Coulomb Potentials by Path Integration

1994

Annalen der Physik

Abstract. The incorporation of two-and three-dimensional δ-function perturbations into the path-integral formalism is discussed. In contrast to the onedimensional case, a regularization procedure is needed due to the divergence of the Green-function G (V ) (x, y; E), (x, y ∈ IR 2 , IR 3 ) for x = y, corresponding to a potential problem V (x). The known procedure to define proper self-adjoint extensions for Hamiltonians with deficiency indices can be used to regularize the path integral, giving a perturbative approach for δ-function perturbations in two and three dimensions in the context of path integrals. Several examples illustrate the formalism.

Section: One-dimensional Examplesmentioning

confidence: 92%

Path integrals for two‐ and three‐dimensional δ‐function perturbations

1994

Annalen der Physik

“…We proceed to the time-transformed path integral K (K III ) (s ′′ ) which has the form Here,k 2 1 = k 2 1 − 2mβE/ 2 ,k 2 2 = k 2 2 − 2mγE/ 2 ,α = α 2 − α 1 E. This path integral for the Coulomb potential has been discussed extensively in literature and the solution in terms of the Green function has been obtained by many authors, e.g. [4,16,9,17,27]. We obtain for the Green function in polar coordinates (λ 1 = 2n ϕ ±k 1 ±k 2 + 1,…”

Section: Koenigs-space K III With Coulomb-potentialmentioning

confidence: 99%

Path integral approach for spaces of nonconstant curvature in three dimensions

2007

Phys. Atom. Nuclei

Self Cite

In this contribution a path integral approach for the quantum motion on three-dimensional spaces according to Koenigs, for short"Koenigs-Spaces", is discussed. Their construction is simple: One takes a Hamiltonian from three-dimensional flat space and divides it by a three-dimensional superintegrable potential. Such superintegrable potentials will be the isotropic singular oscillator, the Holt-potential, the Coulomb potential, or two centrifugal potentials, respectively. In all cases a non-trivial space of non-constant curvature is generated. In order to obtain a proper quantum theory a curvature term has to be incorporated into the quantum Hamiltonian. For possible bound-state solutions we find equations up to twelfth order in the energy E.

“…In order to avoid cumbersome notation, we restrict ourselves to the one-dimensional case. For the general case we refer to DeWitt [18], Fischer, Leschke and Müller [33], Gervais and Jevicki [36], [50,53,55,57], Junker [67], Kleinert [72,73], Pak and Sökmen [91], Steiner [104] and Storchak [106], and references therein. We consider the one-dimensional path integral…”

Section: Formulation Of the Path Integralmentioning

confidence: 99%

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

1994

Fortschr. Phys.

In this paper path integration in two‐ and three‐dimensional spaces of constant curvature is discussed: i.e. the flat spaces R2 and R3, the two‐ and three‐dimensional sphere and the two‐ and three‐dimensional pseudosphere. We are going to discuss all coordinates systems where the Laplace operator admits separation of variables. In all of them the path integral formulation will be stated, however in most of them an explicit solution in terms of the spectral expansion can be given only on a formal level. What can be stated in all cases, are the propagator and the corresponding Green function, respectively, depending on the invariant distance which is a coordinate independent quantity. This property gives rise to numerous identities connecting the corresponding path integral representations and propagators in various coordinate systems with each other.