We first introduce a brief review of the history of Brownian Motion up to the modern experiments where isolated Brownian particles are observed.Later, we introduce a one-space-dimensional wavefunction model of a heavy particle and a collection of light particles that might generate "Brownian-Motion-Like" trajectories as well as diffusive motion (displacement proportional to the square-root of time).This model satisfies two conditions that grant, for the temporal motion of the heavy particle:

(a) An oscillating series with properties similar to those of the Ornstein-Uhlenbeck process;

(b) A best quadratic fit with an "average" non-positive curvature in a proper time interval.

We note that Planck's constant and the molecular mass enter into the diffusion coefficient, while they also recently appeared in experimental estimates;to our knowledge, this is the first microscopic derivation in which they contribute directly to the diffusion coefficient.Finally, we discuss whether cat states are present in the thermodynamic ensembles.

(Joint for with W.D. Wick)

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