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arXiv:quant-ph/0411199 · Physics of Particles and Nuclei

Path integration on Darboux spaces

C. Grosche

Published May 1, 2006• 4 citations
arXivHTMLPDF

Abstract

In this paper, the Feynman path integral technique is applied to two-dimensional spaces of nonconstant curvature: these spaces are called Darboux spaces D I-D IV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases; the exceptions being the quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified Pöschl-Teller potential, and the spheroidal wave functions. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green’s functions and the expansions into the wave functions. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.

DOI
10.1134/s1063779606030051

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In this paper, the Feynman path integral technique is applied to two-dimensional spaces of nonconstant curvature: these spaces are called Darboux spaces D I-D IV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases; the exceptions being the quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified Pöschl-Teller potential, and the spheroidal wave functions. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green’s functions and the expansions into the wave functions. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.
In this paper the Feynman path integral technique is applied to two-dimensional spaces of non-constant curvature: these spaces are called Darboux spaces DI\\DIDI--DIV\\DIVDIV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases, the exceptions being quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified P\\"oschl--Teller potential, and for spheroidal wave-functions, respectively. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green's functions and the expansions into the wave-functions, respectively. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.
In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant curvature, i.e., three-dimensional Darboux spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that, in the two three-dimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In D 3d-I, we find seven coordinate systems which separate the Schrödinger equation. For the second space, D 3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger equation in D 3d-II. I solve the path integral on D 3d-I in the (u, v, w) system and on D 3d-II in the (u, v, w) system and in spherical coordinates.
Towards the Classification of Exactly Solvable Feynman Path Integrals: δδδ-Function Perturbations and Boundary-Problems as Miscellaneous Solvable Models
Invited talk given at the ``International Workshop on `Symmetry Methods in Physics' in memory of Ya.\ A.\ Smorodinsky, 5--10 July 1993, Dubna, Russia; to appear in the proceedings. In this contribution I present further results on steps towards a Table of Feynman Path Integrals. Whereas the usual path integral solutions of the harmonic oscillator (Gaussian path integrals), of the radial harmonic oscillator (Besselian path integrals), and the (modified) Pöschl-Teller potential(s) (Legendrian path integrals) are well known and can be performed explicitly by exploiting the convolution properties of the various types, a perturbative method opens other possibilities for calculating path integrals. Here I want to demonstrate the perturbation expansion method for point interactions and boundary problems in path integrals.
Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.
Abstract A central coordinate system is investigated within the framework of the Wigner-Dunkl deformation of quantum mechanics, treated using Feynman’s path integral approach. This approach involves performing path integration of a Lagrangian in polar coordinates, with an effective potential term that depends on both radial and angular coordinates. The success in separating variables within the two-dimensional Dunkl system facilitated obtaining exact expressions for the propagators associated with the free particle, inverse square potential, harmonic oscillator, and Coulomb potential. The energy eigenvalues and their corresponding wave functions are also determined.
In this lecture a short introduction is given into the theory of the Feynman path integral in quantum mechanics. The general formulation in Riemann spaces will be given based on the Weyl- ordering prescription, respectively product ordering prescription, in the quantum Hamiltonian. Also, the theory of space-time transformations and separation of variables will be outlined. As elementary examples I discuss the usual harmonic oscillator, the radial harmonic oscillator, and the Coulomb potential. Lecture given at the graduate college ''Quantenfeldtheorie und deren Anwendung in der Elementarteilchen- und Festkörperphysik'', Universität Leipzig, 16-26 November 1992.