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Physical Review C

Remarks Concerning the Kerman-Klein Theory of Collective Motion

C.H. Dasso, Abraham Klein

Published Dec 1, 1973• 1 citation
Publisher

Abstract

On the example of rotations in a plane, a concise form of expansion for the density matrix of this theory is presented and applied. It is emphasized again that the method is more general than the usual form of time-dependent Hartree-Fock approximation (or self-consistent cranking method) but contains the latter as a well-defined limiting case.

DOI
10.1103/physrevc.8.2511

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On the example of rotations in a plane, a concise form of expansion for the density matrix of this theory is presented and applied. It is emphasized again that the method is more general than the usual form of time-dependent Hartree-Fock approximation (or self-consistent cranking method) but contains the latter as a well-defined limiting case.
This is the second of a series of papers aimed at exploiting a generalized self-consistent approach to the study of collective excited states in finite many-particle systems. It contains: (i) A restatement of the equations of the method with further discussion of the basic approximations involved; (ii) a study, through the medium of the exactly soluble problem of uniform translational motion, of the inherent limitations in accuracy of the method when applied to the "strong-coupling" limit for which it was designed. A mode of approximation which can be made self-consistent is indicated. (iii) This approximation is applied to the problem of rotational motion in a plane. In the further limit in which one assumes the spectrum of the rigid rotator, one regains the results of the self-consistent cranking model. In contrast to previous work, the possibility of removing this limitation is indicated, but remains to be carried through.
A finite many-particle system can have collective states for which the off-diagonal matrix elements of certain one-particle operators are of the same order of magnitude as the diagonal elements. In such cases it is suggested that the random-phase approximation is in need of generalization. Examples are the uniform translational motion of any system and the rotational motion of deformed nuclei. The generalization is suggested after a review and critical analysis of the Hartree-Fock approximation. The model single-particle wave functions of the latter are replaced by wave functions in a space labeled both by the particle variables and by the quantum numbers of the collective motion. These generalized amplitudes are defined field-theoretically, and a self-consistent scheme for their calculation is obtained from the equations of motion. In addition to the self-consistent potential defined in the enlarged space, the energies of the excited states also turn out to be given by a natural self-consistency requirement. The new calculational scheme is first applied to a systematic restudy of the random-phase approximation where the self-consistency requirement on the energies has previously been overlooked. As a first characteristic application we obtain without "pushing" the total mass of a system in uniform translation, and a reinterpretation of the Hartree-Fock average field.
A classical theory of large amplitude, adiabatic collective motion, developed and applied previously to test examples, is transcribed into the language of time-dependent Hartree-Fock theory in order to initiate the application of the new method to problems in nuclear physics. The formulation which emerges can be characterized as a generalized cranking theory, in the sense that the cranking operator cannot be chosen arbitrarily, as in the conventional formulation of this method, but is instead fully constrained by the formalism. A procedure for obtaining approximate solutions to the new equations is described and illustrated with several simplified many-body models. It is inferred that traditional cranking calculations can serve as starting points for more realistic models. This paper also includes additional discussion concerning the calculation of collective masses and the problem of local stability of collective motion, not covered in our previous work.
We show that to any Lie algebra of observables in quantum mechanics there corresponds a system of densities, of which the one-body densities are just one example. These densities may correspond to a single pure state or to an ensemble of states. Like the one-body densities, they support a Hamiltonian dynamics of the time-dependent Hartree-Fock type, which can be applied, for example, to collective motions where the collective states of interest may be either more or less general than single Slater determinants. The dynamics can also be applied to the average dynamics of an ensemble of states.