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2 results

  • 3 - The Phase Space Formulation of Quantum Mechanics

  • Marcos Mariño, Université de Genève
  • Book:
    Advanced Topics in Quantum Mechanics
    Published online:
    02 December 2021
    Print publication:
    09 December 2021, pp 114-158

  • Summary

    This chapter presents Wigner’s approach to quantum mechanics, based on the Wigner function in phase space. It explains Wigner–Weyl quantization, which makes it possible to associate functions on phase space to wave functions and operators, and it develops the technology to do quantum mechanics in this formalism. This includes the star product, Moyal evolution,and star-eigenvalue equations. It also develops semiclassical methods in this formulation, and it has a section on Berry’s semiclassical formula for the Wigner function in one-dimensional systems.

  • 12 - Spectrogram Geometry 2

  • from Part II - Geometry and Statistics
  • Patrick Flandrin
  • Book:
    Explorations in Time-Frequency Analysis
    Published online:
    22 August 2018
    Print publication:
    06 September 2018, pp 106-115

  • Summary

    Spectrograms are revisited from a Bargmann transform point of view, with the time-frequency plane identified to the complex plane. This permits to establish simple phase-magnitude relationships for the Gaussian STFT and to describe reassignment via a vector field which happens to be the gradient of the associated (log-)spectrogram. This also paves the way to variations such as differential or adjustable reassignment. Within this picture, the whole reassignment process can be described in terms of attractors (maxima), repellers (zeros), and basins of attraction (component domains).