  We give a partial answer to the question whether the Schrodinger equation can
be derived from the Newtonian mechanics of a particle in a potential subject to
a random force. We show that the fluctuations around the classical motion of a
one dimensional harmonic oscillator subject to a random force can be described
by the Schrodinger equation for a period of time depending on the frequency and
the energy of the oscillator. We achieve this by deriving the postulates of
Nelson's stochastic formulation of quantum mechanics for a random force
depending on a small parameter. We show that the same result applies to small
potential perturbations around the harmonic oscillator as long as the total
potential preserves the periodicity of motion with a small shift in frequency.
We also show that the noise spectrum can be chosen to obtain the result for all
oscillator frequencies for fixed mass. We discuss heuristics to generalize the
result for a particle in one dimension in a potential where the motion can be
described using action-angle variables.
