An appearance of classical matter from the self-organizing process of quantum systems

We present a quantum effect where matter follows the classical Hamilton-Jacobi equation, which emerges from quantum systems with Riemannian structures, as in standard quantum systems such as semiconductor heterostructures, quantum plasmas, and quantum dots. The proposed effect is derived from solving a standard elliptic partial differential equation of the radial part of the wave function, which is equivalent to a vanished quantum potential of the system. We then analyze such an effect and examine how the classical matter tends to be denser at the boundary region of the system when the quantum system is given in a finite region in space. While the proposed effect is derived from the hydrodynamical formulation of quantum mechanics, the results are free from any interpretation of quantum mechanics.