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Physical Review

Microscopic Quantum Hydrodynamics for Interacting Bose Systems

Robert L. Fanelli, R. E. Struzynski

Published Sep 5, 1968• 7 citations
Publisher

Abstract

Two equivalent microscopic formulations of quantum hydrodynamics are developed. Commutation rules for hydrodynamical quantities are developed, and in particular, the commutation rule for components of the fluid velocity operator is found to be in disagreement with that derived by Landau. Vortex quantization is found to be a natural consequence of the theory.

DOI
10.1103/physrev.173.248

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Two equivalent microscopic formulations of quantum hydrodynamics are developed. Commutation rules for hydrodynamical quantities are developed, and in particular, the commutation rule for components of the fluid velocity operator is found to be in disagreement with that derived by Landau. Vortex quantization is found to be a natural consequence of the theory.
Landau's quantized hydrodynamics is derived from the microscopic Hamiltonian, which is written in terms of the density and velocity operators. In Euler's equation for the velocity operator, the force per unit mass has a form different from Landau's and is given explicitly in terms of the two-body potential. A quantum pressure term, which does not occur in Landau's theory, also appears. New realizations of the density and velocity operators are given which satisfy the commutation relations, but must act in a Hilbert space larger than Fock space. It is shown that the formalism is without meaning in Fock space owing to the nonexistence of certain operators.
In this work we study the consequences of locality and Galilean covariance for the operators that occur in Landau's quantum hydrodynamics. We specifically consider the following requirements: (1) Galilean covariance of the velocity field, (2) locality of the velocity field, and (3) Landau's assumption that the momentum density is a symmetrized product of the velocity and density operators. It is demonstrated that the density-velocity commutation relation of the Landau theory is essentially a direct consequence of (1) and (2). The addition of (3) is sufficient to determine the velocity-velocity commutation relation, also in agreement with Laudau. We further show that the density-velocity commutation relation, independent of (3) or any specific form for the velocity field, is inconsistent with the nonnegative character of the local density.
Abstract Chapter 2 describes the basic knowledge of Bose-condensed systems in order to study the quantum hydrodynamics for both the superfluid helium and the atomic-gas BECs in the following sections. Starting from the field theoretical formulation, we will first give a self-contained explanation of the derivation of the most fundamental GP equation for weakly interacting Bose gases at zero temperature. We will also highlight the theoretical treatment for incorporating the concept of elementary excitations to describe low-energy excitations of Bose systems, the definition of normal and superfluid components, and eventually the celebrated model of two-fluid hydrodynamics. Finally, the microscopic structure and dynamics of topological excitations obtained from the GP equation are discussed by exemplifying solitons, vortex lines, and vortex rings.
The velocity operator vk studied in a previous paper admits further reduction in a Bose-Einstein condensed system. The reduced expression manifests the longitudinal nature of the operator vk(k≠0), and is proportional to the operator φk canonically conjugate to the density fluctuation, ρk. The value of v0 is the momentum to mass ratio of a condensed particle. As a dynamical variable describing the Hamiltonian, v0 plays peculiar role reminiscent of the super-velocity introduced in phenomenological theories. The collective formalism of condensed boson systems is found to provide the microscopic foundation of the quantum hydrodynamics incorporating the super-velocity.