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Journal of Mathematical Physics

Scattering of a beam of particles by a potential

John D. Dollard, Charles N. Friedman

Published Aug 1, 1979• 2 citations
Publisher

Abstract

We discuss the potential scattering into a cone C in position space of a beam of particles with localized momenta distributed over a set K, and the derivation from time-dependent scattering theory of the formula ∫Kdk̃∫CdΩ σk̃(Ω) [where σk̃(Ω) is the scattering cross section] as a measure of the probability of this scattering. We make use in our discussion of a result of Dollard giving the probability of scattering into a cone for a single wavepacket.

DOI
10.1063/1.524257

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We discuss the potential scattering into a cone C in position space of a beam of particles with localized momenta distributed over a set K, and the derivation from time-dependent scattering theory of the formula ∫Kdk̃∫CdΩ σk̃(Ω) [where σk̃(Ω) is the scattering cross section] as a measure of the probability of this scattering. We make use in our discussion of a result of Dollard giving the probability of scattering into a cone for a single wavepacket.
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.
In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.
In time-dependent scattering theory, the collision problem is described by assuming that a particle was localized outside the interaction domain where V = 0. After collision, there exists a definite probability that the particle is deflected from its impact direction at a certain scattering angle. After collision, there exists a definite probability that the particle is deflected from its impact direction at a certain scattering angle. This probability cannot be directly observed in experiments but is indirectly accessible to measurements through differential cross sections. The unitarity of the operator S is one of the most important properties of the S-matrix theory of scattering. Of course, the study of the S-matrix will greatly be facilitated by the knowledge that the S-operator belongs to a narrower class of the unitary operators.