The behaviour of electrons in nanostructures such as quantum wells is of interest for the design of new electronic and electro-optic devices, and also for exploration of basic many-body physics. This thesis develops and tests improved methods for describing such electronic behaviour. The system used for this work was the parabolic quantum well (PQW), an important special system which has recently attracted much experimental and theoretical attention. We firstly report self-consistent nonlinear groundstate solutions of the Poisson equation together with the Thomas-Fermi (TF) hydrostatic equations. In contrast to most previous solutions, all the electron density profiles were inhomogeneous and continuous. We also added a von Weizsacker term with and without the exchange/exchange-correlation to the above treatment, using a novel numerical approach allowing for wider electron gases than previously possible. We also report for the first time the effects of spatially varying effective mass and dielectric function in theories of this type. To investigate infrared response of these systems, we apply new hydrodynamic theories recently proposed by Dobson. By using this type of theory, we simultaneously satisfy the Harmonic Potential Theorem (extended generalized Kohn theorem) and obtain the correct 2D plasmon dispersion, as well as obtaining the correct spacing of standing plasmons. Other inhomogeneous hydrodynamic theories do not achieve this. We also showed analytically an exact solution for a plasmon mode at the Kohn frequency in addition to one found in the Harmonic Potential Theorem. An open hydrodynamic theory was then developed based on this type of mode. Numerical application of Kohn Frequency Theorem theory was shown and the results were compared with other existing hydrodynamic theories.