Canonical transformations relating the oscillator and Coulomb problems and their relevance for collective motions

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Abstract
The present paper can be viewed from two standpoints. The first is that it derives the canonical transformation that takes the Hamiltonian of the Coulomb problem (in the Fock–Bargmann formulation) into that of the harmonic oscillator, while transforming the angular momenta of both problems into each other. The second is the one in which the solution of the previous problem is required if we wish to find the canonical transformation relating microscopic and macroscopic collective models, where the former is derived from a system of A particles moving in two dimensions and interacting through harmonic oscillator forces. The canonical transformation shows the existence of a U(3) symmetry group in the microscopic collective model corresponding to that of the three-dimensional oscillator which is the Hamiltonian of the macroscopic collective model. The importance of this result rests on the fact that had the motion of the particles taken place in the physical three-dimensional space, rather than the hypothetical two-dimensional one discussed here, the symmetry group would have been U(6) rather than U(3). Thus, the group theoretical structure of an s-d boson picture or, equivalently, of a generalized Bohr–Mottelson approach, is present implicitly in an A-body system interacting through harmonic oscillator forces.