  In this paper we propose an extended particle model whose evolution is
deterministic. In dimension 2, the extended particle is represented by four
points that define a small elastic string that vibrates, alternating between a
creation process and an annihilation process. First we show how the spin and
the Heisenberg uncertainty relations emerge from this extended particle. We
then show how the complex action associated with this extended particle
satisfies, from a generalized principle of least action, a second order complex
Hamilton-Jacobi equation. Third, we show that the wave function, which admits
this action as a complex phase, satisfies the Schr{\"o}dinger equation.
Finally, we show that the gravity center of this extended particle follows the
trajectories proposed by the de Broglie-Bohm interpretation well as the
Schr{\"o}dinger interpretation. This model is built on two new mathematical
concepts we have introduced: complex analytical mechanics on complex-valued
functions and a periodic deterministic process.
