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 Kubos 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...
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