  The most fundamental approach to an understanding of electronic, optical, and
transport phenomena which the condensed matter physics (of conventional as well
as nonconventional systems) offers is generally founded on two experiments: the
inelastic electron scattering and the inelastic light scattering. This work
embarks on providing a systematic framework for the theory of inelastic
electron scattering and of inelastic light scattering from the electronic
excitations in GaAs/Ga$_{1-x}$Al$_{x}$As quantum wells. To this end, we start
with the Kubo's correlation function to derive the generalized nonlocal,
dynamic dielectric function, and the inverse dielectric function within the
framework of Bohm-Pines' random-phase approximation. This is followed by a
thorough development of the theory of inelastic electron scattering and of
inelastic light scattering. The methodological part is then subjected to the
analytical diagnoses which allow us to sense the subtlety of the analytical
results and the importance of their applications. The general analytical
results, which know no bounds regarding, e.g., the subband occupancy, are then
specified so as to make them applicable to practicality. After trying and
testing the eigenfunctions, we compute the density of states, the Fermi energy,
the full excitation spectrum made up of intrasubband and intersubband --
single-particle and collective (plasmon) -- excitations, the loss functions for
all the principal geometries envisioned for the inelastic electron scattering,
and the Raman intensity, which provides a measure of the real transitions
induced by the (laser) probe, for the inelastic light scattering...
