  The relationship between classical and quantum mechanics is explored in an
intuitive manner by the exercise of constructing a wave in association with a
classical particle. Using special relativity, the time coordinate in the frame
of reference of a moving particle is expressed in terms of the coordinates in
the laboratory frame of reference in order to provide an initial spatiotemporal
function to work from in initiating the development of a quantum wave. When
temporal periodicity is ascribed to the particle, a provisional spatiotemporal
function for a particle travelling at constant velocity manifests itself as an
running wave characterized by parameters associated with the moving particle. A
wave description for bidirectional motion is generated based on an average time
coordinate for a combination of oppositely directed elementary running waves,
and the resulting spatiotemporal function exhibits wave behavior characteristic
of a standing wave. Ascribing directional orientation to the intrinsic
periodicity of the particle introduces directional sub-states; variations in
the relative number of sub-states as a function of angle in combined states
lead to spatially varying magnitudes for the associated waves. Further analysis
leads to full mathematical expression for all waves representing free particle
motion. A generalization for particles subject to force fields enables us to
develop a governing differential equation identical in form to the Schroedinger
equation.
