Abstract
An analysis of the Wigner function for identical particles is presented. Four situations have been considered. (i) The first is scattering process between two indistinguishable particles described by a minimum uncertainty wave packets showing the exchange and correlation effects in Wigner phase space. (ii) An equilibrium ensemble of $N$ particles in a one-dimensional box and in a one-dimensional harmonic potential is considered second, showing that the reduced one-particle Wigner function, as a function of the energy defined in the Wigner phase space, tends to the Fermi-Dirac or to the Bose-Einstein distribution function, depending on the considered statistics. (iii) The third situation is reduced one-particle transport equation for the Wigner function, in the case of interacting particles, showing the need for the two-particle reduced Wigner function within the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy scheme. (iv) Finally, the electron-phonon interaction in the two-particle case is considered, showing coparticipation of two electrons in the interaction with the phonon bath.
Highlights
Sophisticated technologies produce physical systems, and in particular semiconductor devices, of very small dimensions, comparable with the electron wavelength or with the electron coherence length
Among the possible different approaches, the Wigner functionWFhas proved to be very useful for studying quantum electron transport,1–4 owing to its strong analogy with the semiclassical picture, since it explicitly refers to variables defined in anr, p Wigner phase space, together with a rigorous description of electron dynamics in quantum terms
Four situations will be analyzed: ͑ia scattering process between two indistinguishable particles described by minimum-uncertainty wave packets, showing the exchange and correlation effects in Wigner phase space; ͑iian equilibrium ensemble of N particles in a box and in a harmonic potential, showing that the value of the WF in points in the Wigner phase space with given energy tends to a FermiDiracFDor to a Bose-EinsteinBEdistribution function, depending on the considered statistics; ͑iiithe transport equation for interacting particles, showing the BogoliubovBorn-Green-Kirkwood-YvonBBGKYhierarchy when the integral, over the degrees of freedom of all the particles but one, is performed,5,6 ͑ivthe electron-phonon interaction in the case of two particles, where new Keldysh diagrams7 appear with respect to the one-electron case
Summary
Sophisticated technologies produce physical systems, and in particular semiconductor devices, of very small dimensions, comparable with the electron wavelength or with the electron coherence length. Under such conditions, semiclassical dynamics is not justified in principle, and interference effects due to the linear superpositions of quantum states have to be considered. Among the possible different approaches, the Wigner functionWFhas proved to be very useful for studying quantum electron transport, owing to its strong analogy with the semiclassical picture, since it explicitly refers to variables defined in anr , p Wigner phase space, together with a rigorous description of electron dynamics in quantum terms. Four situations will be analyzed: ͑ia scattering process between two indistinguishable particles described by minimum-uncertainty wave packets, showing the exchange and correlation effects in Wigner phase space; ͑iian equilibrium ensemble of N particles in a box and in a harmonic potential, showing that the value of the WF in points in the Wigner phase space with given energy tends to a FermiDiracFDor to a Bose-EinsteinBEdistribution function, depending on the considered statistics; ͑iiithe transport equation for interacting particles, showing the BogoliubovBorn-Green-Kirkwood-YvonBBGKYhierarchy when the integral, over the degrees of freedom of all the particles but one, is performed,5,6 ͑ivthe electron-phonon interaction in the case of two particles, where new Keldysh diagrams appear with respect to the one-electron case.
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