Energy-Dispersion of Matter Waves in Schr\"{o}dinger-Poisson Model

The energy dispersion of collective excitations of free electron gas as well as self-gravitating ensemble of uncharged particles is derived using a unified 1D Schr\"{o}dinger-Poisson model. The energy dispersion of self-gravitating system is shown to lead to unique features which are absent in the case of electrostatic excitations. Current mathematical model of collective particle excitations is shown to gives rise to a novel description of the paradoxical wave-particle duality and many intriguing new collective features due to the scale-dependence effective forces in collective interactions, absent in the single particle description of physical systems. It is shown that the excitations in a self-gravitating system lead to a fundamentally different features of collective interaction under gravitational potential than that of electrostatic ones. Particularly, it is found that the total energy of self-gravitating systems can be negative and the effective mass of excitations vary significantly in the whole spectrum of wavelength. The later may be considered as meaningful absence of self-consistent theory of quantum gravity. The significance of the peculiar aspects of the two fundamentally different excitation types is discussed based on their relevance to modern theories.