New aspects of the path integrational treatment of the Coulomb potential

Castrigiano, Domenico P. L.; Stark, Freya

doi:10.1063/1.528513

Cited by 18 publications

(13 citation statements)

References 8 publications

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“…This equality can be proven by elementary but lengthy computations using the polygonial approximation technique for path integrals, see [12]. However there is a short proof using stochastic calculus which in addition has the advantage of displaying the universal character of the new-time transformation most clearly.…”

Section: Intrinsic Clock and Equality Of Measuresmentioning

confidence: 94%

“…We omit the well-known formulas here, which can be looked up e.g. in [12], [8]. Quantum Probability and Related Topics Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 08/27/15.…”

Section: Ve(x) := C(s(x))[ie + V(s(x))] W{x) := C(s(x))w(s(x))mentioning

confidence: 99%

“…As a consequence of this one gets an equality of measures on the spaces of paths : the image of the four-dimensional Wiener measure under the Kustaanheimo-Stiefel transformation, which is no longer a Wiener measure, becomes the three-dimensional Wiener measure just by the new-time transformation. This equality of measures can be regarded as the very content of the reduction of the Coulomb path integral to gaussian ones, see [12], [8].…”

Section: Introductionmentioning

confidence: 99%

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General Aspects of the Space and Time Transformations for the Coulomb Propagator

Castrigiano¹,

Staerk²

1991

Quantum Probability and Related Topics

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IntroductionFor a surprisingly long time the problem of evaluating the path integral for the Coulomb propagator remained unsolved. Only in 1979, Duru/Kleinert [1] presented a solution by reducing the Coulomb path integral to gaussian ones. This reduction was achieved by using just the same two transformations of paths which in the mid-sixties had been invented by Kustaanheimo and Stiefel [2] in order to develop a linear and regular celestial mechanics. One of these transformations, the now called Kustaanheimo -Stiefel transformation consists in a rather special change of variables by which the Kepler trajectories are transferred to a four-dimensional configuration space. The other one works by a pathwise time substitution causing a kind of slow motion in the vicinity of the central-body.In the sequel this second kind of transformations, now called (new -)time transformations, became a general tool for the evaluation of path integrals. Employed for the first time by Duru and Kleinert for solving the Coulomb path integral, Pak/Sokmen [3] developed a general new time formalism for path integrals, and Inomata and collaborators [4] showed in a series of papers the applicability of the new-time technique to several relevant potentials other than the Coulomb one. But also the old attempt of solving the Coulomb path integral by exploiting explicitly its radial symmetry -a general partial wave expansion of path integrals using polar coordinates had been provided by Peak/Inomata [5] -finally succeeded by employing the new-time method, see Inomata [6] and Steiner [7], and also [8]. Soon after the solution of the Coulomb problem by Duru and Kleinert it became clear that the new-time transformation has a more fundamental meaning than to be a mere integration technique. Blanchard/Sirugue [9], Blanchard Quantum Probability and Related Topics Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 08/27/15. For personal use only.

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Section: Intrinsic Clock and Equality Of Measuresmentioning

confidence: 94%

Section: Ve(x) := C(s(x))[ie + V(s(x))] W{x) := C(s(x))w(s(x))mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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General Aspects of the Space and Time Transformations for the Coulomb Propagator

Castrigiano¹,

Staerk²

1991

Quantum Probability and Related Topics

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“…Let us assume that the constraint (see also [76]) has for all admissible path a s" will be path dependent. function G ( E ) :…”

Section: Expanding About Midpoints Yieldsmentioning

confidence: 99%

“…Properties of the Kustaanheimo-Stiefel transformation in connection to the Coulomb problem have been discussed by various authors, e. g. BARUT, SCHNEIDER and WILSON [75], CAHILL [48], CASTRIGIANO and STARK [76], CHEN [77], GRINBERG, MARAGON and VUCETICH [78], GARCIA-BONDIA [79], LAMBERT and KIBLER [SO], KIBLER and NEGADI [Sl] and RINCWOOD [82]. See also HOLAS and MARCH [83] for a generalization of the Runge-Lenz vector.…”

Section: ) Cartesian Coordinatesmentioning

confidence: 99%

Coulomb Potentials by Path Integration

Grosche

1992

Fortschr. Phys.

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The path integral for the potential is evaluated in three different coordinate systems, i. e. in cartesian coordinates, in polar coordinates and in parabolic coordinates. The equivalence between the different approaches is shown and in each case the energyspectrum and the wave functions are explicitly calculated. Furthermore we discuss special cases, including the ring‐potential, the Coulomb potential with an Aharonov‐Bohm solenoid, and the genuine Coulomb problem. We also point out the separability of the path integral formulation of the ring‐potential in spheroidal coordinates, and of the Coulomb potential in spheroidal and spheroconical coordinates, respectively.

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

Grosche

1994

Fortschr. Phys.

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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.

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New aspects of the path integrational treatment of the Coulomb potential

Cited by 18 publications

References 8 publications

General Aspects of the Space and Time Transformations for the Coulomb Propagator

General Aspects of the Space and Time Transformations for the Coulomb Propagator

Coulomb Potentials by Path Integration

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

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