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arXiv:quant-ph/0211195 · Journal of Physics A Mathematical and General

Classical limit for the scattering of Dirac particles in a magnetic field

G Murgu a, M Moreno

Published Feb 26, 2003• 1 citation
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Abstract

We present a relativistic quantum calculation at first order in perturbation theory of the differential cross section for a Dirac particle scattered by a solenoidal magnetic field. The resulting cross section is symmetric in the scattering angle as those obtained by Aharonov and Bohm (AB) in the string limit and by Landau and Lifshitz (LL) for the non-relativistic case. We show that taking pr 0 |sin (θ/2)|/ ≪ 1 in our expression of the differential cross section it reduces to the one reported by AB, and if additionally we assume θ ≪ 1 our result becomes the one obtained by LL. However, these limits are explicitly singular in as opposed to our initial result. We analyse the singular behaviour in and show that the perturbative Planck limit ( → 0) is consistent, in contrast to those of the AB and LL expressions. We also discuss the scattering in a uniform and constant magnetic field, which resembles some features of QCD.

DOI
10.1088/0305-4470/36/10/314

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We present a relativistic quantum calculation at first order in perturbation theory of the differential cross section for a Dirac particle scattered by a solenoidal magnetic field. The resulting cross section is symmetric in the scattering angle as those obtained by Aharonov and Bohm (AB) in the string limit and by Landau and Lifshitz (LL) for the non-relativistic case. We show that taking pr 0 |sin (θ/2)|/ ≪ 1 in our expression of the differential cross section it reduces to the one reported by AB, and if additionally we assume θ ≪ 1 our result becomes the one obtained by LL. However, these limits are explicitly singular in as opposed to our initial result. We analyse the singular behaviour in and show that the perturbative Planck limit ( → 0) is consistent, in contrast to those of the AB and LL expressions. We also discuss the scattering in a uniform and constant magnetic field, which resembles some features of QCD.
We present a relativistic quantum calculation at first order in perturbation theory of the differential cross section for a Dirac particle scattered by the magnetic field of a solenoid. The resulting cross section is symmetric in the scattering angle as those obtained by Aharonov and Bohm (AB) in the string limit and by Landau and Lifshitz (LL) for the non relativistic case. We show that taking pr_0 sin(theta/2) << 1 in our expression of the differential cross section it reduces to that one reported by AB, and if additionally we assume theta << 1 our result becomes the one obtained by LL. However, these limits are explicitly singular in hbar as opposed to our initial result. We analyse the singular behavior in hbar and show that the perturbative Planck limit (hbar -> 0) is consistent, contrarily to those of the AB and LL expressions.
We present a detailed analysis of the scattering of charged particles by the magnetic field of a long solenoid of constant magnetic flux and finite radius. We study the relativistic and non-relativistic quantum and classical scenarios. The classical limit of the perturbative quantum expressions, understood as the Planck's limit (making ℏ\hbarℏ going to zero) is analyzed and compared with the classical result. The classical cross section shows a general non-symmetric behavior with respect to the scattering angle in contradistinction to the quantum calculations performed so far. The various regimes analyzed show that the quantum cross sections do not satisfy the correspondence principle: they do not reduce to the classical result in any considered limit, an argument in favor of the interpretation of the process as a purely quantum phenomenon. We conclude that in order to restore the classical correspondence of the phenomenon, a complete non-perturbative quantum calculation for a finite solenoid radius is required.
A well known example in quantum electrodynamics (QED) shows that Coulomb scattering of unpolarized electrons, calculated to lowest order in perturbation theory, yields a results that exactly coincides (in the non‐relativistic limit) with the Rutherford formula. We examine an analogous example, the classical and perturbative quantum scattering of an electron by a magnetic field confined in an infinite solenoid of finite radius. The results obtained for the classical and the quantum differential cross sections display marked differences. While this may not be a complete surprise, one should expect to recover the classical expression by applying the classical limit to the quantum result. This turn not to be the case. Surprisingly enough, it is shown that the classical result can not be recuperated even if higher order corrections are included. To recover the classic correspondence of the quantum scattering problem a suitable non‐perturbative methodology should be applied.
We consider the Dirac equation in cylindrically symmetric magnetic fields and find its normal modes as eigenfunctions of a complete set of commuting operators. This set consists of the Dirac operator itself, the zzz-components of the linear and the total angular momenta, and of one of the possible spin polarization operators. The spin structure of the solution is completely fixed independently of the radial distribution of the magnetic field which influences only the radial modes. We solve explicitly the radial equations for the uniform magnetic field inside a solenoid of a finite radius and consider in detail the scattering of scalar and Dirac particles in this field. For particles with low energy the scattering cross section coincides with the Aharonov-Bohm scattering cross section. We work out the first order corrections to this result caused by the fact that the solenoid radius is finite. At high energies we obtain the classical result for the scattering cross section.
We consider the Dirac equation in cylindrically symmetric magnetic fields and find its normal modes as eigenfunctions of a complete set of commuting operators. This set consists of the Dirac operator itself, the z-components of the linear and the total angular momenta, and of one of the possible spin polarization operators. The spin structure of the solution is completely fixed independently of the radial distribution of the magnetic field which influences only the radial modes. We solve explicitly the radial equations for the uniform magnetic field inside a solenoid of finite radius and consider in detail the scattering of scalar and Dirac particles in this field. For particles with low energy the scattering cross section coincides with the Aharonov - Bohm scattering cross section. We work out the first-order corrections to this result caused by the fact that the solenoid radius is finite. At high energies we obtain the classical result for the scattering cross section.
The Aharonov-Bohm scattering of relativistic spin-1∕21∕21∕2 particles with an anomalous magnetic moment are studied. The scattering cross sections for unpolarized and polarized particles are obtained by solving the Dirac-Pauli equation. It is somewhat unexpected that the results are in general the same as those for particles without an anomalous magnetic moment. However, when the incident energy takes some special values, the cross section for polarized particles is dramatically changed. In these cases the helicity of scattered particles is not conserved. In particular, the helicity of particles scattered in the backward direction is all reversed. In the nonrelativistic limit, a very simple relation between the polarized directions of the incident and scattered particles is found, for both general and special incident energies. For particles without an anomalous magnetic moment this relation can be drawn from previous results but it appears to be unnoticed.