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 characteristic polynomial for a chain of dipole-dipole coupled two-level atoms with nearest-neighbour and next-nearest-neighbour interactions is developed for the study of eigenvalues and eigenvectors for single-photon excitations. We find the exact form of the polynomial in terms of the Chebyshev polynomials of the second kind that is valid for an arbitrary number of atoms and coupling strengths. We then propose a technique for expressing the roots of the polynomial as a power series in the coupling constants. The general properties of the solutions are also explored, to shed some light on the general properties that the exact, analytic form of the energy eigenvalues should have. A method for deriving the eigenvectors of the Hamiltonian is also outlined.