Wigner function

Wigner functions of Fock states with 0 (a), 1 (b) and 5 (c) photons .

The Wigner function (Wigner quasi- probability distribution ) was introduced by Eugene Wigner in 1932 in order to investigate quantum corrections in classical statistical mechanics . The aim was to replace the wave function of the Schrödinger equation with a probability distribution in phase space . Such a distribution was independently found in 1931 by Hermann Weyl as a density matrix in representation theory . It was discovered again by J. Ville in 1948 as a quadratic (as a function of the signal) representation of the local time-frequency energy of a signal. This distribution is also known under the names “Wigner function”, “Wigner-Weyl transformation” or “Wigner-Ville distribution”. It is used in statistical mechanics, quantum chemistry , quantum optics , classical optics and signal analysis , as well as in a number of areas of electrical engineering , seismology , biology and engine design.

A classical particle has a defined position and a defined momentum and can therefore be represented by a point in phase space. A probability distribution can be defined for an ensemble of particles, which indicates the probability with which a particle is located at a certain location in phase space. However, this is not possible for a quantum particle, which must satisfy the uncertainty principle . Instead, a quasi-probability distribution can be defined, which does not necessarily have all the properties of an ordinary probability distribution. The Wigner distribution can, for example, assume negative values ​​for non-classical states and can therefore be used to identify such states.

The Wigner distribution is defined as: ${\displaystyle P(x,p)}$

${\displaystyle P(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }dy\,\psi ^{*}(x+y)\psi (x-y)e^{2ipy/\hbar }}$

with the wave function and the location and momentum . The latter, however, can also be any pair of conjugate variables (e.g. real and imaginary parts of the electric field or frequency and duration of a signal). The distribution is symmetrical in and : ${\displaystyle \psi }$ ${\displaystyle x}$ ${\displaystyle p}$ ${\displaystyle x}$ ${\displaystyle p}$

${\displaystyle P(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }dq\,\phi ^{*}(p+q)\phi (p-q)e^{-2ixq/\hbar }}$

where is the Fourier transform of . ${\displaystyle \phi }$ ${\displaystyle \psi }$

For a mixed state:

${\displaystyle P(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }dy\,\langle x-y|{\hat {\rho }}|x+y\rangle e^{2ipy/\hbar }}$

where denotes the density matrix. ${\displaystyle {\hat {\rho }}}$

Mathematical properties

1. is real. ${\displaystyle P(x,p)}$

2. The probability distributions of and result from: ${\displaystyle x}$ ${\displaystyle p}$

• ${\displaystyle \int _{-\infty }^{\infty }dp\,P(x,p)=\langle x|{\hat {\rho }}|x\rangle }$. If the system can be described by a pure state , it follows . ${\displaystyle \int _{-\infty }^{\infty }dp\,P(x,p)=|\psi (x)|^{2}}$
• ${\displaystyle \int _{-\infty }^{\infty }dx\,P(x,p)=\langle p|{\hat {\rho }}|p\rangle }$. If the system can be described by a pure state, it follows . ${\displaystyle \int _{-\infty }^{\infty }dx\,P(x,p)=|\phi (p)|^{2}}$
• ${\displaystyle \int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,P(x,p)=\operatorname {Spur} ({\hat {\rho }})}$.
• The trace of is usually equal to 1. ${\displaystyle {\hat {\rho }}}$
• It follows from 1. and 2. that if it is not a coherent state (or a mixture of coherent states) or not a squeezed vacuum state , it is negative in some places . ${\displaystyle P(x,p)}$

3. has the following symmetries: ${\displaystyle P(x,p)}$

• Time reversal: . ${\displaystyle \psi (x)\rightarrow \psi (x)^{*}\Rightarrow P(x,p)\rightarrow P(x,-p)}$
• Space reflection: . ${\displaystyle \psi (x)\rightarrow \psi (-x)\Rightarrow P(x,p)\rightarrow P(-x,-p)}$

4. is Galileo invariant : ${\displaystyle P(x,p)}$

• ${\displaystyle \psi (x)\rightarrow \psi (x+y)\Rightarrow P(x,p)\rightarrow P(x+y,p)}$.
• It is not invariant under the Lorentz transformation .

5. The equation of motion of a point in phase space is classic without forces:

• ${\displaystyle {\frac {\partial P(x,p)}{\partial t}}={\frac {-p}{m}}{\frac {\partial P(x,p)}{\partial x}}}$.

