (or predictable relationship) between waves at different points in space, either lateral or longitudinal. Temporal coherence describes the correlation between waves observed at different moments in time. Both are observed in the Michelson–Morley experiment and Young's interference experiment. Once the fringes are obtained in the Michelson interferometer, when one of the mirrors is moved away gradually from the beam-splitter, the time for the beam to travel increases and the fringes become dull and finally disappear, showing temporal coherence. Similarly, in a double-slit experiment, if the space between the two slits is increased, the coherence dies gradually and finally the fringes disappear, showing spatial coherence. In both cases, the fringe amplitude slowly disappears, as the path difference increases past the coherence length. |
Coherence was originally conceived in connection with Thomas Young's double-slit experiment in optics but is now used in any field that involves waves, such as acoustics, electrical engineering, neuroscience, and quantum mechanics. Coherence describes the statistical similarity of a field (electromagnetic field, quantum wave packet etc.) at two points in space or time. The property of coherence is the basis for commercial applications such as holography, the Sagnac gyroscope, radio antenna arrays, optical coherence tomography and telescope interferometers (astronomical optical interferometers and radio telescopes). |
A precise definition is given in the article on degree of coherence. |
The coherence function between two signals formula_1 and formula_2 is defined as |
The coherence varies in the interval formula_11. If formula_12 it means that the signals are perfectly correlated or linearly related and if formula_13 they are totally uncorrelated. If a linear system is being measured, formula_1 being the input and formula_2 the output, the coherence function will be unitary all over the spectrum. However, if non-linearities are present in the system the coherence will vary in the limit given above. |
These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof. |
In most of these systems, one can measure the wave directly. Consequently, its correlation with another wave can simply be calculated. However, in optics one cannot measure the electric field directly as it oscillates much faster than any detector's time resolution. Instead, one measures the intensity of the light. Most of the concepts involving coherence which will be introduced below were developed in the field of optics and then used in other fields. Therefore, many of the standard measurements of coherence are indirect measurements, even in fields where the wave can be measured directly. |
Temporal coherence is the measure of the average correlation between the value of a wave and itself delayed by τ, at any pair of times. Temporal coherence tells us how monochromatic a source is. In other words, it characterizes how well a wave can interfere with itself at a different time. The delay over which the phase or amplitude wanders by a significant amount (and hence the correlation decreases by significant amount) is defined as the coherence time "τc". At a delay of τ=0 the degree of coherence is perfect, whereas it drops significantly as the delay passes "τ=τc". The coherence length "Lc" is defined as the distance the wave travels in time τc. |
One should be careful not to confuse the coherence time with the time duration of the signal, nor the coherence length with the coherence area (see below). |
The relationship between coherence time and bandwidth. |
It can be shown that the larger the range of frequencies Δf a wave contains, the faster the wave decorrelates (and hence the smaller τc is). Thus there is a tradeoff: |
Formally, this follows from the convolution theorem in mathematics, which relates the Fourier transform of the power spectrum (the intensity of each frequency) to its autocorrelation. |
We consider four examples of temporal coherence. |
The high degree of monochromaticity of lasers implies long coherence lengths (up to hundreds of meters). For example, a stabilized and monomode helium–neon laser can easily produce light with coherence lengths of 300 m. Not all lasers have a high degree of monochromaticity, however (e.g. for a mode-locked Ti-sapphire laser, Δλ ≈ 2 nm - 70 nm). LEDs are characterized by Δλ ≈ 50 nm, and tungsten filament lights exhibit Δλ ≈ 600 nm, so these sources have shorter coherence times than the most monochromatic lasers. |
Holography requires light with a long coherence time. In contrast, optical coherence tomography, in its classical version, uses light with a short coherence time. |
Holography requires temporally and spatially coherent light. Its inventor, Dennis Gabor, produced successful holograms more than ten years before lasers were invented. To produce coherent light he passed the monochromatic light from an emission line of a mercury-vapor lamp through a pinhole spatial filter. |
In February 2011 it was reported that helium atoms, cooled to near absolute zero / Bose–Einstein condensate state, can be made to flow and behave as a coherent beam as occurs in a laser. |
Waves of different frequencies (in light these are different colours) can interfere to form a pulse if they have a fixed relative phase-relationship (see Fourier transform). Conversely, if waves of different frequencies are not coherent, then, when combined, they create a wave that is continuous in time (e.g. white light or white noise). The temporal duration of the pulse formula_19 is limited by the spectral bandwidth of the light formula_20 according to: |
which follows from the properties of the Fourier transform and results in Küpfmüller's uncertainty principle (for quantum particles it also results in the Heisenberg uncertainty principle). |
If the phase depends linearly on the frequency (i.e. formula_22) then the pulse will have the minimum time duration for its bandwidth (a "transform-limited" pulse), otherwise it is chirped (see dispersion). |
Measurement of the spectral coherence of light requires a nonlinear optical interferometer, such as an intensity optical correlator, frequency-resolved optical gating (FROG), or spectral phase interferometry for direct electric-field reconstruction (SPIDER). |
Light also has a polarization, which is the direction in which the electric field oscillates. Unpolarized light is composed of incoherent light waves with random polarization angles. The electric field of the unpolarized light wanders in every direction and changes in phase over the coherence time of the two light waves. An absorbing polarizer rotated to any angle will always transmit half the incident intensity when averaged over time. |
If the electric field wanders by a smaller amount the light will be partially polarized so that at some angle, the polarizer will transmit more than half the intensity. If a wave is combined with an orthogonally polarized copy of itself delayed by less than the coherence time, partially polarized light is created. |
The polarization of a light beam is represented by a vector in the Poincaré sphere. For polarized light the end of the vector lies on the surface of the sphere, whereas the vector has zero length for unpolarized light. The vector for partially polarized light lies within the sphere |
Coherent superpositions of "optical wave fields" include holography. Holographic objects are used frequently in daily life in television and credit card security. |
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