Breaking so.(4) symmetry without degeneracy lift
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We consider on S (Fourmula Presented) the quantum motion of a scalar particle of mass m, perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, ℓ, the 3D orbital angular momentum, and another, an external scale introducing continuous parameter, B. We show that a loss of the geometric hyperspherical so.(4) symmetry of the free motion can occur that leaves intact the unperturbed N2fold degeneracy patterns, with (Fourmula Presented) and n denoting the nodes of the wave function. Our point is that although the number of degenerate states for any N matches dimensionality of an irreducible so.(4) representation space, the corresponding set of wave functions do not transform irreducibly under any so.(4). Indeed, in expanding the Scarf I wave functions in the basis of properly identified so.(4) representation functions, we find power series in the perturbation parameter, B, where 4D angular momenta (Fourmula Presented) contribute up to the order(Fourmula Presented) In this fashion, we work out an explicit example on a symmetry breakdown by external scales that retains the degeneracy. The scheme extends to so.(d %2b 2) for any d. © Springer Japan 2014.
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Angular momentum; Continuous parameters; Degenerate state; Dimensionless parameters; External scale; Perturbation parameters; Power series; Quantum motions; Representation space; Wave functions
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