Wigner Function

The Wigner function ( the Wigner quasi-probability distribution function , the Wigner distribution , the Weyl distribution ) was introduced by Wigner in 1932 to study quantum corrections to classical statistical mechanics . The goal was to replace the wave function that appears in the Schrödinger equation with a probability distribution function in phase space . It was independently derived by Weil in 1931 as a symbol of the density matrix of representation theory in mathematics . The Wigner function is used in statistical mechanics, quantum chemistry , quantum optics , classical optics and signal analysis in various fields, such as electronics , seismology , acoustics , biology . When analyzing signals, the names are the Wigner – Villa transform and the Wigner – Villa distribution .

Content

Physical sense

A classical particle has a certain position and momentum and therefore is represented by a point in phase space . When there is a set ( ensemble ) of particles, the probability of finding a particle in a certain small volume of phase space is given by the probability distribution function. This is not true for a quantum particle due to the uncertainty principle . Instead, a quasi-probability distribution can be introduced that does not have to satisfy all the properties of the normal probability distribution function. For example, the Wigner function becomes negative for states that do not have classical analogues; therefore, it can be used to identify non-classical states.

The Wigner distribution P ( x , p ) is defined as:

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

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

Where $\psi$ ${\ displaystyle \ psi}$ Is the wave function, and $x$ ${\ displaystyle x}$ and $p$ ${\ displaystyle p}$ - a set of conjugate generalized coordinates and momenta . It is symmetrical in $x$ ${\ displaystyle x}$ and $p$ ${\ displaystyle p}$ :

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

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

Where $\phi$ ${\ displaystyle \ phi}$ - Fourier transform function $\psi$ ${\ displaystyle \ psi}$ .

In the case of a mixed state :

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

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

Where $\rho$ ${\ displaystyle \ rho}$ - density matrix .

Mathematical Properties

P ( x , p ) is a real function
The probability distributions over x and p are given by the integrals :
- $\int \limits _{-\infty }^{\infty }dp\,P(x,p)=|\psi (x)|^{2}=\langle x|{\hat {\rho }}|x\rangle$ ${\ displaystyle \ int \ limits _ {- \ infty} ^ {\ infty} dp \, P (x, p) = | \ psi (x) | ^ {2} = \ langle x | {\ hat {\ rho }} | x \ rangle}$
- $\int \limits _{-\infty }^{\infty }dx\,P(x,p)=|\phi (p)|^{2}=\langle p|{\hat {\rho }}|p\rangle$ ${\ displaystyle \ int \ limits _ {- \ infty} ^ {\ infty} dx \, P (x, p) = | \ phi (p) | ^ {2} = \ langle p | {\ hat {\ rho }} | p \ rangle}$
- $\int \limits _{-\infty }^{\infty }dx\int \limits _{-\infty }^{\infty }dp\,P(x,p)=Tr({\hat {\rho }})$ ${\ displaystyle \ int \ limits _ {- \ infty} ^ {\ infty} dx \ int \ limits _ {- \ infty} ^ {\ infty} dp \, P (x, p) = Tr ({\ hat { \ rho}})}$
- Usually trace $\rho$ ${\ displaystyle \ rho}$ equal to 1.
- 1. and 2. suggests that P ( x , p ) is negative elsewhere, with the exception of the coherent state (and mixed coherent states) and squeezed vacuum states .
P ( x , p ) has the following mirror symmetries :
- Temporary Symmetry:

\psi (x)\rightarrow \psi (x)^{*}\Rightarrow P(x,p)\rightarrow P(x,-p)

{\ displaystyle \ psi (x) \ rightarrow \ psi (x) ^ {*} \ Rightarrow P (x, p) \ rightarrow P (x, -p)}

- Spatial symmetry:

\psi (x)\rightarrow \psi (-x)\Rightarrow P(x,p)\rightarrow P(-x,-p)

