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Transport Theory and Statistical Physics

Scattering theory in three-body kinetic equations

J. T. Lowry, William C. Schieve

Published Jan 1, 1971• 3 citations
Publisher

Abstract

The relationship between the collision kernel of the three-body quantum mechanical kinetic equation (quantum version of the Choh-Uhlenbeck equation) and modern scattering theory is examined utilizing the work of Faddeev and Weinberg. Recognizing that the collision kernel is related to the M⊘ller scattering operator it is shown that the collision kernel is “well-defined” for suitable interaction potentials and contains only connected three particle collisions in the sense of Weinberg. The collision kernel is interpreted physically and is shown to contain a generalization of the ordinary Boltzmann collision term to three particle processes plus a distorted-wave-impulse approximation correction to the two particle Boltzmann collision term.

DOI
10.1080/00411457108231448

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The relationship between the collision kernel of the three-body quantum mechanical kinetic equation (quantum version of the Choh-Uhlenbeck equation) and modern scattering theory is examined utilizing the work of Faddeev and Weinberg. Recognizing that the collision kernel is related to the M⊘ller scattering operator it is shown that the collision kernel is “well-defined” for suitable interaction potentials and contains only connected three particle collisions in the sense of Weinberg. The collision kernel is interpreted physically and is shown to contain a generalization of the ordinary Boltzmann collision term to three particle processes plus a distorted-wave-impulse approximation correction to the two particle Boltzmann collision term.
We derive some new equations for the wavefunction for the three-body scattering problem using the ``partial-wavefunction approach.'' All scattering processes are solutions of inhomogeneous coupled integral equations for the scattered parts of the total wavefunction. We show how the Pauli principle is incorporated when some or all of the particles are identical. This leads to a reduction in the number of independent scattered partial wavefunctions when some of the particles are identical. The kernels in the integral equations contain no δ functions but each scattered partial wavefunction does have a self-coupling term. We apply the method to the scattering of three identical bosons interacting through zero-range pair potentials. The method gives the same equation for this problem as obtained by the Faddeev T-matrix approach.
A quantum kinetic equation is obtained which includes scattering events beyond the binary collision process of the Boltzmann equation. The time rate of change of the Wigner function due to collisions is given as the difference of two scattering rates analogous to the collisional side of the Boltzmann equation. These total rates are given in terms of partial rates which are identified with probability amplitudes for scattering events in the medium. Thus, the method is inherently free of the divergence difficulties of theories which calculate the total rates in terms of scattering events in vacuum. Special attention is devoted to the question in what sense does some form of Fermi's golden rule apply to the calculation of the partial rates for a given event. It is shown that certain modifications to this picture are necessary, but the changes may be understood from a simple physical point of view.
The quantum Boltzmann equation (QBE), originally formulated as the quantum extension of Boltzmann’s classical transport theory by Uehling and Uhlenbeck and subsequently developed within the nonequilibrium Green-function formalisms of Kadanoff–Baym and Keldysh, provides the central kinetic framework for describing relaxation and transport in weakly interacting quantum many-body systems. Although its formal structure is well established, conventional derivations often obscure the microscopic equivalence between operator and nonequilibrium Green-function approaches. Wick factorization is typically applied without explicitly enumerating real-time contractions, while Green-function treatments encode the same physics in Born-level self-energies that conceal their operator origin. In this paper, we present a pedagogical microscopic derivation of the quantum Boltzmann equation that makes this equivalence fully explicit. Starting from the ladder-operator formulation, we enumerate all fifteen Wick pairings of the second-order six-field correlator, identifying three Hartree–Fock contributions and twelve real-time Born scattering channels generated by a quartic, number-conserving interaction Hamiltonian. We then establish a one-to-one correspondence between these twelve operator contractions and the Keldysh components of the second-order Born self-energy, demonstrating explicitly how the Uehling–Uhlenbeck collision integral emerges after Wigner transformation and quasiparticle projection. Under controlled Born–Markov and quasiparticle approximations, this work offers a transparent synthesis of operator and Green-function formulations, and clarifies how kinetic fluctuations can be understood as emergent under coarse-graining.
The aim of this thesis is to present a reasonable account of some of the applications that functional analysis, or more specifically operator theory, has in the theory of quantum scattering. We begin by introducing the physical scattering experiment as well as establishing the standard framework for discussing quantum theory, that of a Hilbert space. This leads to a discussion of the properties of operators acting in a Hilbert space and the role played by unbounded selfadjoint operators in quantum physics. We also spend some time discussing the position, momentum and energy operators which are of most relevance here. This in turn leads logically to the problem of formulating the scattering problem mathematically using these operators. The stationary equations of scattering theory are also rigorously derived and their suitability for two-body scattering problems is proven. We show that this analysis does not extend to three-body scattering problems, and introduce the Faddeev formulation for three-body systems to supplement the theory. This thesis concludes with a brief overview of how our scattering equations can be modified to take into account effects such as particle spin and identical particles, before discussing an example of a three-body scattering problem (incorporating these features) and deriving a mathematically satisfactory set of equations which are amenable to numerical solution.
In the linear-response regime, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. We derive a general expression for this kernel and evaluate it explicitly to second order in the interparticle potential. The result is a wave-vector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born-approximation cross section. Our theory is developed in terms of a second-quantized form of the Wigner representation. Convenient expressions are obtained for the commutators and anticommutators of the phase-space density operators, and the equilibrium averages of these operators are analyzed in terms of momentum-dependent generalizations of the classical pair distribution function h(k)h(k)h(k) and direct correlation function c(k)c(k)c(k). The central quantity in this study is a two-particle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ gives the symmetrized scattering function $S(k\ensuremath{\omega})$ by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ to the same order and that the result satisfies the appropriate positivity and symmetry conditions.
The Quantum Mechanical Theory of Collis3ions. I»-In the present course we shall consider the quantum mechanical treatment of problems of particle interactions in which at least one of the particles is unbound and therefore able to travel large distances freely.In effect such particles are able to communicate the results of their interaction directly to macroscopic detection apparatus* The interpretation of the resulting data is in general the most direct way of forming conclusions about the interactions.A considerable variety of physical phenomena involves unbound particles» Among the simplest are scattering processes ;in particular the deflection of particles from a collinated beam.More general collision processes which induce reactions of various sorts also lie within the class we shall examine, Our chief concern willbe with the means of solving problems in which well-defined models have been postulated to describe the basic interactions.We shall restrict ourselves, in fact, to the treatment of fairly simple models, and shall discuss a number of approaches to the problems they raise.We begin by discussing the scattering of particles by a simple potential field, fixed in space» The problem of two free particles with a potential between them of course assumes this form in the center-of-mass system.By imagining that a beam of particles of constant strenght is at all times being scattered by the potential, we cay seek to describe the situation with the stationary-state Schrb'dinger equation.