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Separation of the normal and the superfluid motions in the Schroedinger equation by means of the displacement and collective variables method

I. O. Vakarchuk, I.R. Yukhovskii

Published Jan 1, 1974• 0 citations
OpenAlex

Abstract

With the aid of the displacement method, the wave function of interacting bosons is represented as a product of the ground state function and the function describing the excitations in the system. The equations for these functions are solved in the collective variables representation by means of a special perturbation method free of divergences. The expressions obtained for the wave functions, the energy of the ground state, the energy spectrum and the damping of collective excitations are investigated. (auth)

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With the aid of the displacement method, the wave function of interacting bosons is represented as a product of the ground state function and the function describing the excitations in the system. The equations for these functions are solved in the collective variables representation by means of a special perturbation method free of divergences. The expressions obtained for the wave functions, the energy of the ground state, the energy spectrum and the damping of collective excitations are investigated. (auth)
By introducing a number of auxiliary variables, quantum-mechanical collective and internal motions in a Bose system are described. In a collective representation obtained here, the wave functions are expressed as functions of the auxiliary variables and the internal coordinates. The energy for the ground state and the excitation energies of collective motion and internal motion are obtained for the case of a long-range interaction potential. The lowest energy obtained proves to be smaller than that afforded by the Hartree-Fock approximation. The excitations of the collective motion with smaller wave numbers than a cut-off wave number kc correspond to Landau's phonons. The excitations of the internal motion resemble Landau's rotons; but they are forbidden for the smaller wave numbers K ≦ kc. When one applies this method to the problem of liquid helium, it becomes necessary to take account of the two-body correlation of individual particles. It is pointed out that the phonon energy depends upon the zero-point motion of the individual particles.
The ground-state energy of an interacting many-boson system obtained by Bogoliubov and Zubarev using the collective-coordinate method is shown to be equivalent through second order to the variation-perturbation energy evaluated in the uniform limit by means of the method of correlated basis functions. It is also shown that the wave function derived with a slight modification of the Bogoliubov-Zubarev approach is equivalent to that determined formally from the Rayleigh-Schr\"odinger perturbation theory in the uniform limit.
The method of “auxiliary variables” (collective coordinates) used by Bogolyubov and Zubarev is formulated in a somewhat different manner. Improvements are made in two points: 1) The Hamiltonian is given in an Hermitian form and 2) the number of the auxiliary variables can be limited, if necessary. We calculate the ground state energy and the excitation energy spectrum of the spinless bosons taking account of phonon-phonon interactions. The expression for the ground state energy is shown to be equivalent to the one derived by Bogolyubov and Zubarev.