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Physical Review C

Collective Hamiltonians in the generator coordinate method: A numerical procedure

D. Galetti, Salomon Sylvain Mizrahi

Published May 1, 1982• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.25.2795

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arXiv:2205.13058•May 25, 2022•0 citations
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−qp\ensuremath{-}qp\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.
A procedure is developed for treating a nuclear collective motion via the path integral formalism by defining a collective propagator in the "phase space" of the collective subspace. This collective subspace is projected out of the full many-body Hilbert space by an approach of the Generator Coordinate Method. Under the approximation of quadratic "momentum" dependence we write the propagator in the coordinate representation from which we derive the collective Schrödinger equation in terms of a canonical pair and .
Using a very simple type of wave packet, which is obtained by letting unitary displacement operators having as generators canonical operators $\stackrel{^}{Q}$ and $\stackrel{^}{P}$ in the many-body Hilbert space act on a reference state, we investigate the relationship between the semiclassical and the generator coordinate methods. The semiclassical method is based on the time-dependent variational principle, whereas in the generator coordinate method the wave packets are taken as generator states. To establish the equivalence of the two methods, we examine in detail, using tools developed in previous works, the concept of redundancy of the wave packet and the importance of the zero-point energy effects. We make a numerical application to the case of the Goldhaber-Teller mode in 4He^{4}\mathrm{He}4He.[NUCLEAR STRUCTURE Relationship between the semiclassical and the generator coordinate methods. Redundancy and zero-point energy effects. Numerical application to the Goldhaber-Teller mode in 4He^{4}\mathrm{He}4He.]
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.
We investigate the validity of the canonical quantization of the collective Hamiltonian derived by the selfconsistent-collective-coordinate method. Special attention is paid to the boundary condition for the (η*, η) expansion, which fixes the canonical coordinate system to carry out the quantization procedure. We remove the ambiguity of the boundary condition by requiring that the boson representations obtained by the canonical quantization agree, to the order 1/Ω, with those derived by the modified-Marumori boson-expansion method in a certain limit. Numerical examples for an exactly solvable model show that our quantization procedure provides us with remarkably accurate results.
A many•body technique called the dynamical nuclear field theory is developed by taking the example of pair rotation.The time-dependent one-body Hamiltonian contains the collective coordinates; the magnitude and phase aIigleof the energy gap.Effective forms of the original Hamiltonian and other operators are derived for a direct-product space of the collective states and a restricted set of the particle states defined by the driving Hamiltonian in the intrinsic frame.Pair rotations of even-and odd-A systems are treated in the lowest order approximation.The particle-rotor picture and the cranking model are shown to be equivalent.*) Brief report on DNFT in the Tamm-Dancoff approximation can be found in Ref. 13).
In nuclear theory, the generator coordinate method (GCM), a type of configuration mixing method, is often used for the microscopic description of collective motions. However, the GCM has a problem that a structure of the collective subspace, which is the Hilbert space spanned by the configurations, is not generally understood. In this paper, I investigate the structure of the collective subspace in the dynamical GCM (DGCM), an improved version of the GCM. I then show that it is restricted to a specific form that combines tensor products and direct sums under reasonable conditions. By imposing additional specific conditions that are feasible in actual numerical calculations, it is possible to write the collective subspace as a simple tensor product of the collective part and the others. These discussions are not dependent on the details of the function space used for generating the configurations and can be applied to various methods, including the mean-field theory. Moreover, this analytical technique can also be applied to a variation after projection method (VAP), then which reveals that under a specific condition, the function space of the VAP has an untwisted structure. These consequences can provide powerful tools for discussing the collective motions with the DGCM or the GCM.