Abstract
In order to appreciate the intricacies of quantum-mechanical scattering, you have to know something about classical scattering. When two particles collide and undergo elastic scattering, both the classical and quantum scattering problems can be very difficult to solve. The first step in the analysis of this problem is to make a transformation to the center-of-mass frame of the two particles. Then the interaction can be reduced to the scattering of a particle having reduced mass μ from a center of potential having relative coordinate r. One calculates the scattering in the center-of-mass frame and then must transform back to laboratory coordinates. I will discuss only the problem of scattering in the center-of-mass frame or, equivalently, scattering of a particle having mass μ from a potential V (r) that is assumed to possess spherical symmetry. The particle is incident along the z axis and has energy E.
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Notes
- 1.
If you expand the square root in the denominator of Eq. (18.16) about \(r=r_{\min }\) at the impact parameter corresponding to orbiting, you will find it varies as \(\left ( r-r_{\min }\right ) \), giving a deflection angle that diverges as \(\ln \left [ \left ( r-r_{\min }\right ) /a.\right ] \) Numerically, it is very hard to reproduce this very slow divergence. There are very special values, V 0/E = e 2, B 0/E = 4, \(r_{\min }/a=\sqrt {2}\), for which the second derivative also vanishes when E = E 0; in this case, the divergence is more rapid, varying as \(\left [ \left ( r-r_{\min }\right ) /a\right ] ^{-1/2}\).
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See, for example, R. A. Forber, L. Spinelli, J. E. Thomas, and M. S. Feld, Observation of Quantum Diffractive Velocity-Changing Collisions by Use of Two-Level Heavy Optical radiators, Physical Review Letters 50, 331–334 (1983).
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Berman, P.R. (2018). Scattering: 3-D. In: Introductory Quantum Mechanics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68598-4_18
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DOI: https://doi.org/10.1007/978-3-319-68598-4_18
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