Gradient Expansions and Quantum Mechanical Extensions of the Classical Phase Space

Abstract

One common feature of most derivations of the gradient expansion for the kinetic energy density is the use of a mock phase space in intermediate steps. By mock phase space I mean a space parametrized by two coordinates p and r which in the classical limit reduce to the classical phase space. Of course a real phase space can not be defined quantum mechanically since p and r can not be specified simultaneously. Also there are many different mock phase spaces of which two in particular are frequently used. We will discuss these two in detail later. I will begin by giving a brief derivation of the Thomas-Fermi relation between the kinetic energy density and the spatial density to motivate the use of phase space and then indicate how this must be generalized. I then will discuss the various quantum mechanical generalizations of the phase space and show using the Wigner transform as an example one method of obtaining an ħ expansion. More details can be found in a review article by N.L. Balazs and myself.¹