No 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. § I. IntroductionIn the preceding paper/> we have studied the collective behavior of the one-dimensional Bose gas system. In the present paper, we shall study the theory of the three-dimensional system, which furnishes the Landau theory of quantum hydrodynamics 2 >,s> with a quantum mechanical foundation. _Such attempts were already made by Husimi and Nishiyama 4 > and others. 5 > They, however, suffered from the appearance of the singular operators, i.e. the square root and the inverse of the density operator. The purpose of the present paper is to propose an alternative theory which is free from such difficulties, and to. make clear the relation of the present theory to others.In the case of the one-dimensional system which was treated in the preceding paper, the exact canonical conjugate Tik was introduced by modifying the approximate one 'irk· In the present case, the momentum density operator gk will play the role corresponding to the approximate canonical conjugate rrk, as. was suggested by Landau. 2 > Then we may imagine that the modified momentum density operator which corresponds to the exact canonical conjugate nk in the one-dimensional theory may in anology be. the velocity operator v in hydrodynamics. In § 2, we shall make sure of the conjecture mentioned above, and the commutation relations satisfied by the velocity operator and the density operator will be derived. In § 3, the original Hamiltonian will be rewritten in terms of the collective variables for the case of the irrotational motion. Our results will be discussed and compared with other theories in the final section. § 2. Velocity operatorThe Hamiltonian of our system is given by

No abstract

No abstract

127The definite expression of the phonon-phonon interaction in a Bose $ystem is derived up to the order of N-1 in terms of the density operator and its exact canonical conjugate which was introduced in the previous papers. We find some mathematical ambiguity in the course of calculation, and clarify the reason why the different results were obtained so far. The validity of our formalism is discussed from the physical point of view.