By Thomas L Curtright, David B Fairlie, Cosmas K Zachos

Wigner's quasi-probability distribution functionality in section house is a distinct (Weyl) illustration of the density matrix. it's been priceless in describing quantum delivery in quantum optics; nuclear physics; decoherence, quantum computing, and quantum chaos. it's also vital in sign processing and the maths of algebraic deformation. A extraordinary element of its inner good judgment, pioneered by way of Groenewold and Moyal, has in basic terms emerged within the final quarter-century: it furnishes a 3rd, substitute, formula of quantum mechanics, self sufficient of the traditional Hilbert area, or course necessary formulations.

In this logically whole and self-standing formula, one don't need to decide on aspects - coordinate or momentum area. it really works in complete part area, accommodating the uncertainty precept, and it deals certain insights into the classical restrict of quantum thought. This necessary e-book is a set of the seminal papers at the formula, with an introductory review which supplies a path map for these papers; an intensive bibliography; and easy illustrations, compatible for functions to a huge variety of physics difficulties. it could supply supplementary fabric for a starting graduate direction in quantum mechanics.

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**Additional info for Quantum Mechanics in Phase Space**

**Sample text**

Thus, if the system is characterized by a total of N possible modes of excitation, the associated phase space has 4N dimensions. The positive nature of P (α, α† ) means that whenever the corresponding evolution equation is of the Fokker-Planck type then an equivalent set of Ito stochastic differential equations can be written down which describe the motion of the coordinates α, α† of a fictitious particle whose movement in the phase space is governed by the stochastic equation. After both the reservoirs and the vibrational states have been eliminated from the problem, the residual equations describing the motion of the light field possess an extremely simple and familiar form, except with an enlarged phase-space.

This correction factor is necessary because the Wigner function represents symmetrically ordered operators, which have a diverging vacuum noise term as the cut-off is taken to infinity. Computer simulations[57] of these equations reveal that there is good agreement between this technique, and the earlier technique using the positive P-representation, at large photon number. Similar techniques can be worked out for the parametric case. 4. 4 59 Representing the reservoirs A strong practical limitation on many experiments originates from the refractiveindex fluctuations and Raman scattering in real media.

3) Here, the Raman excitations are treated as an inhomogeneously broadened continuum of modes, localized at each longitudinal location x. GAWBS (Guided Wave Acoustic Brillouin Scattering)[58] is a special case of this, in the low- frequency limit. Since neither Raman nor Brillouin excitations are completely localized, this treatment requires a frequency and wave-number ˆ is slowly varying on the phonon scattering cut- off, so that the field operator Ψ distance scale. The frequency dependent coupling r(x, ω) determines the Raman gain, which from now on is assumed to be uniform in space.