It is pointed out that the Wigner function $f(r, p)$ is $\frac{2}{h}$ times the expectation value of the parity operator that performs reflections about the phase-space point $r$ , $p$ . Thus $f(r, p)$ is proportional to the overlap of the wave function $\ensuremath{\psi}$ with its mirror image about $r$ , $p$ ; this is clearly a measure of how much $\ensuremath{\psi}$ is centered about $r$ , $p$ , and the Wigner distribution function now appears physically more meaningful and natural than it did previously.It is pointed out that the Wigner function $f(r, p)$ is $\frac{2}{h}$ times the expectation value of the parity operator that performs reflections about the phase-space point $r$, $p$. Thus $f(r, p)$ is proportional to the overlap of the wave function $\ensuremath{\psi}$ with its mirror image about $r$, $p$; this is clearly a measure of how much $\ensuremath{\psi}$ is centered about $r$, $p$, and the Wigner distribution function now appears physically more meaningful and natural than it did previously.

Physical Review AFeb 1, 1977372 citations

The Wigner function in the relativistic quantum mechanics

K. Kowalski, Jakub Rembieliński

arXiv:1903.08975Annals of PhysicsOct 8, 20169 citations

Relativistic Wigner operator and its distribution

A. D. Jannussis, Antony Streklas, D. Sourlas +1 more

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoMar 1, 19775 citations

Commuting periodic operators and the periodic Wigner function

J. Žák

Commuting periodic operators (CPO) depending on the coordinate and the momentum operators are defined. The CPO are functions of the two basic commuting operators and , with a being an arbitrary constant. A periodic Wigner function (PWF) w ( x , p ) is defined and it is shown that it is applicable in a normal expectation value calculation to the CPO, as done in the original Wigner paper. Moreover, this PWF is non-negative everywhere, and it can therefore be interpreted as an actual probability distribution. The PWF w ( x , p ) is shown to be given as an expectation value of the periodic Dirac delta function in the phase plane.Commuting periodic operators (CPO) depending on the coordinate and the momentum operators are defined. The CPO are functions of the two basic commuting operators and , with a being an arbitrary constant. A periodic Wigner function (PWF) w ( x , p ) is defined and it is shown that it is applicable in a normal expectation value calculation to the CPO, as done in the original Wigner paper. Moreover, this PWF is non-negative everywhere, and it can therefore be interpreted as an actual probability distribution. The PWF w ( x , p ) is shown to be given as an expectation value of the periodic Dirac delta function in the phase plane.

Journal of Physics A Mathematical and GeneralNov 4, 20043 citations

Wigner Distribution Functions in Relativistic Quantum Mechanics

J. H. Irving

Jan 1, 19650 citations

Generalized parity and quasi-probability density functions

M. G. Benedict, Attila Czirják

By shifting the parity operator in phase space, one obtains a class of operators, which we call the Wigner operator, because its expectation value is equal to the Wigner function. Calculating the commutator of Wigner operators with different arguments, we show that the corresponding observables cannot be measured simultaneously. Introducing a new parameter, we give a trace-class generalization of this operator. The displaced number states of the harmonic oscillator constitute the natural eigenstate basis of the corresponding generalized Wigner operator. Establishing the integral representation of this operator we show that its expectation value is proportional to the Q- and P- and Wigner functions at special values of the parameter. We illustrate this connection with the coherent, number and squeezed states of the harmonic oscillator.By shifting the parity operator in phase space, one obtains a class of operators, which we call the Wigner operator, because its expectation value is equal to the Wigner function. Calculating the commutator of Wigner operators with different arguments, we show that the corresponding observables cannot be measured simultaneously. Introducing a new parameter, we give a trace-class generalization of this operator. The displaced number states of the harmonic oscillator constitute the natural eigenstate basis of the corresponding generalized Wigner operator. Establishing the integral representation of this operator we show that its expectation value is proportional to the Q- and P- and Wigner functions at special values of the parameter. We illustrate this connection with the coherent, number and squeezed states of the harmonic oscillator.

Journal of Physics A Mathematical and GeneralAug 21, 19959 citations

It is pointed out that the Wigner function

f(r, p)

\frac{2}{h}

times the expectation value of the parity operator that performs reflections about the phase-space point

r

p

. Thus

f(r, p)

is proportional to the overlap of the wave function $\ensuremath{\psi}$ with its mirror image about

r

p

; this is clearly a measure of how much $\ensuremath{\psi}$ is centered about

r

p

, and the Wigner distribution function now appears physically more meaningful and natural than it did previously.