Exact solutions to the Schrödinger equation for realistic nanoscale systems are beyond reach, hence, different strategies for approximating the solutions are necessary. Perturbation theory relies on a “zero-order Hamiltonian” to express the desired eigenvalues and eigenvectors of “the full Hamiltonian.” We derive working equations for the Rayleigh–Schrödinger perturbation theory, and the validity of the approach is analyzed for a generic two-level system. Applications are given to atoms perturbed by point charges or static fields, and for electrons in quantum wells. An alternative strategy is the variation method, which replaces exact solutions by their projection on a reduced space of “trial functions,” varied to minimize the associated error. Particularly important is the method of linear variation, which can potentially converge to the exact Hamiltonian eigenstates. The mean-field approximation, commonly used for many-particle systems, is derived by optimizing a trial function in the form of product of single-particle functions.

The previous three chapters cover the elastic behaviour of composites containing aligned fibres that are, in effect, infinitely long. Use of short fibres (or equiaxed particles) creates scope for using a wider range of reinforcements and more versatile processing and forming routes (see Chapter 15). There is thus interest in understanding the distribution of stresses and strains within such composites, and the consequences of this for the stiffness and other mechanical properties. In this chapter, brief outlines are given of two analytical models. In the shear lag treatment, a cylindrical (short fibre) reinforcement is assumed, with stress fields in fibre and matrix being simplified (leading to some straightforward analytical expressions). It introduces important concepts concerning load transfer mechanisms, although it is not very widely used for property prediction. The Eshelby method, on the other hand, is based on the reinforcement being ellipsoidal (anything from a sphere to a cylinder or a plate): the analysis is more rigorous, but with the penalty of greater mathematical complexity. The model is only briefly described here. Its use also introduces an important concept – that of a misfit strain, which is helpful in areas well beyond those of the mechanics of conventional composite materials.