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arXiv:2001.01581

Homogeneous and Inhomogeneous Formulations of Nonrelativistic Potential Scattering

Jan Alexander Grzesik

Published Dec 30, 2019• 0 citations
arXivHTMLPDF

Abstract

Advantage is taken of the arbitrariness in energy reference to consider anew integral transcriptions of Schrodinger's equation in the presence of potentials which at infinity acquire constant, nonvanishing values. It is found possible to present for the probability amplitude $ψ$ a linear integral equation which is entirely devoid of explicit reference to the wave function incident from infinity, and thus differs markedly from the prevailing inhomogeneous formulation. Identity of the homogeneous equation with an inhomogeneous statement which is at the same time available is affirmed in general terms with the aid of the Fourier transformation, and is then still further reinforced by application of both formalisms to the particular example of a spherical potential barrier/well. Identical, closed-form outcomes are gotten in each case for wave function eigenmode expansion coefficients on both scatterer interior and exterior. Admittedly, the solution procedure is far simpler in the inhomogeneous setting, wherein it exhibits the aspect of a direct, leapfrog advance, unburdened by any implicit algebraic entanglement. By contrast, the homogeneous path, of considerably greater length, insists, at each mode index, upon an exterior/interior coefficient entanglement, an entanglement which, happily, is no more severe than that of a non-singular two-by-two linear system. Each such two-by-two linear system reproduces of course the output already gotten under the inhomogeneous route, and is indeed identical to the two-by-two system encountered during the routine procedure wherein continuity is demanded at the barrier/well interface of both $ψ$ and its radial derivative.

DOI
10.48550/arxiv.2001.01581

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Advantage is taken of the arbitrariness in energy reference to consider anew integral transcriptions of Schrodinger's equation in the presence of potentials which at infinity acquire constant, nonvanishing values. It is found possible to present for the probability amplitude ψψψ a linear integral equation which is entirely devoid of explicit reference to the wave function incident from infinity, and thus differs markedly from the prevailing inhomogeneous formulation. Identity of the homogeneous equation with an inhomogeneous statement which is at the same time available is affirmed in general terms with the aid of the Fourier transformation, and is then still further reinforced by application of both formalisms to the particular example of a spherical potential barrier/well. Identical, closed-form outcomes are gotten in each case for wave function eigenmode expansion coefficients on both scatterer interior and exterior. Admittedly, the solution procedure is far simpler in the inhomogeneous setting, wherein it exhibits the aspect of a direct, leapfrog advance, unburdened by any implicit algebraic entanglement. By contrast, the homogeneous path, of considerably greater length, insists, at each mode index, upon an exterior/interior coefficient entanglement, an entanglement which, happily, is no more severe than that of a non-singular two-by-two linear system. Each such two-by-two linear system reproduces of course the output already gotten under the inhomogeneous route, and is indeed identical to the two-by-two system encountered during the routine procedure wherein continuity is demanded at the barrier/well interface of both ψψψ and its radial derivative.
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.
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.
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The completely antisymmetric solution ${\ensuremath{\Psi}}^{A}$ to the problem of the scattering of a fermion by a finite system of identical fermions is studied by means of the expansion ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}{\ensuremath{\phi}}_{\ensuremath{\alpha}}{\ensuremath{\psi}}_{\ensuremath{\alpha}}$, where ${{\ensuremath{\phi}}_{\ensuremath{\alpha}}}$ is a complete set of antisymmetric states for the target and ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are one-particle functions. Coupled equations for the ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are found that obey the proper boundary conditions. This is done by means of the integral equation for ${\ensuremath{\Psi}}^{A}$ and the use of projection operators. The elastic-scattering (optical-model) wave function ${\ensuremath{\psi}}_{0}$ is shown to obey an inhomogeneous differential equation, rather than a Schr\"odinger equation. The homogeneous solution is identical to the elastic wave function obtained when the projectile is distinguishable, while the inhomogeneous solution is due entirely to exchange effects. The function ${\ensuremath{\psi}}_{0}$ is identical to the optical-model wave function found by Bell and Squires, who showed that ${\ensuremath{\psi}}_{0}$ obeys a Schr\"odinger equation with an optical potential containing direct and exchange contributions. It is shown that ${\ensuremath{\psi}}_{0}$ yields the exact elastic amplitude including direct and exchange contributions, and a phase-shift analysis of the exchange term is given. The more standard form of solution ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}\mathcal{a}{{\ensuremath{\phi}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}}$, where a\mathcal{a}a is an antisymmetrizer, is briefly discussed. The extension to the cases of inelastic scattering and deuteron elastic scattering is made.