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arXiv:hep-th/9612058 · Europhysics Letters (EPL)

Whirling waves and the Aharonov-Bohm effect for relativistic spinning particles

H. O Girotti, F. Fonseca Romero

Published Jan 20, 1997• 6 citations
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Abstract

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 an Aharonov-Casher background.

DOI
10.1209/epl/i1997-00127-x

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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

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C. R. Hägen, S. Ramaswamy

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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 15, 1990•11 citations
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 an Aharonov-Casher background.
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
A solution of Schrodinger's equation for a particle in a magnetic field B can be obtained from the wavefunction when B=0 by Dirac's prescription of multiplication by a phase factor. But this solution is often multiple-valued and hence unsatisfactory. It is shown that in the case of the Aharonov-Bohm effect the Dirac prescription can nevertheless be made to yield the single-valued exact wavefunction, provided it is applied not to the total wave with B=0 but to its separate components in a 'whirling-wave' representation. The mth whirling wave at a point r is a contribution that has arrived at r after travelling m times around the region containing B.
The Aharonov-Bohm scattering of massive spin-one particles is computed. Starting from the relativistic wave equation for a charged, massive spin-one particle, a reduction to two spatial dimensions is made, consistent with the standard Aharonov-Bohm (AB) geometry. It is found that the problem consists of the independent scattering of the three vector-meson spins. While the amplitude for the spin-0 mode is trivially identical to the AB amplitude, the same result is obtained for the spin-up and spin-down components, in sharp contrast to the corresponding result for the spin-1/2 AB amplitude. This can be traced to the fact that the Hamiltonian is unbounded as a consequence of the presence of a \ensuremath{\delta}-function term in the potential. In the Galilean limit it is found that the amplitude of the nonzero-spin components is identical to the spin-1/2 amplitude in that limit, in agreement with a general claim made earlier by one of the authors. The Galilean limit of the spin-one AB cross section is found to be modified and calculations for some pure-state polarizations are presented.