Waves and Trajectories

Abstract

The angular differential-cross section calculations in the previous section were based on a classical model of the collision. This was a classical model in that a well-defined trajectory was used to describe the locations of the colliding particles as a function of time. Such a model could be modified and made statistical by associating a number of trajectories with every impact parameter and by weighting each trajectory, as in Eq. (2.32); nevertheless the model remains classical. In the first half of this century it became clear that beams of atoms and molecules do exhibit statistical behavior. However, the statistical properties exhibited were those associated with wave-like phenomenon, e.g., interference and diffraction. It is well established now that the wave-mechanical description of atomic interactions is the more fundamental description. The particle or trajectory (i.e., classical) model of atomic interactions bears roughly the same relationship to the wave-mechanical description of atomic collisions that geometric optics (ray description of light) bears to the wave description of the interaction of light with matter. The criteria for the validity of trajectory and ray approximations to their respective wave-mechanical phenomena are analogous. If the wavelength of light is small compared to the distances over which changes in the material occur, then the passage of light through the material can be described quite accurately using rays. On the other hand, when the changes in the material are abrupt, connection formulas, like Snell’s law in geometric optics, can be developed between rays in adjacent regions. The ray description is also useful if the source of light is very incoherent and/or the detectors (e.g., the eye) are insensitive to small spatial or time differences, that is, significant averaging occurs.