The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle $\ensuremath{\theta}$ and angular momentum $p$ is solved. While a key element for the construction of Wigner functions for the planar phase space ${(q,p)\ensuremath{\in}{\mathbb{R}}^{2}}$ is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space ${(\ensuremath{\theta},p)\ensuremath{\in}{S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}}$ is the Euclidean group $E(2)$ of the plane and its unitary representations. Here the angle $\ensuremath{\theta}$ is replaced by the pair $(cos\ensuremath{\theta},sin\ensuremath{\theta})$, which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role is played by the $sinc$ function, which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function well known from interpolation and sampling theories. The quantum mechanical marginal distributions for the angle (continuous) and angular momentum (discontinuous) are, as usual, uniquely obtained by appropriate integrations of the $(\ensuremath{\theta},p)$ Wigner function. Among the examples discussed is an elementary system of simple cat states.The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle $\ensuremath{\theta}$ and angular momentum $p$ is solved. While a key element for the construction of Wigner functions for the planar phase space ${(q,p)\ensuremath{\in}{\mathbb{R}}^{2}}$ is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space ${(\ensuremath{\theta},p)\ensuremath{\in}{S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}}$ is the Euclidean group $E(2)$ of the plane and its unitary representations. Here the angle $\ensuremath{\theta}$ is replaced by the pair $(cos\ensuremath{\theta},sin\ensuremath{\theta})$, which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role is played by the $sinc$ function, which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function well known from interpolation and sampling theories. The quantum mechanical marginal distributions for the angle (continuous) and angular momentum (discontinuous) are, as usual, uniquely obtained by appropriate integrations of the $(\ensuremath{\theta},p)$ Wigner function. Among the examples discussed is an elementary system of simple cat states.

arXiv:1601.02520Physical Review ADec 15, 201629 citations

Linear Perturbations of the Wigner Transform and the Weyl Quantization

Dominik Bayer, Elena Cordero, Karlheinz Gröchenig +1 more

arXiv:1906.02503Applied and numerical harmonic analysisJan 1, 202026 citations

Wigner functions, signed particles, and the harmonic oscillator

Jean Michel Sellier, Ivan Dimov

Journal of Computational ElectronicsJul 16, 20154 citations

Analytic representations of quantum systems with Theta functions

Pavlos Evangelides

Quantum systems in a d-dimensional Hilbert space are considered, where \nthe phase spase is Z(d) x Z(d). An analytic representation in a cell S in the \ncomplex plane using Theta functions, is defined. The analytic functions have \nexactly d zeros in a cell S. The reproducing kernel plays a central role in \nthis formalism. Wigner and Weyl functions are also studied. \nQuantum systems with positions in a circle S and momenta in Z are also \nstudied. An analytic representation in a strip A in the complex plane is also \ndefined. Coherent states on a circle are studied. The reproducing kernel is \ngiven. Wigner and Weyl functions are considered.Quantum systems in a d-dimensional Hilbert space are considered, where \nthe phase spase is Z(d) x Z(d). An analytic representation in a cell S in the \ncomplex plane using Theta functions, is defined. The analytic functions have \nexactly d zeros in a cell S. The reproducing kernel plays a central role in \nthis formalism. Wigner and Weyl functions are also studied. \nQuantum systems with positions in a circle S and momenta in Z are also \nstudied. An analytic representation in a strip A in the complex plane is also \ndefined. Coherent states on a circle are studied. The reproducing kernel is \ngiven. Wigner and Weyl functions are considered.

Jan 1, 20150 citations

Weyl quantization and Wigner distributions on phase space

R. Wezeman

This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.

Jan 1, 20142 citations

From quantization in phase space to functions of quantum numbers

Konstantin V. Kazakov

Physics Letters AAug 18, 20260 citations

The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle $\ensuremath{\theta}$ and angular momentum

p

is solved. While a key element for the construction of Wigner functions for the planar phase space ${(q,p)\ensuremath{\in}{\mathbb{R}}^{2}}$ is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space ${(\ensuremath{\theta},p)\ensuremath{\in}{S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}}$ is the Euclidean group

E(2)

of the plane and its unitary representations. Here the angle $\ensuremath{\theta}$ is replaced by the pair $(cos\ensuremath{\theta},sin\ensuremath{\theta})$, which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role is played by the

sinc

function, which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function well known from interpolation and sampling theories. The quantum mechanical marginal distributions for the angle (continuous) and angular momentum (discontinuous) are, as usual, uniquely obtained by appropriate integrations of the $(\ensuremath{\theta},p)$ Wigner function. Among the examples discussed is an elementary system of simple cat states.