Functional Derivative Approach

Walter Dittrich, Martin Reuter

Daniel P. Joubert

Homogeneity and isotropy of space lead to two hierarchies of equations for the independent Fermion kinetic-energy functional Ts[n]. The hierarchies link the mth functional derivative of Ts[n] to the (m+1)th functional derivative. The constraints imposed on Ts[n] are different for nonlocal functionals but equivalent for local functionals. As a consequence of the first hierarchy, Ts[n] can be expressed in terms of any of its functional derivatives.

Walter Dittrich, Martin Reutera

Siamak Khademi

Wigner distribution function has much importance in quantum statistical mechanics. It finds applications in various disciplines of physics including condense matter, quantum optics, to name but a few. Wigner distribution function is introduced by E. Wigner in 1932. However, there is no analytical derivation of Wigner distribution function in the literatures, to date. In this paper, a simple analytical derivation of Wigner distribution function is presented. Our derivation is based on two assumptions, these are A) by taking the integral of Wigner distribution function, with respect to configuration space, the momentum space distribution function is obtained B) WDF is real. Similarly, and in addition to Wigner distribution function, the distribution function of Sobouti-Nasiri, which is imaginary, is also derived.

Walter Dittrich, Martin Reuter

M. A. Pokrant, F. A. Stevens

The ground-state wave function for a quantum electron gas is approximated by $\ensuremath{\psi}=\ensuremath{\Phi}D$, where $D$ is the ground-state ideal-gas wave function and $\ensuremath{\Phi}=\mathrm{exp}[\ensuremath{-}(\frac{1}{2})\ensuremath{\Sigma}{i<j}^{}u({r}_{\mathrm{ij}})]$. Three-body correlation functions in the energy are approximated so that the energy is a functional of only pair-correlation functions. The variational principle is applied by utilizing the functional derivative of the pair-correlation function with respect to $u(r)$. The three- and four-body correlation functions in the functional derivative are approximated in a consistent manner. Use of the Kirkwood superposition approximation for the many-body correlation functions in the functional derivative gives unphysical results. A new approximation to the functional derivative is obtained by using the convolution approximation for the many-body terms. This gives a reasonable result for the asymptotic form of the wave function. The hypernetted-chain functional derivative combined with the convolution approximation for the energy gives the same asymptotic form. When the random-phase approximation to the energy is used with the Gaskell-Broyles-Sahlin-Carley functional derivative, the equation may be solved analytically, giving the same result obtained by Gaskell. The asymptotic form is the same as the two previous methods, namely, $u\ensuremath{\sim}(\frac{e}{\ensuremath{\hbar}})\frac{{(\frac{m}{\ensuremath{\pi}\ensuremath{\rho}})}^{\frac{1}{2}}}{r}$.