Path integrals with a periodic constraint: The Aharonov–Bohm effect
01 Apr 1981-Journal of Mathematical Physics (American Institute of Physics)-Vol. 22, Iss: 4, pp 715-718
TL;DR: In this article, the Aharonov-Bohm effect is formulated in terms of a constrained path integral, which is explicitly evaluated in the covering space of the physical background to express the propagator as a sum of partial propagators corresponding to homotopically different paths.
Abstract: The Aharonov–Bohm effect is formulated in terms of a constrained path integral. The path integral is explicitly evaluated in the covering space of the physical background to express the propagator as a sum of partial propagators corresponding to homotopically different paths. The interference terms are also calculated for an infinitely thin solenoid, which are found to contain the usual flux dependent shift as the dominant observable effect and an additional topological shift unnoticeable in the two slit interference experiment.
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TL;DR: In this paper, path integral formulations for the Smorodinsky-Winternitz potentials in two-and three-dimensional Euclidean space are presented, where path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave-functions are discussed.
Abstract: 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.
136 citations
TL;DR: In this article, the interaction of point particles in a gauge theory for gravity in 2 + 1 dimensions with particular emphasis on the effects of spin is analyzed, and it is shown that the known space-time solution for spinning sources in Einstein gravity exhibits torsion at the location of the sources.
126 citations
TL;DR: In this paper, an exact path integral treatment of the hydrogen atom was proposed based on the bijective transformation of Kustaanheimo and Stiefel, which reduced the radial path integral for the hydrogen atoms into that for an oscillator in R 3 by one-to-one mapping.
56 citations
TL;DR: In this article, the motion of a nonrelativistic charged particle in a plane, multiply-connected region is studied within the framework of Feynman's pathintegral approach to quantum mechanics.
Abstract: The motion of a nonrelativistic charged particle in a plane, multiply-connected region is studied within the framework of Feynman's path-integral approach to quantum mechanics. In particular, the authors study the simple case when the multiply-connected region is obtained by excluding a disc from the plane. If a nonzero magnetic flux is confined inside the disc, this is shown to include a change in the one-dimensional unitary representation of the fundamental group of the space which enters a proper definition of the path-integral. In this way, a simple explanation of the Aharonov-Bohm effect is shown to arise.
39 citations
TL;DR: In this article, the authors evaluate the quantum propagator for systems with boundaries and topological constraints using the Streit-Hida formulation where the Feynman path integral is realized in the framework of white noise analysis.
Abstract: Using the Streit–Hida formulation where the Feynman path integral is realized in the framework of white noise analysis, we evaluate the quantum propagator for systems with boundaries and topological constraints. In particular, the Feynman integrand is given as generalized white noise functionals for systems with flat wall boundaries and periodic constraints. Under a suitable Gauss–Fourier transform of these functionals the quantum propagator is obtained for: (a) the infinite wall potential; (b) a particle in a box; (c) a particle constrained to move in a circle; and (d) the Aharonov–Bohm system. The energy spectrum and eigenfunctions are obtained in all four cases.
24 citations
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Book•
01 Jan 1965
TL;DR: Au sommaire as discussed by the authors developed the concepts of quantum mechanics with special examples and developed the perturbation method in quantum mechanics and the variational method for probability problems in quantum physics.
Abstract: Au sommaire : 1.The fundamental concepts of quantum mechanics ; 2.The quantum-mechanical law of motion ; 3.Developing the concepts with special examples ; 4.The schrodinger description of quantum mechanics ; 5.Measurements and operators ; 6.The perturbation method in quantum mechanics ; 7.Transition elements ; 8.Harmonic oscillators ; 9.Quantum electrodynamics ; 10.Statistical mechanics ; 11.The variational method ; 12.Other problems in probability.
8,141 citations
TL;DR: In this paper, the authors formulated non-relativistic quantum mechanics in a different way and showed that the probability of an event which can happen in several different ways is the absolute square of a sum of complex contributions, one from each alternative way.
Abstract: Non-relativistic quantum mechanics is formulated here in a different way. It is, however, mathematically equivalent to the familiar formulation. In quantum mechanics the probability of an event which can happen in several different ways is the absolute square of a sum of complex contributions, one from each alternative way. The probability that a particle will be found to have a path x(t) lying somewhere within a region of space time is the square of a sum of contributions, one from each path in the region. The contribution from a single path is postulated to be an exponential whose (imaginary) phase is the classical action (in units of ℏ) for the path in question. The total contribution from all paths reaching x, t from the past is the wave function ψ(x, t). This is shown to satisfy Schroedinger's equation. The relation to matrix and operator algebra is discussed. Applications are indicated, in particular to eliminate the coordinates of the field oscillators from the equations of quantum electrodynamics.
3,678 citations
TL;DR: In this paper, an intrinsic and complete description of electromagnetism in a space-time region is formulated in terms of a nonintegrable phase factor, and connections on principal fiber bundles are identified.
Abstract: Through an examination of the Bohm-Aharonov experiment an intrinsic and complete description of electromagnetism in a space-time region is formulated in terms of a nonintegrable phase factor. This concept, in its global ramifications, is studied through an examination of Dirac's magnetic monopole field. Generalizations to non-Abelian groups are carried out, and result in identification with the mathematical concept of connections on principal fiber bundles.
1,121 citations
TL;DR: In this article, the authors discuss the significance of potentials in the quantum theory and answer a number of arguments that have been raised against the conclusions of their first paper on the same subject.
Abstract: In this article, we discuss in further detail the significance of potentials in the quantum theory, and in so doing, we answer a number of arguments that have been raised against the conclusions of our first paper on the same subject. We then proceed to extend our treatment to include the sources of potentials quantum-mechanically, and we show that when this is done, the same results are obtained as those of our first paper, in which the potential was taken to be a specified function of space and time. In this way, we not only answer certain additional criticisms that have been made of the original treatment, but we also bring out more clearly the importance of the potential in the expression of the local character of the interaction of charged particles and the electromagnetic field.
262 citations
TL;DR: In this article, the use of polar coordinates is examined in performing summation over all Feynman histories, and several relationships for the Lagrangian path integral and the Hamiltonians path integral are derived in the central force problem.
Abstract: Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central‐force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse‐square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.
202 citations