Quantum non‐locality is here put at variance with spatial contacts, as modelled within the Wigner‐Function formalism: we shall show how these two concepts are indeed intrinsically incompatible when applied to transport through quantum devices. More specifically, by means of an exactly‐solvable semiconductor model, we show that the application of the usual Up‐Wind boundary scheme to the Wigner transport equation may produce highly non‐physical results, like boundary‐driven negative probability distributions.
Quantum non‐locality is here put at variance with spatial contacts, as modelled within the Wigner‐Function formalism: we shall show how these two concepts are indeed intrinsically incompatible when applied to transport through quantum devices. More specifically, by means of an exactly‐solvable semiconductor model, we show that the application of the usual Up‐Wind boundary scheme to the Wigner transport equation may produce highly non‐physical results, like boundary‐driven negative probability distributions.
We shall revisit the conventional treatment of open quantum devices based on the Wigner-Function formalism. Our analysis will show that the artificial spatial separation between device active region and external reservoirs -properly defined within a semiclassical simulation scheme- is intrinsically incompatible with the non-local character of quantum mechanics. More specifically, by means of an exactly-solvable semiconductor model, we shall show that the application of the conventional boundary-condition scheme to the Wigner transport equation may produce highly non-physical results, like thermal injection of coherent state superpositions and boundary-driven negative probability distributions.
Recently, quantum models of carrier transport in semiconductors have been proposed on the basis of the Wigner formulation of quantum mechanics. Such models are demanded by the semiconductor industry due to the fast transition from microelectronics to nanoelectronics. However, the Wigner function cannot be interpreted as a quantum analog of a distribution function, since it is not a probability density. In this work we show how this difficulty could be overcome by introducing a non standard quantum theory, based on the concept of Positive-Operator-Valued (POV) measures [3–5,7].Such a theory is demanded by the semiconductor industry due to the fast transition from microelectronics to nanoelectronics. The Wigner function cannot be strictly interpreted as a quantum analog of a distribution function, because it is not a probability density. In this communication we show that this difficulty can be overcome by introducing a non standard quantum theory, based on the concept of Positive-Operator-Valued (POV) measures rather than on Projector-Valued (PV) measures. The introduction of this more general class of mathematical objects accounts for the stochastic effect of measure errors in the effective evaluation of observables described by PV measures [1, 2]. Following Prugovecki [3], we show that, on an appropriately defined stochastic phase space, it is possible to introduce a “quantum distribution function” whose marginals give the desired macroscopic quantities, and which does not present the disadvantages of the Wigner function. We present the resulting quantum Boltzmann equations for an ensemble of free carriers, and discuss extensions to the case of carriers in a semiconductor.
Quantum mechanics in phase space (or deformation quantization) appears to fail as an autonomous quantum method when infinite potential walls are present. The stationary physical Wigner functions do not satisfy the normal eigen equations, the *-eigen equations, unless an ad hoc boundary potential is added [Dias-Prata]. Alternatively, they satisfy a different, higher-order, ``*-eigen-* equation'', locally, i.e. away from the walls [Kryukov-Walton]. Here we show that this substitute equation can be written in a very simple form, even in the presence of an additional, arbitrary, but regular potential. The more general applicability of the -eigen- equation is then demonstrated. First, using an idea from [Fairlie-Manogue], we extend it to a dynamical equation describing time evolution. We then show that also for general contact interactions, the -eigen- equation is satisfied locally. Specifically, we treat the most general possible (Robin) boundary conditions at an infinite wall, general one-dimensional point interactions, and a finite potential jump. Finally, we examine a smooth potential, that has simple but different expressions for x positive and negative. We find that the -eigen- equation is again satisfied locally. It seems, therefore, that the -eigen- equation is generally relevant to the matching of Wigner functions; it can be solved piece-wise and its solutions then matched.
The Wigner function of quantum systems is an effective instrument for constructing an approximate classical description of the systems for which the classical approximation is possible. In recent years, the Wigner function formalism has been applied as well to seek indications of specific quantum properties of quantum systems leading to the impossibility of constructing the classical approximation. Used most of all as such an indication is the existence of negative values in the Wigner function for specific states of the quantum system being studied. The existence of such values itself prejudices the probabilistic interpretation of the Wigner function, though for an arbitrary observable depending jointly on the coordinate and the momentum of the quantum system just the Wigner function provides an effective instrument for calculating the average value and the other statistical characteristics. In this paper, a probabilistic interpretation of the Wigner function based on coordinating a theoretical probabilistic definition of the probability density, with restrictions to a physically small domain of phase space due to the uncertainty principle, is proposed.
The use of the Wigner function for the study of quantum transport in open systems present severe criticisms. Some of the problems arise from the assumption of infinite coherence length of the electron dynamics outside the system of interest. In the present work the theory of the Wigner function is revised assuming a finite coherence length. A new dynamical equation is found, corresponding to move the Wigner momentum off the real axis, and a numerical analysis is performed for the case of study of the onedimensional potential barrier.