LINEAR COLLISIONS WITH HARMONIC-OSCILLATOR FORCES - INVERSE SCATTERING PROBLEM

A particular kind of elastic scattering is studied between three bodies of equal mass. Two of the bodies are at rest and in contact with each other. The third body impinges on these two. The motion of the bodies is confined to a straight line as in a linear air track. The scattering is mediated by nearest-neighbor Hooke's law forces having force constants and k1k2. The final velocities of all three bodies are studied as a function of β (= k2/k1). It is shown that the possible final scattering states may be divided into two classes. In one class it is possible to determine β from a knowledge of the final velocities of the three bodies. In the other class no such determination is possible. A formula is derived that yields an infinite set of values of 13, all of which produce the same velocities in the final state. Other measurements are proposed which allow a determination of β and the individual spring constants k1 and k2 as well as the size of the springs. These further measurements involve time correlations between incident and scattered particles. No measurements are permitted during the time of interaction. The correspondence principle indicates that the same ambiguity will be present in the analogous one-dimensional quantum mechanical problem necessitating time correlation measurements to solve the inverse scattering problem. © 1979, American Association of Physics Teachers. All rights reserved.