Vlasov’s equation for plasmas can be derived from a model of quantum N-electron dynamics: one considers ”jellium” electrons with a truncated Coulomb interaction and derives the classical Vlasov equation in the thermodynamic limit [1]. Recent work indicates that the time-dependent Hartree equation (a weakly nonlinear Schroedinger equation) provides a kind of semiclassical correction to Vlasov’s equation in the same thermodynamic limit

[2]. Thus, essentially classical behavior arises in the thermodynamic limit. However, if there is some confinement of electrons at the nanoscale, as there is in semiconductor quantum wells, then quantum behavior must survive the limit N tends to infinity. In practice, time-dependent Schroedinger-Poisson equations are used to simulate such systems. Although there is still no fully satisfactory derivation of such approximations from an N-electron model, the Schroedinger-Poisson equation can be derived rigorously if a mean field scaling is assumed

[3]. I will review the results of [1] and [2], and speculate upon the application of [3] to semiconductor quantum wells.

[1] G. Sewell, Quantum statistical derivation of the hydrodynamics of a plasma model. J. Math. Phys., 26 (9): 2324 2334 (1985)

[2] A. Elgart, L. Erdos, B. Schlein, H.T. Yau. Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. Preprint: http://lanl.arxiv.org/abs/mathph/0311043

[3] C. Bardos, L. Erdos, F. Golse, N.J. Mauser and H.T. Yau. Derivation of the Schroedinger-Poisson equation from the quantum N-particle Coulomb problem. C. R. Acad. Sci. Ser. I Math, 334: 515–520 (2002)