In this paper we study the stationary scattering problem of the Aharonov–Bohm (AB) effect. To achieve this goal we construct a Hamiltonian from the most general representations of the Heisenberg algebra. Such representations are defined on a non-simply-connected manifold which we set as the flat circular annulus. By means of the von Neumann's self-adjoint extensions formalism, the scattering data are then provided. No solenoid is considered in this paper. The corresponding Hamiltonian is based on a topological quantum degree of freedom inherent in such representations. This variable stands for the magnetic vector gauge potential at quantum level. Our outcomes confirm the topological nature of this effect.
In this paper we study the stationary scattering problem of the Aharonov–Bohm (AB) effect. To achieve this goal we construct a Hamiltonian from the most general representations of the Heisenberg algebra. Such representations are defined on a non-simply-connected manifold which we set as the flat circular annulus. By means of the von Neumann's self-adjoint extensions formalism, the scattering data are then provided. No solenoid is considered in this paper. The corresponding Hamiltonian is based on a topological quantum degree of freedom inherent in such representations. This variable stands for the magnetic vector gauge potential at quantum level. Our outcomes confirm the topological nature of this effect.
We consider the Hamiltonian operator associated with planar sections of infinitely long cylindrical solenoids and with a homogeneous magnetic field in their interior.First, in the Sobolev space H 2 , we characterize all generalized boundary conditions on the solenoid border compatible with quantum mechanics, i.e. the boundary conditions, so that the corresponding Hamiltonian operators are self-adjoint.Then we study and compare the scattering of the most usual boundary conditions, that is, Dirichlet, Neumann and Robin.
The Aharonov–Bohm (AB) effect demonstrates that a charged particle can acquire an observable phase shift in a region where the local magnetic field identically vanishes provided the configuration space is multiply connected and the electromagnetic connection exhibits non-trivial holonomy. We present a gauge-covariant physico-mathematical framework that systematically disentangles three regimes: (i) the standard ideal AB effect governed by the holonomy of a flat but globally inexact connection on ΩR=R2∖DR with DR={r≤R}; (ii) the path-integral decomposition into homotopy sectors indexed by the winding number, which is constructed on the universal cover; and (iii) non-topological corrections arising from evanescent wave-function penetration when the solenoid is modeled as a finite radial barrier. The ideal AB phase is taken as the reference topological holonomy, while the finite-barrier extension developed here separates the measured phase, under explicit weak-penetration assumptions, as Δφtotal=ΔφABhol+δφfinite+O(Teff2). Here, ΔφABhol=qΦ/ħ is the standard topological holonomy, whereas δφfinite=O(e−2κℓb) is a small dynamical, microphysics-dependent correction controlled by the barrier height, particle energy, and effective forbidden length ℓb. Within Laskin’s fractional quantum mechanics, we use the fractional extension as a diagnostic test: the topological AB phase is independent of the Lévy index α, whereas the finite-barrier correction acquires an anomalous, α-dependent penetration length. The numerical material provides gauge-covariant diagnostics: a Peierls-substituted split-operator scheme, a sector-resolved Monte Carlo path integral benchmarked against the exact Aharonov–Bohm propagator of Gerry and Singh, a WKB/transfer-matrix finite-barrier estimate tracking δφfinite→0, and Berry-phase computations used as consistency checks.
The celebrated Aharonov-Bohm effect is perhaps the first example in which the the interplay between classical topology and quantum theory was explored. This connection has continued to shed light on diverse areas of physics like quantum statistics, anomalies, condensed matter physics, and gauge theories. Several attempts were made to generalize the Aharonov-Bohm effect by modifying the simple solenoidal current distribution used by them to the case of multiple solenoids, and a toroidal solenoid, for example. A particularly ambitious task is to confine the magnetic flux to the interior of a knotted solenoid. While it is to be expected that a non-trivial phase factor will be picked up by the wave function of a charged particle travelling in the complement of the knot in three-dimensional space, the lack of symmetry defied attempts to explicitly solve the associated scattering problem. In this paper we report on a way to make progress towards this problem based on multipole expansions. The vector potential produced by a knot is obtained by making a multipole expansion, which is then used to calculate the S-matrix for the scattering of the charged particle, in the Born approximation. It is found that the S-matrix carries an imprint of the knottedness at the octopole order. For the case of a torus knot, a curious factorization property is seen to hold.
We add a scalar potential to the 2D Aharonov-Bohm (AB) model which properly diverges both at the solenoid border and at infinity so that the resulting operator is essentially self-adjoint and has a discrete spectrum; the former property is interpreted as no contact of the particle with the solenoid border since there is no need of boundary conditions. We study gauge transformations to get the usual periodic behavior of the AB properties as a function of the magnetic flux. The presence of the AB effect is proven through the ground state energy, which is shown to be smooth in case it is simple and with a nonzero derivative if the ground state is real valued; such properties are verified in the case of circular solenoids, for which it is shown to be a nonconstant periodic function with a minimum at integer and a maximum at half-integer circulations (at half-integer circulations, it is doubly degenerated).
This thesis reviews fibre-bundle theory for the purpose of showing its equivalence to gauge theory and, in particular, utilizing fibre bundles to explain the origin of a quantum mechanical effect named the Aharonov-Bohm (AB) effect. Initially, the mathematical preliminaries necessary to review fibre-bundle theory are presented. This includes topological manifolds, smooth manifolds, vector fields and 1-forms on smooth manifolds as well as Lie groups and their Lie algebras. Thereafter, fibre bundles are introduced. In particular, principal bundles, connections on principal bundles, parallel transport and associated bundles are studied. The electrons in the setup of the AB effect experience the effect of an electromagnetic potential and form an interference pattern on the screen they hit. The AB effect is described by a principal U(1)-bundle over R2\{0} and an associated complex line bundle with bundle space P xρ C, where P is the bundle space in the principal bundle and ρ a representation of U(1) on C defined ρ(x) = xq, for x in U(1) and q in Z. The vector potential A, on a simply connected subset U ⊂ R2\{0}, is given by A = ∇χU, for a smooth real-valued function χU. The complex scalar field, i.e. wave function, ψU: U → C of an electron with charge q has a local solution ψU = eiqχUψ0, where ψ0 satisfies the free Schrödinger equation. Consequentially, the holonomy is eiqΦ, where Φ is the total flux of the magnetic field.