Path Integration and Separation of Variables in Spaces of Constant Curvature in Two and Three Dimensions

Grosche, Christian

doi:10.1002/prop.2190420602

Cited by 11 publications

(18 citation statements)

References 102 publications

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“…which fixes the functions g i . Introducing the (new) momentum operators P ξ i =h i (∂ξ i + 1 2 f ′ i /f i ) we obtain according to the general theory the following identity in the path integral [58] by means of the space-time transformation technique (Duru [28], Fischer et al [40], Refs. [59,61], Kleinert [85], and Pak and Sökmen [110]…”

Section: Separation Of Coordinates In the Path Integralmentioning

confidence: 99%

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space

1995

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Path integral formulations for the Smorodinsky‐Winternitz potentials in two‐ and three‐dimensional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky‐Winternitz potentials and state the corresponding path integral formulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave‐functions.

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Section: Separation Of Coordinates In the Path Integralmentioning

confidence: 99%

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space

1995

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“…[19]- [22] and references therein) studied the quantum oscillator on curved spaces. We also mention that the path integral formulation has been also studied in curved spaces [23]- [24].…”

Section: Introductionmentioning

confidence: 99%

The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach

Cariñena

Rañada

Santander

2011

Journal of Mathematical Physics

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The quantum free particle on the sphere S 2 κ (κ > 0) and on the hyperbolic plane H 2 κ (κ < 0) is studied using a formalism that considers the curvature κ as a parameter. The first part is mainly concerned with the analysis of some geometric formalisms appropriate for the description of the dynamics on the spaces (S 2 κ , IR 2 , H 2 κ ) and with the the transition from the classical κ-dependent system to the quantum one using the quantization of the Noether momenta. The Schrödinger separability and the quantum superintegrability are also discussed. The second part is devoted to the resolution of the κ-dependent Schrödinger equation. First the characterization of the κ-dependent 'curved' plane waves is analyzed and then the specific properties of the spherical case are studied with great detail. It is proved that if κ > 0 then a discrete spectrum is obtained. The wavefunctions, that are related with a κ-dependent family of orthogonal polynomials, are explicitly obtained.Keywords: Quantization. Quantum mechanics on Spaces of constant curvature. Plane waves on spaces of constant curvature. Orthogonal polynomials.Running title: The quantum free particle on spaces with curvature.

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“…In comparison to Ref. [36] the notation and notion of the coordinate systems will be improved. In the third and in the fourth Section we discuss the Smorodinsky-Winternitz potentials on the two-and three-dimensional sphere, respectively.…”

Section: Introductionmentioning

confidence: 99%

Path Integral Discussion for Smorodinsky-Winternitz Potentials: II. The Two- and Three-Dimensional Sphere

1995

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Steps towards path integral formulations for Smorodinsky‐Winternitz potentials, respectively systems with accidental degeneracies, on the two‐ and three‐dimensional sphere, and a complete classification of super‐integrable systems on spaces of constant curvature are presented. We mention all coordinate systems which separate the Smorodinsky‐Winternitz potentials on a sphere, and state the corresponding path integral formulations. Whereas in many coordinate systems explicit path integral solutions are not possible, we list in all soluble cases the path integral in terms of the propagator, respectively the spectral expansion into the wave functions and the energy spectrum.

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Path Integration and Separation of Variables in Spaces of Constant Curvature in Two and Three Dimensions

Cited by 11 publications

References 102 publications

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space

The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach

Path Integral Discussion for Smorodinsky-Winternitz Potentials: II. The Two- and Three-Dimensional Sphere

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