2016/01/31 by H.A. Kastrup, H. A. Kastrup · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Angular momentum #Angular momentum coupling #Angular momentum operator #Azimuthal quantum number #Classical mechanics #Euclidean space #Interpolation (computer graphics) #Mathematical analysis #Mathematical functions and polynomials #Mathematical physics #Mathematics #Orbital Angular Momentum in Optics #Phase space #Physics #Quantum #Quantum mechanics #Sinc function #Total angular momentum quantum number #Unit circle #Wigner distribution function #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1103/physreva.94.062113
published as Physical Review A 94, 062113 (2016) · 15 pages, 3 figures. Considerably expanded update of the previous Letter version, including discussions of the marginal distributions and of several examples. Accepted for publication in Phys. Rev. A
arxiv created 2016/12/13 · openalex publication_date 2016/12/15 · arxiv updated 2016/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle \ensuremathθ and angular momentum p is solved. While a key element for the construction of Wigner functions for the planar phase space (q,p)\ensuremath∈ℝ2 is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space (\ensuremathθ,p)\ensuremath∈S1\ifmmode×\else\texttimes\fiℝ is the Euclidean group E(2) of the plane and its unitary representations. Here the angle \ensuremathθ is replaced by the pair (cos\ensuremathθ,sin\ensuremathθ), 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 (\ensuremathθ,p) Wigner function. Among the examples discussed is an elementary system of simple cat states.