  Using the Schr\"odinger-Poisson system in this paper the basic quantum
features of plasmon excitations in a free noninteracting electron gas with
arbitrary degeneracy is investigated. The standing wave solution of the free
electron gas is derived from the corresponding linearized pseudoforce system
with appropriate boundary conditions. It is shown that the plasmon excitation
energies for electron gas confined in an infinite potential well are quantized
eigenvalues of which are obtained. It is found that any arbitrary degenerate
quantum electron gas possesses two different characteristic length scales,
unlike the classical dilute electron gas, with the smaller length scale
corresponding to the single particle oscillation and the larger one due to the
collective Langmuir excitations. The probability density of the free electron
gas in a box contains fine structures which are modulated over a larger
pattern. The envelope probability density profile for the electron Fermi gas
confined in an impenetrable well in different energy states are found to be
quite similar to that of the free electron confined to an infinite potential
well. However, the illustrative features of the plasmon theory presented in
this research can be further elaborated in order to illuminate a wide range of
interesting physical phenomenon involving both single particle as well as the
collective features.
