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Journal of Mathematical Physics

Rigorous Derivation of the Phase Shift Formula for the Hilbert Space Scattering Operator of a Single Particle

T. A. Green, Oscar E. Lanford

Published Mar 1, 1960• 47 citations
Publisher

Abstract

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.

DOI
10.1063/1.1703644

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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.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. The integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. The relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. The chapter concludes with a consideration of identical particle scattering and an application to the proton–proton case, where a dramatic manifestation of the underlying substructure emerges at high energies.
We derive and explain the key ideas behind a time-dependent formulation of quantum scattering theory, applicable generally to systems with a finite-range scattering potential. The scattering is initiated and probed by plane wave packets, which are localized just outside the range of the potential. The asymptotic limits of conventional scattering theory (initiation in the remote past; detection in the remote future) are not taken. Instead, the differential cross section (DCS) is obtained by projecting the scattered wave packet onto the probe plane wave packets. The projection also yields a time-dependent version of the DCS. Cuts through the wave packet, just as it exits the scattering potential, yield time-dependent and time-independent angular distributions that give a close-up picture of the scattering which complements the DCS. We have previously applied the theory to interpret experimental cross sections of chemical reactions [e.g., S. C. Althorpe, F. Fern\'andez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede, Nature (London) 416, 67 (2002)]. This paper gives the derivation of the theory, and explains its relation to conventional scattering theory. For clarity, the derivation is restricted to spherical-particle scattering, though it may readily be extended to general multichannel systems. We illustrate the theory using a simple application to hard-sphere scattering.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. ƒe integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. ƒe relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. ƒe chapter concludes with a consideration of identical particle scattering and an application to the protonproton case, where a dramatic manifestation of the underlying substructure emerges at high energies.
The elastic scattering of spinless particles from a spherically symmetric stationary target is considered. The main idea is to describe the incident beam of particles as a certain statistical ensemble of one-particle wave packets, and to define the differential cross section in terms of the average scattering of the particles in this ensemble. The physical significance of this mode of description is discussed, and arguments are presented as to why it should be regarded as a realistic description of typical scattering experiments. The discussion is general, and not restricted to potential scattering. It is based on the notion of the S matrix, and the most basic ideas of the S matrix are discussed. A simple derivation of the partial wave expansion is given within the framework of S matrix theory. The aim of this paper is mainly pedagogical, and the final results obtained are old and well established.