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Revista Mexicana de Física

Descripción causal de la dispersión potencial de Bargmann

Manuel Jesús Maldonado Lozano

Published May 10, 2009• 1 citation
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Abstract

IN APREVIOUS PAPER, WE HAVE STUDIED THE TIME DEPENDENT DESCRIPTION OF THE SCATTERING BY A GENERAL POTENTIAL, AND SHOWN THAT THE LAPLACE TRANSFORM OF THE TIME DEPENDENT GREEN FUNCTION OF THE PROBLEM SATISFIES DISPERSION RELATIONS. IN THIS PAPER WE OBTAIN THE EXPLICIT FORM OF THE TIME DEPENDENT GREEN FUNCTION FOR A POTENTIAL OF THE BARGMANN TYPE, AND DISCUSS THE RELATION BETWEEN THE TIME DPENDENT BEHAVOIR OF THE GREEN FUNCTION AND THE POSITION OF THE POLES AND ZEROS OF S(K) IN THE COMPLEX WAVE NUMBER PLANE.

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Revista Mexicana de Física•May 10, 2009•0 citations
IN APREVIOUS PAPER, WE HAVE STUDIED THE TIME DEPENDENT DESCRIPTION OF THE SCATTERING BY A GENERAL POTENTIAL, AND SHOWN THAT THE LAPLACE TRANSFORM OF THE TIME DEPENDENT GREEN FUNCTION OF THE PROBLEM SATISFIES DISPERSION RELATIONS. IN THIS PAPER WE OBTAIN THE EXPLICIT FORM OF THE TIME DEPENDENT GREEN FUNCTION FOR A POTENTIAL OF THE BARGMANN TYPE, AND DISCUSS THE RELATION BETWEEN THE TIME DPENDENT BEHAVOIR OF THE GREEN FUNCTION AND THE POSITION OF THE POLES AND ZEROS OF S(K) IN THE COMPLEX WAVE NUMBER PLANE.
IN THIS PAPER WE GIVE THE TIME DEPENDENT DESCRIPTION OF THE SCATTERING OF A WAVE PACKET BY A SHORT RANGE POTENTIAL. WE SHOW THAT WHEN THE WAVE PACKET IS INITIALLY OUTSIDE THE SCATTERER, IT IS POSIBLE TO INTRODUCE AN APPROPIATE TIME DEPENDENT GREEN FUNCTION TO DESCRIBE THE SCATTERING PROCESS. THIS GREEN FUNCTION WILL BE EVALUATED EXPLICITLY IN TERM OF ERROR INTEGRAL FUNCTIONS, AND THE ONLY PARAMETERS THAT APPEAR IN THE GREEN FUNCTION WILL BE THE POLES OF THE S MATRIX. FROM THE ASYMPTOTIC BEHAVIOR OF THE GREEN FUNCTIOR WHEN T, WE OBTAIN RESTRICIONS ON THE POSITION OF THE POLES OF THE S MATRIX WHICH AGREE WITH THOSE DUE TO THE DEFINITION OF THE S IN TERMS OF THE DERIVATIVE MATRIX R. THE PRESENT ANALYSIS IS CARRIED OUT FOR ARBITRARY ANGULAR MOMENTUM.
THE PRESENT NOTE ANALYSES THE DYNAMICAL DESCRIPTION OF THE SCATTERING DUE TO A SHORT RANGE POTENTIAL. IN PARTICULAR, IT DEVELOPS A CLASS OF TRANSFORMS THAT ARE USEFUL FOR THIS DYNAMICAL DESCRIPTION. A TIME-DEPENDENT GREEN FUNCTION FOR THE SCATTERING PROCESS IS DISCUSSED, THAT TAKES US FROM THE INITIAL STATE (PLANE WAVE) TO THE FINAL STATE (PLANE PLUS SCATTERED WAVES). THIS GREEN FUNCTION CAN BE USED TO GIVER THE MOTION OF ANY WAVE PACKET THAT INITIALLY WAS OUTSIDE THE POTENTIAL. THE PRESENT PAPER COMPLEMENTS A PREVIOUS DYNAMICAL DESCRIPTION OF A SCATTERING PROCESS, IN WHICH THE INTERACTION WAS REPRESENTED BY APPROPRIATE BOUNDARY.
THE PRESENT NOTE ANALYSES THE DYNAMICAL DESCRIPTION OF THE SCATTERING DUE TO A SHORT RANGE POTENTIAL. IN PARTICULAR, IT DEVELOPS A CLASS OF TRANSFORMS THAT ARE USEFUL FOR THIS DYNAMICAL DESCRIPTION. A TIME-DEPENDENT GREEN FUNCTION FOR THE SCATTERING PROCESS IS DISCUSSED, THAT TAKES US FROM THE INITIAL STATE (PLANE WAVE) TO THE FINAL STATE (PLANE PLUS SCATTERED WAVES). THIS GREEN FUNCTION CAN BE USED TO GIVER THE MOTION OF ANY WAVE PACKET THAT INITIALLY WAS OUTSIDE THE POTENTIAL. THE PRESENT PAPER COMPLEMENTS A PREVIOUS DYNAMICAL DESCRIPTION OF A SCATTERING PROCESS, IN WHICH THE INTERACTION WAS REPRESENTED BY APPROPRIATE BOUNDARY.
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.
THE PURPOSE OF THE PRESENT IS TO FIND THE RELATION BETWEEN THE S-MATRIX FORMALISM AND THE DESCRIPTION OF INTERACTIONS BY MEANS OF BOUNDARY CONDITIONS. IT IS SHOWN THAT IN THE ABSENCE OF AN ESSENTIAL SINGULARITY AT -- FOR ----, THE DYNAMICAL DESCRIPTION SUGESTED IN A NATURAL WAY BY THE S-FORMALISM, GIVES THE SAME TIME-DEPENDENT WAVE FUNCTIONS THAT ARE OBTAINED WHEN THE INTERACTIONS ARE REPRESENTED BY BOUNDARY CONDITIONS. THE DYNAMICAL DESCRIPTION OF THE S-FORMALISM IS EXTENDED TO THE CASE WHERE ----- MAY HAVE AN ESSENTIAL SINGULARITY AT ---.