Quantum mechanics on spaces of nonconstant curvature: The oscillator problem and superintegrability

Journal Article · Mon Aug 15 04:00:00 EDT 2011 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2011.03.002· OSTI ID:21583315

Ballesteros, Angel ^[1]; Enciso, Alberto ^[2]; Ragnisco, Orlando ^[3]; Riglioni, Danilo ^[3]

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1. Departamento de Fisica, Universidad de Burgos, 09001 Burgos (Spain)
2. Departamento de Fisica Teorica II, Universidad Complutense, 28040 Madrid (Spain)
3. Dipartimento di Fisica, Universita di Roma Tre and Istituto Nazionale di Fisica Nucleare sezione di Roma Tre, Via Vasca Navale 84, 00146 Roma (Italy)

Highlights: > Quantization of Hamiltonians on spaces of nonconstant curvature is addressed. > Our approach is based on superintegrability requirements. > The procedure is applied to a nonlinear classical superintegrable oscillator. > Schroedinger, Laplace-Beltrami and PDM quantizations are worked out. > The quantum system is solved by obtaining the spectrum and the eigenfunctions. - Abstract: The full spectrum and eigenfunctions of the quantum version of a nonlinear oscillator defined on an N-dimensional space with nonconstant curvature are rigorously found. Since the underlying curved space generates a position-dependent kinetic energy, three different quantization prescriptions are worked out by imposing that the maximal superintegrability of the system has to be preserved after quantization. The relationships among these three Schroedinger problems are described in detail through appropriate similarity transformations. These three approaches are used to illustrate different features of the quantization problem on N-dimensional curved spaces or, alternatively, of position-dependent mass quantum Hamiltonians. This quantum oscillator is, to the best of our knowledge, the first example of a maximally superintegrable quantum system on an N-dimensional space with nonconstant curvature.

• OSTI ID: 21583315
• Journal Information: Annals of Physics (New York), Journal Name: Annals of Physics (New York) Journal Issue: 8 Vol. 326; ISSN 0003-4916; ISSN APNYA6
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DEFORMATION
EIGENFUNCTIONS
ELECTRONIC EQUIPMENT
ENERGY
EQUIPMENT
FUNCTIONS
HAMILTONIANS
KINETIC ENERGY
MASS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
NONLINEAR PROBLEMS
OSCILLATORS
QUANTIZATION
QUANTUM MECHANICS
QUANTUM OPERATORS
SPACE
SPECTRA
TRANSFORMATIONS