Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Canadian Journal of Physics

Collective motions in nuclei and the spectrum generating algebras T5 × SO(3), GL(3,R), and CM(3)

P. Gulshani, DAVID J. ROWE

Published May 1, 1976• 67 citations
Publisher

Abstract

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.

DOI
10.1139/p76-115

Related papers

Geometry of collective motions

P. Gulshani, Anatole B. Volkov

A general projection method for decomposing the kinetic energy of an N-particle system into collective and intrinsic parts defined respectively on the orbits and the orbit space of a Lie transformation group is given. Specific targets of the application of the method are the kinematical group GL + (3,R) and the quotient set GL + (3,R)/SO(3) for their importance in microscopic formulation of nuclear collective motions. For these two cases the orbit spaces in the particle configuration space are shown to be identifiable with the Grassman and Stiefel manifolds of 3-planes and 3-frames respectively. Some problems related to expressing the kinetic energy in terms of vector fields on these manifolds are resolved. In particular, non-integrable coordinates previously used by one of the present authors is shown to arise from the imposition of unacceptable conditions. Finally the authors consider the corresponding decomposition of the N-particle Hilbert space. It is proposed that an appropriate basis function for the GL + (3,R) collective model is provided by an irreducible representation of the boson SU(6) group.

Journal of Physics A Mathematical and General•Jan 1, 1982•5 citations

Concept of Ideal Collective Coordinate as the Foundation for a Phenomenological Theory of Nuclear Collective Motion: Basic Ideas and Relation to Other Phenomenological Methods

Abraham Klein, R. M. Dreizler, Robert E. Johnson

The concept of an ideal collective coordinate is introduced by means of the following example: Consider a one-dimensional vibration of a many-body system in the sense that a large subset of states $|n〉$ of the system exhibits an energy spectrum and relative transition probabilities following the laws of the (in general anharmonic) oscillator described by $\mathcal{H}$ (${p}_{\ensuremath{\alpha}}, \ensuremath{\alpha}$) $(\ensuremath{\alpha}|n)={\ensuremath{\omega}}_{n}(\ensuremath{\alpha}|n)$. We suppose the set of many-body states $|n〉$ to extend indefinitely, and we take the transform $|\ensuremath{\alpha}〉=\ensuremath{\Sigma}|n〉$ ($n|\ensuremath{\alpha}$) to define a many-body generating state of the band which is precisely localized in $\ensuremath{\alpha}$ space. The basic assumption of collectivity, that changing the state of at most a few particles cannot much alter the value of $\ensuremath{\alpha}$, is shown to be sufficient to derive a phenomenological theory from the many-body starting point. The phenomenological aspects of a recent theory of rotations due to Villars is seen to be contained in the above formulation as a special case. A brief review is given of the generator coordinate and similar projection methods in order to exhibit their relationship with the present method.

Physical Review•Jul 20, 1968•21 citations

A Classical Theory of the Collective Description

Yôiti Watanabe

An elementary theory is developed of a classical treatment with the ‘Collective Motion’ of many-particle systems. The fundamental concept of collective motion can be reduced into a certain transformation property of Hamiltonian. Regarded as the dynamical variables, the continuous parameters of such a transformation describe the collective motion of the system, while the individual motions are partly suppressed by the same numbers of subsidiary conditions, or constraints, logically derived in, as the numbers of transformation parameters. Then the general procedure of formulation is illustrated; as an example, the incompressible motion with three modes can be systematically formulated. As another example, the unified treatment is performed also with the vortex motion. The digression in the approximating procedure is also presumed.

Progress of Theoretical Physics•Jul 1, 1956•16 citations

A Microscopic Theory of Collective and Independent-Particle Motions

M. Yamamura, A. Kuriyama

A microscopic theory of collective and independent-particle motion in many-fermion system is developed in the framework of the classical theory. The basic idea is an extension of the conventional time-dependent Hartree-Fock method with the use of fermion coherent state representation. Equation to determine collective path and constraints to govern the collective and the independent-particle variables are given. Hamiltonian and other physical quantities given with the original fermion variables are rewritten to the forms expressed in terms of the collective and the independent-particle variables.

Progress of Theoretical Physics•Feb 1, 1981•17 citations

Collective Hamiltonians in the generator coordinate method: A numerical procedure

D. Galetti, Salomon Sylvain Mizrahi

A numerical procedure is introduced which allows us to extract a collective Hamiltonian, expressed in terms of a collective canonical pair $\stackrel{^}{p}$ and $\stackrel{^}{q}$. The starting point is a microscopic many-body approach, namely the generator coordinate method, and the Weyl formalism permits the description of the collective dynamics in the $p\ensuremath{-}q$ space. As an illustration we compare numerical calculations with exact results, obtained analytically, for the quadratic energy kernel with the Gaussian overlap approximation case and the Goldhaber-Teller dipole model applied to the $^{4}\mathrm{He}$ nucleus.NUCLEAR STRUCTURE Numerical derivation of collective Hamiltonians. Generator coordinate method, Weyl procedure. Comparison with analytical results.

