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American Journal of Physics

Quantum hydrodynamics and virial theorems

Peter J. Price

Published May 1, 1995• 1 citation
Publisher

Abstract

This paper elucidates how many-particle quantum hydrodynamic equations underlie the virial formula, including its surface pressure term. It shows the relation of the field variables to the particle Wigner function, proposes a many-particle ‘‘liquid’’ model to approximate these variables, and discusses its application to electrons in semiconductor nanostructures.

DOI
10.1119/1.17909

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This paper elucidates how many-particle quantum hydrodynamic equations underlie the virial formula, including its surface pressure term. It shows the relation of the field variables to the particle Wigner function, proposes a many-particle ‘‘liquid’’ model to approximate these variables, and discusses its application to electrons in semiconductor nanostructures.
The Heisenberg dynamics of the energy, momentum, and particle densities for fermions with short-range pair interactions is shown to converge to the compressible Euler equations in the hydrodynamic limit. The pressure function is given by the standard formula from quantum statistical mechanics with the two-body potential under consideration. Our derivation is based on a quantum version of the entropy method and a suitable quantum virial theorem. No intermediate description, such as a Boltzmann equation or semi-classical approximation, is used in our proof. We require some technical conditions on the dynamics, which can be considered as interesting open problems in their own right.
A formulation of quantum statistical mechanics is given, in terms of distribution functions. It is shown that all quantities of interest are obtained directly from nonlocal distribution functions ${\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta}), {\ensuremath{\gamma}}^{(2)}({{x}_{1}}^{\ensuremath{'}}, {{x}_{2}}^{\ensuremath{'}}, {x}_{1}{x}_{2}, \ensuremath{\beta}), \ensuremath{\cdots}.$ In a uniform system the Fourier transform of ${\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta})$ is the distribution of particles in kkk space, $n(k, \ensuremath{\beta})$. Various schemes for calculating these distribution functions directly are discussed. An approximate integral equation for $n(k, \ensuremath{\beta})$ is found, which can be solved by a converging iteration process. Some remarks are included on the application of the virial theorem.
In the first part of this paper, the many-particle quantum hydrodynamics (MPQHD) equations for a system containing many particles of different sorts are derived exactly from the many-particle Schrödinger equation. It includes the derivation of the many-particle continuity equations (MPCE), many-particle Ehrenfest equations of motion (MPEEM), and many-particle quantum Cauchy equations (MPQCE) for any of the different particle sorts and for the total particle ensemble. The new point in our analysis is that we consider a set of arbitrary particles of different sorts in the system. In MPQCEs, there appears a quantity called pressure tensor. In the second part of this paper, we analyze two versions of this tensor in depth -- the Wyatt pressure tensor and the Kuzmenkov pressure tensor. There are different versions because there is a gauge freedom for the pressure tensor similar to that for potentials. We find that the interpretation of all quantities contributing to the Wyatt pressure tensor is understandable but for the Kuzmenkov tensor, it is difficult. Furthermore, the transformation from Cartesian coordinates to cylindrical coordinates for the Wyatt tensor can be done in a clear way, but for the Kuzmenkov tensor, it is rather cumbersome.