7. Conclusions

This chapter illustrates how solutions of a simple quantum mechanical problem can be used for the description of certain interesting phenomena of nanophysics. Specifically, we referred to the exact solutions of the eigenvalue equations for the eigenenergy of the bound states of a particle in a rectangular well. If the physical problem is elementary, and the wave functions are simply written in terms of elementary functions, the equations for the eigenvalues of energy (or of the wave vector) are transcendental—and highly nontrivial. We obtain both exact solutions (series expansions) of these transcendental equations and approximate ones—with various degrees of complexity and accuracy. The value of the Fermi wave vector of the electrons in the metallic film, calculated for the finite well model, differs drastically from those calculated with the infinite well one.

Our results for the one-electron wave functions of the finite barrier model can be used as Kohn-Sham state in the self-consistent calculations of surface energy [34], for more accurate calculations of the stability of the films [1] and of other QSEs [33]. They can be also used as zero-order approximations for more realistic potentials, e.g., with rounded walls or undulate bottom—in a Rayleigh-Schroedinger or Dalgarno-Lewis perturbation theory [35].

Using the analogy between the movement of electrons in time-independent potentials and propagation of electromagnetic waves in dielectrics or metallic wave guides [18], mathematically, they are identical Sturm-Liouville problems; our results can be extended to several problems of electromagnetism and optics. This analogy can be easily developed for planar dielectric waveguides, namely, for "step-index" dielectrics, consisting of a slab of higher refractive index (core), sandwiched between two half spaces of lower refractive index (cladding). In such a situation, the quantum counterpart of the dielectric guide is a square well. This issue is discussed in detail by Casey and Panish in the context of heterostructure lasers [36]. It is easy to notice that the eigenvalue equations for transverse electric and magnetic modes, (2.4– 45, 54, 60, 66) in [36], are essentially identical with our Eqs. (8) and (9).
