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Canadian Journal of Physics

Remarks on the microscopic derivation of the collective model

Tadashi Toyoda, K. Wildermuth

Published Oct 1, 1984• 0 citations
Publisher

Abstract

The rotational part of the phenomenological collective model of Bohr and Mottelson and others is derived microscopically, starting with the Schrödinger equation written in projection form and introducing a new set of "relative Euler angles." In order to derive the local Schrödinger equation of the collective model, it is assumed that the intrinsic wave functions give strong peaking properties to the overlapping kernels.

DOI
10.1139/p84-134

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The rotational part of the phenomenological collective model of Bohr and Mottelson and others is derived microscopically, starting with the Schrödinger equation written in projection form and introducing a new set of "relative Euler angles." In order to derive the local Schrödinger equation of the collective model, it is assumed that the intrinsic wave functions give strong peaking properties to the overlapping kernels.
The Schrödinger equation is derived classically assuming that particles present local \nrandom spatial fluctuations compatible with the presence of the zero-point field. With- \nout specifying the forces arising from this permanent matter-field interaction but ex- \nploring its fundamental properties (homogeneity, isotropy and random aspect) to justify \nthe emergence of the continuity equation in one-particle context, these fluctuations are \ndescribed in terms of the probability density. Specifically, the starting point is the as- \nsumption that the local activities, which turn the path followed by the particle totally \nunpredictable, must be associated with an energy proportional to \n@ \nP \n=@ \nt \n. The polar form \nof the wave function, which connects the obtained classical equations with the corre- \nsponding quantum equation, emerges as a by-product of the approach.
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.
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.
Euler angles prescribing a body-fixed coordinate system are introduced by the use of the degeneracy of Hartree-Bogolyubov field in space orientation. To express such a degeneracy, a method of pair operators is applied. The Euler angles are quantized so as to recover the breaking of rotation symmetry. Collective rotational and intrinsic excitation modes are separated in a similar way to an RPA treatment. The importance of the non-canonical nature of a transformation to the body-fixed coordinate system is remarked. For simplicity, a single-j shell model with pairing-plus-quadrupole force is adopted.