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Canadian Journal of Physics

A QUANTUM MECHANICAL DESCRIPTION OF COLLECTIVE MOTIONS

Ray Skinner

Published Sep 1, 1956• 10 citations
Publisher

Abstract

A rigorous and general formalism is presented for the quantum mechanical description of the collective motions of a system containing a finite number of particles. The dynamical equations are expressed in terms of collective and "internal" coordinates. The internal coordinates are symmetrical in the particles, and correspond to the laboratory coordinates with the collective motions subtracted off. The formalism is applied to center of mass motion.

DOI
10.1139/p56-100

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Physical Review•Feb 1, 1956•16 citations
A rigorous and general formalism is presented for the quantum mechanical description of the collective motions of a system containing a finite number of particles. The dynamical equations are expressed in terms of collective and "internal" coordinates. The internal coordinates are symmetrical in the particles, and correspond to the laboratory coordinates with the collective motions subtracted off. The formalism is applied to center of mass motion.
An elementary theory of quantum-mechnaical collective motion of many-particle systems is developed. The fundamental idea is illustrated by taking an example of surface oscillation of an incompressible system of particles. The essential point of the theory is, roughly speaking, the neglect of 1/√N against unity, N being the number of particles of the system. The method is a natural generalization of separation of translational motion from the internal relative motion by the use of centre of mass coordinates. A description in terms of field concept is briefly discussed.
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.
Cusson's classical treatment of the collective rotations of a discrete system of N particles is extended to the full quantum mechanical system by means of a straightforward generalization of Villars' canonical transformation. In this manner, Bohr's collective Hamiltonian, with various values for the rotational mass, is microscopically derived. The nature and criteria for the existence of various collective flows in a many-body system are also given. The collective parts of the Hamiltonian are then separately expressed in original particle coordinates and momenta and in this manner the possibility of microscopic calculations for the collective motions is suggested. Finally appropriate microscopic Hamiltonians for the S.G.A.'s T5 × SO(3), GL(3,R), and CM(3) are determined.
A theory of collective submanifold, recently, developed by the present authors (M. Y. and A. K.) is applied to a classical many-body system, the Hamiltonian of which depends on the relative coordinates. The solution enables the separation of the total degrees of freedom into the collective and the intrinsic ones, which may be called the generalized center of mass and the relative motions. The results can be applied to the large amplitude collective motion and its couplings with the intrinsic degrees of freedom, which cannot be given in the HF plus RPA method.
The dynamical behavior of NNN interacting particles may be simplified by the introduction of 3N3N3N collective coordinates. A geometric interpretation of collective coordinates is described which is analogous to a type of random walk problem. Considerations on the validity of the transformation from ordinary spatial coordinates to symmetrical collective coordinates are discussed. The difficult problem of the boundary of collective coordinate space is touched upon.In the second half of the paper, the Lagrangian formulation for the dynamical problem is carried out by means of collective coordinates. Conditions for the equivalence of the physical Lagrangian and the collective coordinate Lagrangian are established. This leads to the problem of the Fourier representation of a potential by a finite number of terms. Finally, preliminary remarks on a modified Dirac $\ensuremath{\delta}$ function are presented.