  Wallstrom's criticism of existing formulations of stochastic mechanics is
that they fail to derive the empirical predictions of orthodox quantum
mechanics because they require an ad hoc quantization condition on the
postulated velocity potential, \emph{S}, in order to derive Schr\"odinger wave
functions. We propose an answer to this criticism by modifying the Nelson-Yasue
formulation of non-relativistic stochastic mechanics for a spinless particle
with the following hypothesis: a spinless Nelson-Yasue particle of rest mass
$m$ continuously undergoes a driven steady-state oscillation of
`zitterbewegung' (\emph{zbw}) frequency,
$\omega_{c}=\left(1/\hbar\right)mc^{2}$, in its instantaneous mean forward (and
backward) translational rest frame. With this hypothesis we show that, in the
lab frame, \emph{S} arises from imposing the constraint of conservative
diffusions on the time-symmetrized steady-state phase of the \emph{zbw}
particle, satisfies the required quantization condition, and evolves in time by
the Hamilton-Jacobi-Madelung equations (when generalized to describe a
statistical ensemble of \emph{zbw} particles). From the mathematical
equivalence of Schr\"odinger's equation with the Hamilton-Jacobi-Madelung
equations plus the quantization condition, Schr\"odinger wave functions for a
spinless particle (in and excluding external fields) are thereby recovered. We
also apply this `zitterbewegung stochastic mechanics' (ZSM) to the case of a
central potential and show that it predicts angular momentum quantization. This
paper sets the foundation for Part II, which will (primarily) work out the
many-particle version of ZSM.
