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Progress of Theoretical Physics

Collective Description of a System of Interacting Bose Particles. II

Sigenobu Sunakawa, Hiroshi Nakatani, Yoshimatsu Yokoo

Published Mar 1, 1962• 11 citations
Publisher

Abstract

The collective behavior of the three-dimensional Bose system is investigated. Along the line of reasoning of the preceding paper, the momentum density operator is modified and it is shown that the modified momentum density operator is equivalent to the velocity operator of the quantum hydrodynamics. The Hamiltonian is described in terms of the collective variables for the case of irrotational motion. The result is compared with other theories.

DOI
10.1143/ptp.27.600

Related papers

Collective Description of a System of Interacting Bose Particles. I

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The collective description of the one-dimensional interacting Bose system is studied. The exact canonical conjugate of the density operator is given explicitly. The Hamiltonian of the system is expressed in terms of the exact collective variables.

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Density and Current Density as Coordinates for a System of Interacting Bosons at Absolute Zero

B. Tsu-Shen Chang

The quantum field Hamiltonian expressed in terms of density and current density variables has been employed together with the equal-time commutation relations among these variables to find the ground state energy and the density fluctuation excitation spectrum of a system of interacting bosons at T = 0°K. The approximation involved consists in assuming that the density fluctuation in space is small compared with the average density. The results easily obtained in the lowest-order approximation agree with those of Bogoliubov. However, in our treatment no condensation of particles in zero-momentum state is assumed or apparent. A connection between the present treatment and the quantum hydrodynamic approach to the irrotational flow of a Bose liquid has been made.

Journal of Mathematical Physics•Jun 1, 1971•8 citations
The collective behavior of the three-dimensional Bose system is investigated. Along the line of reasoning of the preceding paper, the momentum density operator is modified and it is shown that the modified momentum density operator is equivalent to the velocity operator of the quantum hydrodynamics. The Hamiltonian is described in terms of the collective variables for the case of irrotational motion. The result is compared with other theories.
A new well-defined velocity operator is proposed for the collective description of a many-boson system. It is shown that the new velocity operator is irrotational. The Hamiltonian of the system is successfully described in terms of the density-fluctuation operator and the velocity operator. The interaction term in the order of N-1 has a different form from that in previous papers. The range of validity of the present formalism is discussed in the final section in answer to the questions raised so far on the collective formalism of the many-boson system.
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.
An approach to the description of kinetics which taking into account the large-scale hydrodynamic transport processes for quantum Bose system is proposed. The nonequilibrium statistical operator which consistently describes both the kinetic and nonlinear hydrodynamic fluctuations in quantum liquid is calculated. Using this operator the coupled equations for quantum one-particle distribution function and functional of hydrodynamic variables: densities of momentum, energies and number of particles are obtained.
In the preceeding paper, introducing [Formula: see text]-dependent density operators and defining exact momenta (collective variables), we could get an exact canonically momenta approach to a one-dimensional (1D) neutron–proton (NP) system. In this paper, we attempt at a velocity operator approach to a 3D NP system. Following Sunakawa, after introducing momentum density operators, we define velocity operators, denoting classical fluid velocities. We derive a collective Hamiltonian in terms of the collective variables.
The quantum field Hamiltonian expressed in terms of density and current density variables has been employed together with the equal-time commutation relations among these variables to find the ground state energy and the density fluctuation excitation spectrum of a system of interacting bosons at T = 0°K. The approximation involved consists in assuming that the density fluctuation in space is small compared with the average density. The results easily obtained in the lowest-order approximation agree with those of Bogoliubov. However, in our treatment no condensation of particles in zero-momentum state is assumed or apparent. A connection between the present treatment and the quantum hydrodynamic approach to the irrotational flow of a Bose liquid has been made.