It is shown that one can find an orthogonal transformation that will enable one to split the motion of a many-body system to a center-of-mass motion and an internal motion. A particular orthogonal transformation has been chosen which retains the independent-particle aspect of a harmonic-oscillator shell-model Hamiltonian. It is suggested that one could easily study and eliminate the states with spurious motions of the center of mass by a direct transformation of the shell-model wave function into the new coordinate system.
It is shown that one can find an orthogonal transformation that will enable one to split the motion of a many-body system to a center-of-mass motion and an internal motion. A particular orthogonal transformation has been chosen which retains the independent-particle aspect of a harmonic-oscillator shell-model Hamiltonian. It is suggested that one could easily study and eliminate the states with spurious motions of the center of mass by a direct transformation of the shell-model wave function into the new coordinate system.
An explicit construction is found for a unitary operator which insures the free motion of the center of mass of any many-particle wave function on which it is allowed to act. The transformation is used to calculate recoil correction terms for the internal energy and external interactions of nuclei, and some numerical evaluations are given for cases of interest. The many-body harmonic-oscillator problem is exactly soluble when one uses the transformation, and one is thus enabled to give a more general discussion of the spurious states.
Two methods to calculate the density matrix of one• body or more•than•one•body which does not include the spurious center•of•mass excitation are presented.One is the method to calculate the density matrix for the many-body wave function of general type and the other is applicable to the case in which the center~of-mass motion separates from the internal motion and is in the (Os) orbit of harmonic oscillator.The applications of these methods are illustrated for some many-body wave function.§ 1. IntroductionTo represent the characteristic features of the many-body wave function of nuclei, the density matrices are often utilized.Experimentally, for example, in the electron scattering from nuclei, the nuclear structure is analyzed by the terms of the nuclear density and the current density, which are the special forms of the one-body density matrix.Theoretically, the one-body density matrix plays a crucial role in the current nuclear theories based on the mean field approximation, such as Hartree-Fock theory.And, in recent nuclear studies, interest is taken in not only the nuclear density which is the diagonal part of the one-body density matrix, but the non-local part of it.The non-local part of the one-body density matrix is often visualized with the use of the Wigner transform of it. 1),2)Especially, in the time-dependent Hartree-Fock theory, the Wigner transform of the one-body density matrix is interpreted as the classical distribution function and obeys the Vlasov equation approximately.3)These works clarify the importance of the non-local part of the one-body density matrix in the dynamics.Usually the spatial coordinate of the density matrix is measured from some spatially fixed origin.Due to the finiteness of the nuclear system, however, we need to measure the coordinate from the center of mass of the system; otherwise the spurious center-of-mass motion affects the density matrix.This effect of center-of-mass motion becomes important for light systems.For the diagonal part of the one-body transition density matrix, the effect of center-of-mass motion was discussed by Tassie and Barker for the case of harmonic oscillator shell mode1. 4) And for the diagonal part of the one-body density matrix in momentum representation, that is the nuclear momentum distribution, the effect of spurious center-of-mass motion has been discussed. 5) For the general non-local density matrix, however, there seems not to be presented any method to treat the spurious center-of-mass effect.In this paper, we present two methods to get the center-of-mass free density matrix.One is the method, applicable to the general wave function, which is especially suitable for the case where the wave function is a (linear superposition of) Slater determinant.The other is applicable to the cases in which the center-of-mass motion separates from other internal motions and is in the (Os) orbit of harmonic oscillator.For such wave functions, the latter method is more easy to calculate though the former method is also applicable.
A formal technique is developed to study internal properties of finite many-particle systems possessing translational invariance. The basic objective is to discover a method of calculating, in a modified shell model, average values of operators referred to the center of mass, particularly those which have to do with the surface of the system, such as the density of particles. The formalism is based on the method of redundant variables, in which the shell-model wave function describes the internal degrees of freedom of the system, while a superfluous variable is introduced to satisfy the conservation law and to handle the collective motion. The spurious degeneracy which results from the introduction of the extra variable is removed by suitable subsidiary conditions. With the aid of these restricted solutions, it is shown that the density with respect to the center of mass can be calculated in terms of a one-body operator. The dynamics of the system is then formulated in terms of density matrices suitably defined from these restricted solutions. Within the context of the modified shell model, the density matrices consist of the Dirac density matrices of the shell model plus correction terms which depend nonlinearly on these Dirac matrices and matrix elements of the coordinate X. The single-particle states of the shell model are finally chosen to minimize the expectation of the Hamiltonian. The resulting variational equation contains a homogeneous part which has the structure of the conventional Hartree-Fock equation together with an inhomogeneous term depending nonlinearly on the single-particle states. The solution of this equation is finally expanded about the nontranslationally invariant Hartree-Fock solution to first order in perturbation theory.
The method of generating a family of new solutions starting from any wave function satisfying the nonlinear Schrödinger equation in a harmonic potential proposed recently [ J. J. Garc\'ıa-Ripoll, V. M. Pérez-Garc\'ıa, and V. Vekslerchik, Phys. Rev. E {\bf 64}, 056602 (2001)] is extended to many-body theory of mutually interacting particles. Our method is based on a generalization of the displacement operator known in quantum optics and results in the separation of the center of mass motion from the internal dynamics of many-body systems. The center of mass motion is analyzed for an anisotropic rotating trap and a region of instability for intermediate rotational velocities is predicted.
The method of generating a family of new solutions starting from any wave function satisfying the nonlinear Schr\"odinger equation in a harmonic potential proposed recently [J. J. Garc\'{\i}a-Ripoll, V. M. P\'erez-Garc\'{\i}a, and V. Vekslerchik, Phys. Rev. E 64, 056602 (2001)] is extended to the many-body theory of mutually interacting particles. Our method is based on a generalization of the displacement operator known in quantum optics, and results in the separation of the center-of-mass motion from the internal dynamics of many-body systems. The center-of-mass motion is analyzed for an anisotropic rotating trap and a region of instability for intermediate rotational velocities is predicted.
The most general transformations of coordinates that allow for the exact separation of the kinetic energy operator of a quantum many-body system into total center of mass kinetic energy and internal kinetic energy are found and discussed. It is found (i) that the suitable transformations, depending on the number of particles, have a certain number of free parameters and this allows for the generalization of the Jacobi coordinates to a much larger class of coordinates with the same properties and (ii) that there is a new, uncommon, additive group structure hidden in the transformation matrices that is connected to certain geometric properties of the set of coordinates.