{\ displaystyle \ psi (x) \ rightarrow \ psi (-x) \ Rightarrow P (x, p) \ rightarrow P (-x, -p)}

P ( x , p ) is an invariant with respect to Galilean transformations :
- $\psi (x)\rightarrow \psi (x+y)\Rightarrow P(x,p)\rightarrow P(x+y,p)$ ${\ displaystyle \ psi (x) \ rightarrow \ psi (x + y) \ Rightarrow P (x, p) \ rightarrow P (x + y, p)}$
- It is not invariant with respect to Lorentz transformations .
The equations of motion for each point in the phase space are classical in the absence of forces :
${\frac {\partial P(x,p)}{\partial t}}={\frac {-p}{m}}{\frac {\partial P(x,p)}{\partial x}}$ ${\ displaystyle {\ frac {\ partial P (x, p)} {\ partial t}} = {\ frac {-p} {m}} {\ frac {\ partial P (x, p)} {\ partial x}}}$
State overlap is calculated as:
$|\langle \psi |\theta \rangle |^{2}=2\pi \hbar \int \limits _{-\infty }^{\infty }dx\,\int \limits _{-\infty }^{\infty }dp\,P_{\psi }(x,p)P_{\theta }(x,p)$ ${\ displaystyle | \ langle \ psi | \ theta \ rangle | ^ {2} = 2 \ pi \ hbar \ int \ limits _ {- \ infty} ^ {\ infty} dx \, \ int \ limits _ {- \ infty} ^ {\ infty} dp \, P _ {\ psi} (x, p) P _ {\ theta} (x, p)}$
Operators and averages are calculated as:
- $A(x,p)=\int \limits _{-\infty }^{\infty }dy\,\langle x-y/2|{\hat {A}}|x+y/2\rangle e^{ipy/\hbar }$ ${\ displaystyle A (x, p) = \ int \ limits _ {- \ infty} ^ {\ infty} dy \, \ langle xy / 2 | {\ hat {A}} | x + y / 2 \ rangle e ^ {ipy / \ hbar}}$
- $\langle \psi |{\hat {A}}|\psi \rangle =Tr({\hat {\rho }}{\hat {A}})=\int \limits _{-\infty }^{\infty }dx\,\int \limits _{-\infty }^{\infty }dpP(x,p)A(x,p)$ ${\ displaystyle \ langle \ psi | {\ hat {A}} | \ psi \ rangle = Tr ({\ hat {\ rho}} {\ hat {A}}) = \ int \ limits _ {- \ infty} ^ {\ infty} dx \, \ int \ limits _ {- \ infty} ^ {\ infty} dpP (x, p) A (x, p)}$
In order for P ( x , p ) to represent physical density matrices, it is necessary:
$\int \limits _{-\infty }^{\infty }dx\,\int \limits _{-\infty }^{\infty }dp\,P(x,p)P_{\theta }(x,p)\geq 0$ ${\ displaystyle \ int \ limits _ {- \ infty} ^ {\ infty} dx \, \ int \ limits _ {- \ infty} ^ {\ infty} dp \, P (x, p) P _ {\ theta} (x, p) \ geq 0}$ where $|\theta \rangle$ ${\ displaystyle | \ theta \ rangle}$ - pure condition .

Wigner function measurement

Tomography

Literature

EP Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (June 1932) 749-759.
H. Weyl, Z. Phys. 46, 1 (1927).
H. Weyl, Gruppentheorie und Quantenmechanik (Leipzig: Hirzel) (1928).
H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931).
J. Ville, Théorie et Applications de la Notion de Signal Analytique, Cables et Transmission, 2A: (1948) 61-74.
W. Heisenberg, Über die inkohärente Streuung von Röntgenstrahlen, Physik. Zeitschr. 32, 737-740 (1931).
PAM Dirac, Note on exchange phenomena in the Thomas atom, Proc. Camb. Phil. Soc. 26, 376-395 (1930).
C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space (World Scientific, Singapore, 2005).

Links

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