  In this research we study the effect of matter-wave instability on electron
beam transport with arbitrary degree of degeneracy. Particular class of
solutions of the Schr\"{o}dinger-Poisson system is used to model the
electron-beam transport at constant speed. It is shown that such electron-beam
is described by a coupled driven pseudoforce system solution of which leads to
plasmon excitations with dual wave-particle character. The fundamental quantum
mechanical de Broglie relation is found to be due to the resonant interaction
of particle-like plasmon excitation branch with the electron beam drift. We
further obtain a generalized double lengthscale de Broglie wave-particle
relation through which various beam-plasmon instability is studied in this
model. The quantum charge screening and the chemical potential effects on the
matter-wave formation and instabilities are discussed in detail and the
well-known Aharonov-Bohm effect is revisited in current quantum hydrodynamic
model. Current research may further illuminate the origin of matter-wave in
quantum mechanics and lead to clear understanding of novel wave-particle
interactions.
