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Wavepacket Scattering Theory for the Coulomb Potential

Scott E. Hoffmann

Published Apr 14, 2017• 0 citations
OpenAlex

Abstract

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.

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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.
Abstract We calculate the nonrelativistic scattering of a wavepacket from a Coulomb potential and find deviations from the Rutherford formula in all cases. These generally occur only at low scattering angles, where they would be obscured by the part of the incident beam that emerges essentially unscattered. For a model experiment, the scattering of helium nuclei from a thin gold foil, we find the deviation region is magnified for low incident energies (in the keV range), so that a large shadow zone of low probability around the forward direction is expected to be measurable. From a theoretical perspective, the use of wavepackets makes partial wave analysis applicable to this infinite-range potential. It allows us to calculate the everywhere finite probability for a wavepacket to wavepacket transition and to relate this to the differential cross section. Time delays and advancements in the detection probabilities can be calculated. We investigate the optical theorem as applied to this special case.
A previous paper [J. Phys. B: At. Mol. Opt. Phys. 50, 215302 (2017)] showed that partial wave analysis becomes applicable to nonrelativistic Coulomb scattering if wavepackets are used. The scattering geometry considered was special: that of a head-on collision between the wavepacket and the centre of the potential. Our results predicted, in this case, a shadow zone of low probability for small angles around the forward direction for the description of alpha scattering from a gold foil. In this paper we generalize the results to the case of a nonzero impact parameter, a displacement of the wavepacket centre perpendicular to the average momentum direction. We predict a large flux in the forward direction from events with large impact parameters. We find a significant probability of scattering into the deviation region for impact parameters of order the spatial width of the wavepacket. Averaging over impact parameters produces predictions in excellent agreement with the Rutherford formula down to lower angles than for the zero impact parameter prediction. We consider issues that would arise in a real experiment and discuss the possibility of measuring a deviation from the Rutherford formula.
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.
A wave-packet formalism is presented for obtaining the differential cross section for the elastic scattering of a single particle by a potential. The method avoids difficulties inherent in the more common formulations such as interference between incident and scattered waves, box normalization, and density of final states. The time sequence of events in the scattering process is pictured in an intuitively appealing manner. The initial form of the wave packet is specified explicitly, and the time development of the wave packet is determined from the time-translation operator. The asymptotic form of the wave packet is investigated after a sufficiently long time lapse, and the cross section is obtained from a calculation of particle flux densities.
For a single nonrelativistic particle moving in a spherically symmetric potential, the existence of the Hilbert space wave operators and S operator is proved and phase shift formulas for these operators are deduced. The probability, P(Ω), for scattering into the solid angle Ω is obtained from the time dependent theory. The relation between P(Ω) and the R matrix of the standard plane wave formulation of scattering theory is established. For collimated incoming packets, it is shown that P(Ω) can be expressed as an energy average of the differential cross section.
A simple plane wave solution of the Schrödinger–Helmholtz equation is a quantum eigenfunction obeying both energy and linear momentum correspondence principles. Inclusion of the outgoing wave with scattering amplitude f obeys unitarity and the optical theorem. By closely considering the standard asymptotic development of the plane wave, we show that there is a problem with angular momentum when we consider forward scattering at the point of closest approach and at large impact parameter given semiclassically by ( l + 1/2)/ k where l is the azimuthal quantum number and may be large (J Leech et al , Phys. Rev. Lett. 88 257901 (2002)). The problem is resolved via non-uniform, non-standard analysis involving the Heaviside step function, unifying classical, semiclassical and quantum mechanics, and the treatment is extended to the case of pure Coulomb scattering.