WIGNER FUNCTION IN THE SYMMETRIC GAUGE: DE HAAS-VAN ALPHEN OSCILLATIONS, MAGNETIC FIELD LOCALIZATION AND UNCERTAINTY PRINCIPLE

Abstract

Since the Wigner function (WF) is related to a Lindard-constant type linear dielectric function derived in the symmetric gauge,7it is expected to show de Haas-van Alphen (dHvA) oscillations. Starting with the symmetric eigenfunctions, we derived the pure-state WF in a magnetic field, whose plots in phase space and in term of B-1for increasing n are consistent with the dHvA effect. Furthermore the asymptotic expansion of WF at large n show periodic oscillations with a period related to the Fermi energy. The phase space plots of WF also show that dHvA and similar oscillations could be a consequence of Nature's strategy for increasing the effective spatial range without violating the uncertainty principle. Properties of the symmetric eigenfunctions were derived. The dynamics of WF can be obtained from the solution of the time-dependent Schrödinger equation (SE). A new method to solve the SE in a magnetic field in the interaction picture based on expansion in term of symmetric eigenfunctions has been developed. The matrix element for a Gaussian potential were derived explicitly, plotted against B-1, and showed oscillations. The total WF was shown to be a linear combination of the diagonal pure-state WF's by using the orthogonality for symmetric eigenfunctions. The no-special-point property for WF was confirmed, which is important for the construction of a numerical algorithm to solve the SE in a magnetic field.