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Journal of Physics A Mathematical and General

On the most general boundary conditions for the Aharonov-Bohm scattering of a Dirac particle: helicity and Aharonov-Bohm symmetry conservation

Vanilse S. Araujo, Francisco Antônio Bezerra Coutinho, J. Fernando Perez

Published Oct 15, 2001• 11 citations
Publisher

Abstract

We show that the Hamiltonian H and the helicity operator Λ of a Dirac particle moving in two dimensions in the presence of an infinitely thin magnetic flux tube each admit a four-parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BCs) to be satisfied by the eigenfunctions at the origin. Although the actions of these two operators commute before specification of BCs, to ensure helicity conservation it is not sufficient to take the same BCs for both operators. We show that, given certain relations between the parameters of the extensions, it is possible to write down the most general domain where both operators H and Λ are self-adjoint with helicity conservation and also Aharonov-Bohm symmetry (ϕ→ϕ + 1) preserved, where ϕ is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results imply that neither helicity conservation nor Aharonov-Bohm symmetry by itself solves the problem of choosing the `physical' BCs for this system.

DOI
10.1088/0305-4470/34/42/310

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We show that the Hamiltonian H and the helicity operator Λ of a Dirac particle moving in two dimensions in the presence of an infinitely thin magnetic flux tube each admit a four-parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BCs) to be satisfied by the eigenfunctions at the origin. Although the actions of these two operators commute before specification of BCs, to ensure helicity conservation it is not sufficient to take the same BCs for both operators. We show that, given certain relations between the parameters of the extensions, it is possible to write down the most general domain where both operators H and Λ are self-adjoint with helicity conservation and also Aharonov-Bohm symmetry (ϕ→ϕ + 1) preserved, where ϕ is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results imply that neither helicity conservation nor Aharonov-Bohm symmetry by itself solves the problem of choosing the `physical' BCs for this system.
It is well known that the formal Aharonov–Bohm Hamiltonian operator, describing the interaction of a charged particle with a magnetic vortex, has a four-parameter family of self-adjoint extensions, which reduces to a two-parameter family if one requires that the Hamiltonian commutes with the angular momentum operator. The question we study here is which of these self-adjoint extensions can considered as limits of regularized Aharonov–Bohm Hamiltonians, that is Pauli Hamiltonians in which the magnetic field corresponds to a flux tube of nonzero diameter. We show that not all the self-adjoint extensions in this two-parameter family can be obtained by these approximations, but only two one-parameter subfamilies. In these two cases we can choose the gyromagnetic ratio in the approximating Pauli Hamiltonian in such a way that we get convergence in the norm resolvent sense to the corresponding self-adjoint extension.
The formulation of Berry for the Aharonov-Bohm effect is generalized to the relativistic regime. Then, the problem of finding the self-adjoint extensions of the (2+1)-dimensional Dirac Hamiltonian, in an Aharonov-Bohm background potential, is solved in a novel way. The same treatment also solves the problem of finding the self-adjoint extensions of the Dirac Hamiltonian in a background Aharonov-Casher. PACS: 03.65.Bz, 03.65.Pm Typeset using REVTEX 1 A novel systematics for the computation of the energy eigenfunctions of a spinless electrically charged nonrelativistic particle in the presence of an Aharonov-Bohm [1] external potential, was put forward by Berry [2] sometime ago. We propose here a generalization of this systematics to the case where the particle is a relativistic electron. As will be shown, the set of eigenfunctions found through this generalization serves as a basis for writing any function in the domain of self-adjointness of the corresponding Dirac Hamiltonian. This corroborates the correctness of our proposal, which can also be used to obtain the selfadjoint extensions of the Dirac Hamiltonian in an Aharonov-Casher [3] background field. The extension of Berry’s ideas to the relativistic regime is the original contribution in this work. To make this paper self contained, we start by reviewing the main features of Berry’s formulation [2]. As known, the Hamiltonian operator H(⃗r, ⃗p) describing the dynamics of a spinless two-dimensional nonrelativistic particle (mass m and electric charge e) in the presence of an external static magnetic field B (B = ⃗ ∇ × ⃗ ( A) is H(⃗r, ⃗p) = H0 ⃗r, ⃗p − e ⃗ A(⃗r)) where ⃗r ≡ (x1, x2) are the particle coordinates, ⃗p ≡ (p1, p2) are the corresponding canonically conjugate momenta and H0(⃗r, ⃗p) is the free particle Hamiltonian. The energy eigenfunctions of the operator H can formally be constructed in terms of those of H0 as follows [4] ψ(⃗r) = ψ0(⃗r) exp ie ⃗r
The most general admissible boundary conditions are derived for an idealized Aharonov–Bohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on self-adjoint extensions yields a four-parameter family of boundary conditions; the other two parameters of the model are the Aharonov–Bohm flux and the homogeneous magnetic field. The generalized boundary conditions may be regarded as a combination of the Aharonov–Bohm effect with a point interaction. Spectral properties of the derived Hamtonians are studied in detail.
We consider the question of the physically correct choice of the boundary conditions, and therefore of the dynamics, for the scattering of spin-\textonehalf{} particles by a thread of magnetic flux. It is shown that even when the problem is approached by first considering a source of finite radius RRR, the resulting dynamics is a consequence of how much penetration of the wave function inside the tube one chooses to allow. (The indeterminacy in the boundary condition occurs for one value of the total angular momentum for a given value of the flux.) If a Coulomb interaction of the particle with the sources is introduced in the problem, the above-mentioned indeterminacy is enlarged. We show that for a given value of the flux there are now at least two values of the angular momentum nnn for which the indeterminacy occurs.
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.