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Pan Stanford eBooks

Wigner Function

David K. Ferry

Published Dec 14, 2017• 0 citations
Publisher

Abstract

The Wigner function has been studied for more than eight decades. Historically, it was developed from the Schrodinger wave equation, although it could have been developed on its own right. The use of the Wigner function is particularly important in scattering problems and clearly shows the transition to the classical world. In general, the Wigner function is not a positive definite function. One of the interesting aspects of the Wigner function is the ability to transfer many of the results of classical transport theory into quantum approaches merely by replacing the Boltzmann distribution with the Wigner function. The Wigner function for either of the two oscillators can be found by integrating out the coordinates of the other oscillator. Modern approaches with the Wigner function provide a distinct formulation that is recognized as equivalent to, but is a different alternative to normal operator-based quantum mechanics.

DOI
10.1201/9781315206226-8

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The Wigner function has been studied for more than eight decades. Historically, it was developed from the Schrodinger wave equation, although it could have been developed on its own right. The use of the Wigner function is particularly important in scattering problems and clearly shows the transition to the classical world. In general, the Wigner function is not a positive definite function. One of the interesting aspects of the Wigner function is the ability to transfer many of the results of classical transport theory into quantum approaches merely by replacing the Boltzmann distribution with the Wigner function. The Wigner function for either of the two oscillators can be found by integrating out the coordinates of the other oscillator. Modern approaches with the Wigner function provide a distinct formulation that is recognized as equivalent to, but is a different alternative to normal operator-based quantum mechanics.
The Wigner function was formulated in 1932 by Eugene Paul Wigner, at a time when quantum mechanics was in its infancy. In doing so, he brought phase space representations into quantum mechanics. However, its unique nature also made it very interesting for classical approaches and for identifying the deviations from classical behavior and the entanglement that can occur in quantum systems. What stands out, though, is the feature to experimentally reconstruct the Wigner function, which provides far more information on the system than can be obtained by any other quantum approach. This feature is particularly important for the field of quantum information processing and quantum physics. However, the Wigner function finds wide-ranging use cases in other dominant and highly active fields as well, such as in quantum electronics—to model the electron transport, in quantum chemistry—to calculate the static and dynamical properties of many-body quantum systems, and in signal processing—to investigate waves passing through certain media. What is peculiar in recent years is a strong increase in applying it: Although originally formulated 86 years ago, only today the full potential of the Wigner function—both in ability and diversity—begins to surface. This review, as well as a growing, dedicated Wigner community, is a testament to this development and gives a broad and concise overview of recent advancements in different fields.
The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.
The tomographic invertable map of the Wigner function onto the positive probability distribution function is studied. Alternatives to the Schrödinger evolution equation and to the energy level equation written for the positive probability distribution are discussed. Instead of the transition probability amplitude (Feynman path integral) a transition probability is introduced. A new formulation of the conventional quantum mechanics (without wave function and density matrix) based on the “probability representation” of quantum states is given. An equation for the propagator in the new formulation of quantum mechanics is derived. Some paradoxes of quantum mechanics are reconsidered.
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.