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Physical Review

Outgoing Boundary Condition in Rearrangement Collisions

E. Gerjuoy

Published Mar 1, 1958• 59 citations
Publisher

Abstract

The boundary condition on the solution to the nonrelativistic time-independent Schr\"odinger equation for arbitrarily complicated rearrangements of spinless particles is carefully examined. For real energies $E$ it is shown that the outgoing boundary condition on the scattered wave $\ensuremath{\phi}$ need not imply that $\ensuremath{\phi}$ is "every-where outgoing." This and similar considerations make apparent the fact remarked by Foldy and Tobocman, namely that the Lippmann-Schwinger integral equation need not have a unique solution for real energies. The relationship of this result to the added fact that solutions to the Lippmann-Schwinger integral equation are unique for complex energies $E+i\ensuremath{\epsilon}$, $\ensuremath{\epsilon}>0$ is discussed, as is also the relationship of the usual operator manipulations to the outgoing boundary condition.

DOI
10.1103/physrev.109.1806

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The boundary condition on the solution to the nonrelativistic time-independent Schr"odinger equation for arbitrarily complicated rearrangements of spinless particles is carefully examined. For real energies EEE it is shown that the outgoing boundary condition on the scattered wave \ensuremathϕ\ensuremath{\phi}\ensuremathϕ need not imply that \ensuremathϕ\ensuremath{\phi}\ensuremathϕ is "every-where outgoing." This and similar considerations make apparent the fact remarked by Foldy and Tobocman, namely that the Lippmann-Schwinger integral equation need not have a unique solution for real energies. The relationship of this result to the added fact that solutions to the Lippmann-Schwinger integral equation are unique for complex energies E+i\ensuremathϵE+i\ensuremath{\epsilon}E+i\ensuremathϵ, \ensuremathϵ>0\ensuremath{\epsilon}>0\ensuremathϵ>0 is discussed, as is also the relationship of the usual operator manipulations to the outgoing boundary condition.
It is generally accepted that with suitable boundary conditions the Lippmann-Schwinger (LS) integral equation is equivalent to the Schr\"odinger equation, even in systems of more than two particles. It also is generally accepted that although in two-particle systems the scattering solutions to the LS equation are uniquely defined by that equation without need for boundary conditions, the same cannot be said of systems containing more than two particles; in such many-particle systems, it is generally agreed, unique scattering solutions to the LS equation are not obtained unless boundary conditions are imposed. However, these assertions have not been verified heretofore in any system of three or more particles, because exact closed-form solutions of the Schr\"odinger or LS equation for such systems are so difficult to achieve. We have examined these questions in a one-dimensional three-body model first discussed by McGuire, wherein three equal-mass particles interact via equal and finite strength attractive \ensuremath{\delta}-function potentials. With this model, the scattering solutions to the Schr\"odinger equation can be written exactly, in closed form. Thereby we are able to demonstrate explicitly that in this model the scattering solutions to the Schr\"odinger equation do satisfy the LS equation; we also demonstrate explicitly that the LS equation's scattering solutions really are nonunique unless boundary conditions are imposed. This latter result strongly suggests that, in three-particle systems at any rate, recent criticisms of the aforementioned generally accepted nonuniqueness thesis are not well taken.
Time-independent wave-packet forms of the Schr\"odinger equation (TIWSE) and Lippmann-Schwinger equation (TIWLSB) have been derived by a partial time-to-energy Fourier transform of L2{\mathit{L}}^{2}L2 wave-packet solutions to the time-dependent Schr\"odinger equation. The new equations retain the initial wave packet \ensuremath{\chi}(t0{\mathit{t}}_{0}t0​) as a ``universal source'' of scattered waves, which applies for all collision energies E contained in the initial wave packet. The relationship between the solution ${\mathrm{\ensuremath{\Psi}}}_{\mathit{t}0}$(E) of the TIWSE or TIWLSE and the scattering solution ${\mathrm{\ensuremath{\Psi}}}^{(+)}$(E) of the standard time-independent Lippmann-Schwinger or Schr\"odinger equation is given and the method illustrated by a computational application.
In this paper we reexamine, and analyze solutions to, the recently derived time-independent wave packet-Schrödinger (TIWS) and time-independent wave packet-Lippmann–Schwinger (TIWLS) equations. These equations are so named because they are inhomogeneous, with the inhomogeneity being the initial L2 wave packet from an underlying time-dependent treatment of the dynamics. We explicitly show that a particular solution of the homogeneous Schrödinger equation can be constructed out of two particular solutions of the inhomogeneous TIW equation satisfying causal and anticausal boundary conditions. The structure of this solution of the homogeneous equation is shown to depend sensitively on the nature of the initial wave packet inhomogeneity, but, as we demonstrate, correct scattering information can be obtained even when the initial wave packet is nonzero only in the target region. It thus becomes possible to carry out quantum scattering calculations in which one need not propagate the wave packet from the noninteracting to the interacting region. The method is illustrated by calculations for two 1-D scattering problems, namely the transmission of an electron through a single barrier and through a double barrier. The latter is especially challenging because of the occurrence of long-lived resonances with the electron trapped inside the double barrier. In addition, we show that the method can still make use of absorbing potentials to decrease grid size, and we show how the formalism can be used to treat inelastic and reactive scattering, and radial scattering variables, thereby constituting a general approach to time-independent wave packet quantum scattering.
The application of the "causality condition" to the SSS matrix for nonrelativistic particles encounters several difficulties: (a) there is no maximum velocity; (b) the interference of ingoing and outgoing waves has to be taken into account; (c) wave packets with a sharp front do not exist. The condition is therefore reformulated as follows: At any time the total probability of finding the particle outside the scattering center shall not be greater than 1, for every form of the incident wave packet. From this follows for spherical waves that SSS, as a function of the momentum ppp, is analytic and holomorphic in the first quadrant and that e2iapS(p){e}^{2iap}S(p)e2iapS(p) (where aaa is the radius of the scattering center) has an imaginary part \ensuremath{\leqslant} 1. That suffices to give an explicit integral representation and a product expansion for SSS, but these permit a more general form for SSS than is usually envisaged. If, however, the usual symmetry relation $S(\ensuremath{-}p)=S{(p)}^{*}$ is assumed in addition to the causality condition, more specific equations can be derived, which are direct generalizations of those in Part I. In particular, integral relations between the real and imaginary parts of SSS, and the properties that Wigner found for the RRR matrix can be deduced.
The problem of determining the Schr\"odinger wave function of a nonrelativistic particle that is either scattered by a potential of a finite range or that is bound to it is reformulated in a novel way. It is shown that in either case the wave function must satisfy a certain boundary condition on the surface that delimits the effective range of the potential. For scattering states the boundary condition is analogous to the mathematical formulation of the Ewald-Oseen extinction theorem of classical electromagnetic theory. The new formulation is illustrated by determining the scattering states and the bound states for a central potential. It is also shown that a boundary condition that is used in band-structure calculations in solids is an immediate consequence of our quantum-mechanical extinction theorem for bound states.