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Physical Review A

Three-particle scattering rates and singularities of the T matrix. I.

Vladimir S. Potapov, John R. Taylor

Published Dec 1, 1977• 16 citations
Publisher

Abstract

We construct a quantum ensemble to describe nonrelativistic three-particle scattering, and derive expressions for the observable scattering rates. These expressions are seen to reduce to the familiar expressions involving the modulus squared of the $T$ matrix, provided the final momenta are chosen to avoid the disconnected and double-scattering singularities. We show that our expressions remain finite in the neighborhood of the disconnected singularities, and that the predicted rates are consistent with the physical interpretation of the disconnected processes.

DOI
10.1103/physreva.16.2264

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We evaluate the contributions of double-scattering singularities to three-particle scattering rates. It is shown that these contributions are finite and consistent with the physical interpretation of double-scattering processes. The "double-scattering regions" of momentum space where these processes contribute are calculated, and it is shown that in these regions the double-scattering contributions dominate and the regular part of the $T$ matrix can be neglected.

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Physical Review•Jul 1, 1957•11 citations
We construct a quantum ensemble to describe nonrelativistic three-particle scattering, and derive expressions for the observable scattering rates. These expressions are seen to reduce to the familiar expressions involving the modulus squared of the TTT matrix, provided the final momenta are chosen to avoid the disconnected and double-scattering singularities. We show that our expressions remain finite in the neighborhood of the disconnected singularities, and that the predicted rates are consistent with the physical interpretation of the disconnected processes.
We evaluate the contributions of double-scattering singularities to three-particle scattering rates. It is shown that these contributions are finite and consistent with the physical interpretation of double-scattering processes. The "double-scattering regions" of momentum space where these processes contribute are calculated, and it is shown that in these regions the double-scattering contributions dominate and the regular part of the TTT matrix can be neglected.
In the scattering of three free incident particles, a finite cross section for the double scattering has to be defined differently from the processes other than the double scattering. That is, we want to maintain that, in order to obtain a definite quantity being connected to the experimental observation, a particular definition of the cross section has to be required for the double scattering owing to a particular time dependence of its scattering rate. Our definition differs from the ones given recently by others in a form depending on some macroscopic quantities.1), 2) In the previous paper (refered as I in the following)3), we have attained to a finite cross section by working on the basis of the wave packet formalism, while we want to give here a clear-cut derivation in a simple and familear form.
We have shown earlier how the virial coefficients can be expressed in terms of traces taken over products of multiparticle (on-energy-shell) scattering matrices. The traces are to be taken in the angular-momentum representation instead of the momentum representation to avoid the infinite forward N-particle scattering amplitude for N > 2 caused by certain singular diagrams. Here, we analyze and sum the contribution of the singular diagrams to the third virial coefficient. It is also shown that one can get the same result if off-shell amplitudes in the momentum representation are utilized.
In a recent paper by this author, the singular wave function matrix of Mo/ller was used to obtain the scattering solution for the nonrelativistic Coulomb operator. In the present paper this singular wave function operator is defined differently. This definition is deduced from the relation of the characteristic form of the Green's function to the branch cut in the continuous part of the spectrum of the Green's function. As a consequence of this interpretation, the calculation difficulties of the previous paper do not occur. This method is also used to obtain the scattering solution for the attractive square-well potential.
We study scattering of three equal-mass particles in one dimension. Integrable interactions are synonymous with nondiffractive scattering, meaning that the set of incoming momenta for any scattering event coincides with the set of outgoing momenta. A system is integrable if the two-particle scattering matrix obeys the Yang-Baxter equation. Nonintegrable interactions correspond to diffractive scattering, where the set of outgoing momenta may take on all values consistent with energy and momentum conservation. Such processes play a vital role in the kinetics of one-dimensional gases, where binary collisions are unable to alter the distribution function. When integrability is broken weakly, the result is a small diffractive scattering amplitude. Our main result is a simple formula for the diffractive part of the scattering amplitude, when the violation of the Yang-Baxter equation is small. Although the derivation is given for $\ensuremath{\delta}$-function interactions, the result depends only on the two-particle scattering matrix, and should therefore also apply to finite-range interactions close to integrable.
It is shown that if the wave function of a nonrelativistic particle satisfies the Schr\"odinger equation with a velocity-independent potential, then its scattering amplitudes (and the SSS matrix in general) satisfy the same dispersion formulas as those derived for the scattering of light. In the present derivation, the validity of a perturbation expansion and certain integrability of the potential are assumed, and the requirement of the outgoing wave Green's function replaces the condition of strict causality for the scattering of light. The scattering amplitude for fixed momentum transfer is also shown to satisfy the dispersion relations. Together with the unitarity of the SSS matrix, the complete SSS matrix is determined by the Fourier transform of the potential through an iterative procedure using the dispersion relations. If the potential possesses bound states, then all the dispersion formulas are modified to include residue terms corresponding to singularities on the positive imaginary axis of the momentum plane. The necessity of these modifications is related to the divergence of the perturbation series for small wave numbers. Essential singularities of the SSS matrix due to exponentially damped potentials give no additional contribution to the dispersion formulas.