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Journal of the Mathematical Society of Japan

Asymptotic behavior of scattering amplitudes in magnetic fields at large separation

Hiroshi T. ITO, Hideo Tamura

Published Jul 1, 2001• 0 citations
Publisher

Abstract

Magnetic potentials have a direct significance to the motion of particles in quantum mechanics. This is known as the Aharonov-Bohm effect. We study this quantum property in the scattering by two magnetic fields at large separation in two dimensions. We derive the asymptotic formula of scattering amplitude when the distance between the centers of two fields goes to infinity. The result depends on the fluxes of fields and on incident and final directions. We here consider only the simple case that at least one of two fluxes is zero.

DOI
10.2969/jmsj/05330645

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Magnetic potentials have a direct significance to the motion of particles in quantum mechanics. This is known as the Aharonov-Bohm effect. We study this quantum property in the scattering by two magnetic fields at large separation in two dimensions. We derive the asymptotic formula of scattering amplitude when the distance between the centers of two fields goes to infinity. The result depends on the fluxes of fields and on incident and final directions. We here consider only the simple case that at least one of two fluxes is zero.
According to the Aharonov–Bohm effect, magnetic potentials have a direct significance to the motion of particles in quantum mechanics. We study this quantum effect through the scattering by several point‐like magnetic fields at large separation in two dimensions. We derive the asymptotic formula for scattering amplitudes as the distances between centers of fields go to infinity. The result heavily depends on the location of centers. A special emphasis is placed on the case of scattering by fields with centers on an even line. The obtained formula depends on fluxes of fields and on ratios of distances between adjacent centers. We figure the approximate values of differential cross sections to see how the pattern of interferences changes with flux parameters.
We study the Aharonov–Bohm (AB) effect through resonances for magnetic scattering in two dimensions. The scattering system consists of three scatterers, one bounded obstacle, and two scalar potentials with compact supports at large separation, where the obstacle is placed between two supports and the support of the magnetic field is completely shielded by the obstacle. The field does not influence particles from a classical mechanical point of view, but quantum particles are influenced by the corresponding vector potential which does not necessarily vanish outside the obstacle. This quantum phenomenon is called the AB effect. The resonances are shown to be generated near the real axis by the trajectories oscillating between two supports of the scalar potentials as the distances between the three scatterers go to infinity. The location is described in terms of the backward amplitudes for scattering by each of the scalar potentials and by the obstacle, and it depends heavily on the magnetic flux of the field.
In this paper we consider the scattering of charged particles by magnetic strings for several configurations of the incident state and of the enclosed flux. We analyze the scattering of a plane wave by an infinite magnetic string and show that the scattering contribution arises from flux-dependent phase shifts followed by quantum diffusion. Then we report that the amplitude for the scattering of a wave packet of finite extension by a magnetic string becomes exponentially small as the width of the packet goes to zero. At the same time the phase at the center of the incident wave packet is progressively shifted by an amount proportional to the enclosed flux, as predicted on the basis of the quasiclassical approximation. We consider further the scattering by a pair of parallel strings carrying opposite fluxes and show that the probability pattern is affected by the magnetic fluxes mainly in the vicinity of a finite surface spanning the two strings. We finally discuss the relevance for the Aharonov-Bohm effect of the multivalued gauge transformations which eliminate the potentials from the field-free region. The results reported in this paper confirm the reality of the quantum effects of remote fluxes, and prove that the state of the electromagnetic field is not completely determined by the local knowledge of the field strengths.
We study the Aharonov-Bohm effect (the AB effect) in quantum resonances for scattering by three solenoids in two dimensions under the situation that the centers of the solenoids are placed almost in line.The resonances are shown to be generated near the real axis by the trapping trajectories when the centers are largely separated from one another.We analyze how the AB effect is reflected in the location of these resonances.The results are described by use of the backward scattering amplitude by one solenoid and depend on the ratio of the distances between the solenoids as well as on the magnetic fluxes.The method makes a full use of the information from the scattering system by one solenoid, which is known to be exactly solvable in quantum mechanics.We also discuss the case of four solenoids.
The exact wave functions that describe scattering of a charged particle by a confined magnetic field (Aharonov-Bohm effect) and by a Coulomb field are analyzed. It is well known that the usual procedure of finding asymptotic forms of these functions which admit a separation into a superposition of an incident plane wave and a circular or spherical scattered wave is problematic near the forward direction. It thus appears to be impossible to express the conservation of probability by means of an optical theorem of the usual kind. Both the total cross section and the forward scattering amplitude appear to be infinite. To address these difficulties we find a new representation for the asymptotic form of the Aharonov-Bohm wave function that is valid for all angles. Rather than try to define a cross section at forward angles, however, we work instead with the probability current and find that it is quite well behaved. The same is true for Coulomb scattering. We trace the usual difficulties to a nonuniformity of limits.
We briefly review the scattering theory for the Schrödinger operator related to the Aharonov-Bohm effect, by exhibiting some explicitly solvable models; an infinitesimally thin solenoid in mathbb{R}^{2}, two infinitesimally thin solenoids with the quantized magnetic fluxes in mathbb{R}^{2}, and an infinitesimally thin ring solenoid with the quantized magnetic flux in mathbb{R}^{3}. We give explicit formulas for the incoming wave solution and the scattering amplitude for these models in terms of some special functions, with the graphs obtained by a numerical method. We also review known high-energy asymptotic formulas related to these examples.