6. The overlap of two states is calculated as:

• ${\displaystyle |\langle \psi |\theta \rangle |^{2}=2\pi \hbar \int _{-\infty }^{\infty }dx\,\int _{-\infty }^{\infty }dp\,P_{\psi }(x,p)P_{\theta }(x,p)}$.

7. Operators and expected values ​​(mean values) are calculated as follows:

• ${\displaystyle A(x,p)=\int _{-\infty }^{\infty }dy\,\langle x-y/2|{\hat {A}}|x+y/2\rangle e^{ipy/\hbar }}$.
• ${\displaystyle \langle \psi |{\hat {A}}|\psi \rangle =Tr({\hat {\rho }}{\hat {A}})=\int _{-\infty }^{\infty }dx\,\int _{-\infty }^{\infty }dpP(x,p)A(x,p)}$.

8. In order for physical (positive) density matrices to describe, the following must apply: ${\displaystyle P(x,p)}$

• ${\displaystyle \int _{-\infty }^{\infty }dx\,\int _{-\infty }^{\infty }dp\,P(x,p)P_{\theta }(x,p)\geq 0}$

where is a pure state. ${\displaystyle |\theta \rangle }$

Application outside of quantum mechanics

• In the modeling of optical systems such as telescopes or glass fibers in telecommunication devices, the Wigner function fills the gap between simple ray tracing and the complete wave analysis of the system. In the approximation, small angles (paraxial) are replaced by. In this context, the Wigner function is the best approximation for a description of the system with the help of rays with the location and angle including interference effects. If this assumes negative values ​​at any point, the system cannot be described with the simple ray tracing method. ${\displaystyle p/\hbar }$ ${\displaystyle k=|k|\sin \theta \approx |k|\theta }$ ${\displaystyle x}$ ${\displaystyle \theta }$
• In signal analysis, a time-dependent electrical signal, mechanical vibrations or sound waves are represented by the Wigner function. It is replaced by the time and the angular frequency . Here denotes the usual frequency. ${\displaystyle x}$ ${\displaystyle p/\hbar }$ ${\displaystyle \omega =2\pi f}$ ${\displaystyle f}$
• In the field of ultrafast optics, short laser pulses are characterized by the Wigner function using the same substitution of frequency and time. Certain pulse properties such as a chirp (change in frequency over time) can be represented by the Wigner function.
• In quantum optics, and are replaced by and quadratures, which denote the real and imaginary part of the electric field (see coherent state ). ${\displaystyle x}$ ${\displaystyle p/\hbar }$ ${\displaystyle X}$ ${\displaystyle P}$

Measurement

• tomography
• Homodyne detection
• FROG Frequency-resolved optical gating

Further quasi-probability distributions

The Wigner distribution was the first quasi-probability distribution, but many more followed with various advantages, including:

• Glauber P representation ,
• Husimi Q representation .

Historical note

As stated in the introduction, the Wigner function has been found many times independently in different contexts. In fact, it seems that Wigner did not know that this function, even within quantum theory, had previously been introduced by Heisenberg and Dirac . However, they did not see its meaning and believed that this function was only an approximation of the exact quantum mechanical description of the system. Incidentally, Dirac later became Wigner's brother-in-law (see literature).

literature

• EP Wigner: On the quantum correction for thermodynamic equilibrium . In: Phys. Rev. Band 40 , June 1932, p. 749-759 .
• H. Weyl: Quantum Mechanics and Group Theory . In: Z. Phys. tape 46 , June 1927, p. 1-46 .

• H. Weyl: Group theory and quantum mechanics . S. Hirzel, Leipzig 1928.

• H. Weyl: The Theory of Groups and Quantum Mechanics . Dover, New York 1931.
• J. Ville: Theory et Applications de la Notion de Signal Analytique . In: Cables et Transmission . tape 2 , no. 1 , 1948, p. 61-74 .
• W. Heisenberg : About the incoherent scattering of X-rays . In: Physics. Magazine tape 32 , 1931, pp. 737-740 .

• PAM Dirac : Note on exchange phenomena in the Thomas atom . In: Proc. Camb. Phil. Soc. tape 26 , 1930, pp. 376-395 .

• C. Zachos, D. Fairlie, T. Curtright: Quantum Mechanics in Phase Space . World Scientific, Singapore 2005, pp. 737-740 .

Web links

• A gallery of Wigner functions at the Department of Physics and Astronomy, University of Calgary
• A Gallery of Quantum States
• Wigner distribution