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- W2186660596 abstract "A superintegrable Hamiltonian system is a physical system which exhibits maximal symmetry, though the symmetry may be of a very subtle nature. In n dimensions, a system is superintegrable if it admits 1 2n constants of the motion. Such a system may be either classical, in which there exist 1 2n integrals of the motion, functions of position and momentum which are polynomial in the momenta, or quantum, in which the constants of the motion are commuting observables, or symmetry operators. The salient feature of a superintegrable system is that it may be solved algebraically by considering the algebra generated by the constants of the motion. We give examples of such systems, including the Kepler-Coulomb system and its quantum analogue in various geometries and show how superintegrability leads to exact algebraic solutions for the classical trajectories and the quantum energy levels. In particular, we use superintegrability methods to analyze and obtain an energy spectrum for a quantum system in a spherical geometry whose Hamiltonian approaches the Kepler-Coulomb in a particular flat space limit. We show explicitly that this energy spectrum is identical to the hydrogen atom in the same limit. Finally, we note a surprising result in that the (real) energy spectrum of our hydrogen system admits certain complex potentials. This relates our problem of study to the emergent field of PT -symmetric quantum mechanics, which broadens the definition of the Hamiltonian operator by dropping the conventional Hermiticity requirement H H= † in favor of a weaker condition that the Hamiltonian be invariant under simultaneous parity and time reversal, while maintaining the requirement that the spectrum of the Hamiltonian be real." @default.
- W2186660596 created "2016-06-24" @default.
- W2186660596 creator A5045950971 @default.
- W2186660596 date "2011-01-01" @default.
- W2186660596 modified "2023-09-24" @default.
- W2186660596 title "Analysis of a Hydrogen Atom Analgoue in Non-Euclidean Space of Constant Curvature using Superingrability Methods" @default.
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