Physical Review C•May 1, 1982•3 citations

Lie algebra projectors and the kinematics of collective motions

P. Gulshani

Direct-sum decompositions of the Euclidean space of particle momenta into collective and intrinsic subspaces are achieved using projectors on so(3) and gl(3,R) algebra spaces. The separation of the N-particle kinetic energy into the corresponding collective and intrinsic components is then simply obtained. In the case of gl(3,R) the resulting intrinsic kinetic energy is expressed in terms of an appropriate subset of the generators of the direct product group SO(N)*SO(N). A detailed comparison of these results with those of other authors is given.

Journal of Physics A Mathematical and General•Jan 1, 1981•6 citations
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 general projection method for decomposing the kinetic energy of an N-particle system into collective and intrinsic parts defined respectively on the orbits and the orbit space of a Lie transformation group is given. Specific targets of the application of the method are the kinematical group GL + (3,R) and the quotient set GL + (3,R)/SO(3) for their importance in microscopic formulation of nuclear collective motions. For these two cases the orbit spaces in the particle configuration space are shown to be identifiable with the Grassman and Stiefel manifolds of 3-planes and 3-frames respectively. Some problems related to expressing the kinetic energy in terms of vector fields on these manifolds are resolved. In particular, non-integrable coordinates previously used by one of the present authors is shown to arise from the imposition of unacceptable conditions. Finally the authors consider the corresponding decomposition of the N-particle Hilbert space. It is proposed that an appropriate basis function for the GL + (3,R) collective model is provided by an irreducible representation of the boson SU(6) group.
The concept of an ideal collective coordinate is introduced by means of the following example: Consider a one-dimensional vibration of a many-body system in the sense that a large subset of states ∣n〉|n〉∣n〉 of the system exhibits an energy spectrum and relative transition probabilities following the laws of the (in general anharmonic) oscillator described by H\mathcal{H}H (${p}_{\ensuremath{\alpha}}, \ensuremath{\alpha}$) $(\ensuremath{\alpha}|n)={\ensuremath{\omega}}_{n}(\ensuremath{\alpha}|n)$. We suppose the set of many-body states ∣n〉|n〉∣n〉 to extend indefinitely, and we take the transform $|\ensuremath{\alpha}〉=\ensuremath{\Sigma}|n〉$ ($n|\ensuremath{\alpha}$) to define a many-body generating state of the band which is precisely localized in $\ensuremath{\alpha}$ space. The basic assumption of collectivity, that changing the state of at most a few particles cannot much alter the value of $\ensuremath{\alpha}$, is shown to be sufficient to derive a phenomenological theory from the many-body starting point. The phenomenological aspects of a recent theory of rotations due to Villars is seen to be contained in the above formulation as a special case. A brief review is given of the generator coordinate and similar projection methods in order to exhibit their relationship with the present method.
An elementary theory is developed of a classical treatment with the ‘Collective Motion’ of many-particle systems. The fundamental concept of collective motion can be reduced into a certain transformation property of Hamiltonian. Regarded as the dynamical variables, the continuous parameters of such a transformation describe the collective motion of the system, while the individual motions are partly suppressed by the same numbers of subsidiary conditions, or constraints, logically derived in, as the numbers of transformation parameters. Then the general procedure of formulation is illustrated; as an example, the incompressible motion with three modes can be systematically formulated. As another example, the unified treatment is performed also with the vortex motion. The digression in the approximating procedure is also presumed.
A microscopic theory of collective and independent-particle motion in many-fermion system is developed in the framework of the classical theory. The basic idea is an extension of the conventional time-dependent Hartree-Fock method with the use of fermion coherent state representation. Equation to determine collective path and constraints to govern the collective and the independent-particle variables are given. Hamiltonian and other physical quantities given with the original fermion variables are rewritten to the forms expressed in terms of the collective and the independent-particle variables.
A numerical procedure is introduced which allows us to extract a collective Hamiltonian, expressed in terms of a collective canonical pair $\stackrel{^}{p}$ and $\stackrel{^}{q}$. The starting point is a microscopic many-body approach, namely the generator coordinate method, and the Weyl formalism permits the description of the collective dynamics in the $p\ensuremath{-}q$ space. As an illustration we compare numerical calculations with exact results, obtained analytically, for the quadratic energy kernel with the Gaussian overlap approximation case and the Goldhaber-Teller dipole model applied to the 4He^{4}\mathrm{He}4He nucleus.NUCLEAR STRUCTURE Numerical derivation of collective Hamiltonians. Generator coordinate method, Weyl procedure. Comparison with analytical results.
Direct-sum decompositions of the Euclidean space of particle momenta into collective and intrinsic subspaces are achieved using projectors on so(3) and gl(3,R) algebra spaces. The separation of the N-particle kinetic energy into the corresponding collective and intrinsic components is then simply obtained. In the case of gl(3,R) the resulting intrinsic kinetic energy is expressed in terms of an appropriate subset of the generators of the direct product group SO(N)*SO(N). A detailed comparison of these results with those of other authors is given.