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Physical Review

Commutators in Quantum Hydrodynamics for Interacting Bose Systems

Robert L. Fanelli, R. E. Struzynski

Published Jun 5, 1969• 6 citations
Publisher

Abstract

It is shown that the curl of the velocity operator in the second quantized version of quantum hydrodynamics for interacting Bose systems is equal to zero, thus resolving the apparent discrepancies between various existing formulations.
DOI
10.1103/physrev.182.363

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Physical Review A•Apr 1, 1973•20 citations
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.
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.