Identical motion for classical and quantum mechanics is re-examined in order to include the effects of energy-dependent potentials in the equations of motion. Only free particles permit such motion.
Special relativity combined with the stochastic vacuum flux impact model lead to an explicit interpretation of many of the phenomena of elementary quantum mechanics. We examine characteristics of a repetitively impacted submicroscopic particle in conjunction with examination of the ways in which effects associated with the particle's behavior appear in moving frames of reference. As seen from relatively moving frames of reference, the time and location of impacts and recoils automatically exhibit wave behavior. This model leads to free particle waves with frequencies proportional to the energy and wavelengths inversely proportional to the momentum. As seen from relatively moving frames of reference, impacts and their associated recoils can appear to an observer to take place simultaneously at multiple locations in space. For superposed free particle waves corresponding to bidirectional motion, an amplitude that varies sinusoidally with distance results. A governing equation identical in form to the Schroedinger equation is developed that describes the behavior of the impacts and their associated recoils. This approach permits many features of quantum mechanics to be examined within an intuitively visualizable framework.
The free propagation in time of a normalisable wave packet is the oldest problem of continuum quantum mechanics. Its motion from microscopic to macroscopic distance is the way in which most quantum systems are detected experimentally. Although much studied and analysed since 1927 and presented in many text books, here the problem is re-appraised from the standpoint of semi-classical mechanics. Particular aspects are the emergence of deterministic trajectories of particles emanating from a region of atomic dimensions and the interpretation of the wave function as describing a single particle or an ensemble of identical particles. Of possible wave packets, that of gaussian form is most studied due to the simple exact form of the time-dependent solution in real and in momentum space. Furthermore, this form is important in laser optics. Here the equivalence of the time-dependent Schroedinger equation to the paraxial equation for the propagation of light is demonstrated explicitly. This parallel helps to understand the relevance of trajectory concepts and the conditions necessary for the perception of motion as classical.