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Physical Review A

Current-Algebra Approach to Landau's Quantized Hydrodynamics

Donald H. Kobe, Grant C. Coomer

Published Apr 1, 1973• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.7.1312

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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.
For building up a theory of superfluid Helium-4, Lev Landau ingeniously unified the principles of quantum mechanics with the principles of hydrodynamics. By introducing a velocity operator he was able to derive a quantum analogue of the mass density flux equation and a quantum Euler equation. The article shows that it is possible to formulate a consistent quantum theory of the ideal liquid based on Landau's Quantum Hydrodynamics. This means that it is added the quantum analogue of the momentum density flux equation, the energy density flux equation, the entropy density flux equation, and a quantum vorticity equation. These are all manifestly in agreement with the classical limit of the flux equations of classical ideal liquids. It is also shown that the various aspects that have been criticised about Landau's Quantum Hydrodynamics - e.g. the application of the inverse of the mass density operator - are not justified. Such a consistent quantum theory of an ideal liquid may therefore provide a starting point of a complementary approach in the theoretical research of superfluid Helium 4.
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.
The Madelung equations express the Schrödinger equation as a continuity equation and modified Hamilton–Jacobi equation. These equations are equivalent to the Euler equations for a compressible, potential flow, when classical pressure per unit density is replaced by the quantum potential per unit mass. We extend this hydrodynamic interpretation by quantising a single, spinless, non-relativistic particle constrained to a surface wave with small slope. The wave is distinct from the wave function and, in order to reproduce the Schrödinger equation, it must satisfy the kinematic boundary condition for a free surface advected by twice the Madelung velocity field.
Differential conservation equations are derived for the mass-, momentum-, and energy-density operators for a 1-component simple fluid of Bose or Fermi particles with arbitrary pairwise interactions. These equations are used in a statistical mechanical derivation of exact equations of motion for the expectations of these operators. The equations of motion are coupled to equations relating these expectations to the local temperature, chemical potential, and fluid velocity. The coupled equations are closed in the sense that the expectations and their thermodynamic conjugates listed above are the only unknowns, although some of the dependence in the equations on the conjugates is expressed only implicitly. The equations of motion are memory-retaining nonlocal generalizations of the classical hydrodynamic equations and apply to a normal fluid arbitrarily far from equilibrium. The formalism is not carried as far as has the corresponding classical formalism because the local equilibrium expectation of the momentum density here does not equal the fluid velocity times the expectation of the mass density as is true in classical statistical mechanics.