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Journal of Physics A Mathematical and General

On the classical and quantum Coulomb scattering

Dimitri R. Yafaev

Published Oct 7, 1997• 21 citations
Publisher

Abstract

We develop the scattering theory for the Schrödinger operator with the Coulomb potential in the space of an arbitrary dimension d. In particular, we calculate the scattering matrix and show that its spectrum covers the whole unit circle. We also compute the differential cross section and show that it coincides with the classical Coulomb scattering cross section in the dimension d = 3 only.

DOI
10.1088/0305-4470/30/19/032

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We develop the scattering theory for the Schrödinger operator with the Coulomb potential in the space of an arbitrary dimension d. In particular, we calculate the scattering matrix and show that its spectrum covers the whole unit circle. We also compute the differential cross section and show that it coincides with the classical Coulomb scattering cross section in the dimension d = 3 only.
We treat the nonrelativistic quantum mechanical scattering of a wavepacket (for a single, charged, spinless particle) of well-resolved momentum incident on a fixed Coulomb potential. To implement this, we calculate the incoming and outgoing probability amplitudes for normalized state vectors on the basis of eigenvectors of total momentum magnitude and total angular momentum. We show that the method of partial wave analysis becomes applicable for this long range potential with the use of wavepackets. We find a relation between the finite probability and the differential cross section that is seen to hold sufficiently far from the singularity of the latter in the forward direction. The resulting differential cross section reduces to the familiar Rutherford form for scattering angles sufficiently far from the forward direction. We find that the interaction causes small time shifts in the detection probabilities. For our particular scattering geometry, we see, at low energies, a shadow zone of low probability for small angles around the forward direction, a phenomenon that cannot be predicted by the standard treatment.
We investigate the scattering of a charged quantum particle in a helically twisted background that induces an effective Coulomb-like interaction, in the presence of an Aharonov-Bohm (AB) flux. Starting from the nonrelativistic Schrödinger equation in the twisted metric, we derive the radial equation and show that, after including the AB potential, it can be mapped onto the same Kummer-type differential equation that governs the planar 2D2D2D Coulomb +++ AB problem, with a geometry-induced Coulomb strength and the azimuthal quantum number shifted as m→m−λm\to m-λm→m−λ. We construct the exact scattering solutions, obtain closed expressions for the partial-wave SSS matrix and phase shifts, and derive the corresponding scattering amplitude, differential cross section, and total cross section. We also show that the pole structure of the SSS matrix is consistent with the bound-state quantization previously obtained for the helically twisted Coulomb-like problem.
We establish the general form of the evolution equation and study scattering and tunneling in real quantum mechanics. We prove an analogue of Stone's theorem for a real Kahler space and derive the corresponding general evolution equation, whose evolution operator is both orthogonal and symplectic. We formulate scattering theory in terms of real wave operators and the associated S-matrix. For the class of potentials considered, we show that the differential scattering cross section is equivalent to that obtained in standard complex quantum mechanics. We also show that the tunneling probability in real quantum mechanics is identical to that in complex quantum mechanics.