Abstract
A thorough study of regular and quasi-regular polyhedra shows that the symmetries of these polyhedra identically describe the quantization of orbital angular momentum, of spin, and of total angular momentum, a fact which permits one to assign quantum states at the vertices of these polyhedra assumed as the average particle positions. Furthermore, if the particles are fermions, their wave function is anti-symmetric and its maxima are identically the same as those of repulsive particles, e.g., on a sphere like the spherical shape of closed shells, which implies equilibrium of these particles having average positions at the aforementioned maxima. Such equilibria on a sphere are solely satisfied at the vertices of regular and quasi-regular polyhedra which can be associated with the most probable forms of shells both in Nuclear Physics and in Atomic Cluster Physics when the constituent atoms possess half integer spins. If the average sizes of the constituent particles are known, then the average sizes of the resulting shells become known as well. This association of Symmetry with Quantum Mechanics leads to many applications and excellent results.
Highlights
Geometry in the form of symmetry has been extensively applied in many areas of physics and chemistry
The regular and semi-regular polyhedra were employed to derive the relationship between their symmetries and quantum mechanics
We showed that the symmetries of these polyhedra inherently possess the quantization of l, s, and j expected from Quantum Mechanics for central forces
Summary
Geometry in the form of symmetry has been extensively applied in many areas of physics and chemistry. The relationship of Polyhedral Symmetry to Quantum Mechanics here is on the level of identity Another important feature of the present work which is absent from any other model employing polyhedra [17] is that here the nucleons are not considered point particle, but as particles having finite size as predicted from particle physics for neutrons and protons. The motivation of the present work is to make clear that when one deals with regular and semi-regular polyhedra, Quantum Mechanics is inherently involved. This fact permits us to have a pure quantum mechanical treatment of a problem and at the same time to have a geometrical representation. In particular in nuclear physics, the present work provides a way to unite the two milestone models of the field, the Independent Particle Shell Model and the Collective Model, under one common assumption
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