Wigner quasiprobability distribution
The Wigner function ( Wigner quasi- probability distribution ) was introduced in 1932 by Eugene Wigner to investigate quantum corrections to the classical statistical mechanics. The objective was to replace the wave function of the Schrödinger equation by a probability distribution in the phase space. Such a distribution was found in 1931 regardless of Hermann Weyl as the density matrix in the representation theory. Another time she was discovered by J. Ville in 1948 as a square (as a function of the signal) representation of the local time-frequency energy of a signal. This distribution is also known under the name " Wigner function " known " Wigner -Weyl transformation" or " Wigner- Ville distribution ". It is used in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in a number of fields such as electrical engineering, seismology, biology and engine design.
A classic particles has a defined location and momentum and therefore can be represented by a point in phase space. For an ensemble of particles can define a probability distribution that represents the probability that a particle is in a specific place in the phase space. However, this is not possible for a quantum particle, which must satisfy the uncertainty relation. Instead can define a quasi- probability distribution, which necessarily does not have all the properties of a normal probability distribution. The Wignerverteilung, for example, for non-classical states assume negative values and can therefore be used to identify those states.
Wignerverteilung the P (x, p) is defined as:
ψ the wave function and the location, x, and momentum, p. However, the latter could also use any pair of conjugate variables (eg real and imaginary parts of the electric field or the frequency and duration of a signal ). The distribution is symmetric in x and p:
Where Φ is the Fourier transform of Ψ.
For a mixed state:
Where ρ denotes the density matrix.
Mathematical properties
1 P ( x, p ) is real
2 The probability distributions of x and p are due to:
- . If you can describe by a pure state of the system follows.
- . If you can describe by a pure state of the system follows
- The trace of ρ is usually equal to 1,
- From 1 and 2 it follows that P ( x, p) in some places is negative, if it is not a coherent state (or a mixture of coherent states ) or a squeezed vacuum state is.
3 P (x, p) has the following symmetries:
- Time reversal:
- Space reflection:
4 P ( x, p) is Galilean - invariant:
- It is not invariant under the Lorentz transformation.
5 The equation of motion of a point in phase space is classic without forces:
6 The overlap of two states is calculated as:
7 Operators and expectation values (mean values) are calculated as follows:
8 Thus, P ( x, p) physical ( positive ) density matrices describing must apply:
Where | θ > is a pure state.
Application of the Wigner function outside of quantum mechanics
- In the modeling of optical systems such as telescopes or glass fibers in devices for telecommunications, the Wigner function fills the gap between the simple ray tracing and full wave analysis of the system. K | | sin? ≈ | k | Here in the approximation of small angle ( paraxial ) by k = θ is replaced. In this context, the Wigner is the best approximation to a description of the system with the aid of radiation with the position x and angle θ, including interference effects. If this takes negative values at any point, the system can not be described by a simple ray-tracing method.
- In the signal analysis, a time-dependent electrical signal, mechanical vibrations or sound waves is shown by the Wigner. It will be replaced = 2? F x by time and by the angular frequency ω. Here, f denotes the ordinary frequency.
- In the field of ultra-fast optical laser pulses are characterized by the substitution of the same by means of the Wigner frequency and time. Certain features such as a pulse chirp ( change in frequency with time) can be represented by the Wigner function.
- In the x and quantum optics is replaced by X and P quadratures which denote the real and imaginary parts of the electric field (see coherent state ).
Measurement of the Wigner function
- Tomography
- Homodyne detection
- Frequency -resolved optical gating FROG
Other quasi- probability distributions
The Wignerverteilung was the first quasi- probability distribution but many more with various advantages followed, including:
- Glauber P representation
- Husimi Q representation
Historical Note
As stated in the introduction, the Wigner function many times was found independently in different contexts. In fact, it seems that Wigner did not know that this feature was previously introduced even within the quantum theory of Heisenberg and Dirac. However, these did not see their importance, and believed that this feature was only an approximation to the exact quantum- mechanical description of the system. Incidentally, Dirac later became the brother in law of Wigner (see literature).