This chapter starts out by introducing the energy eigenvalue equation – the time-independent Schrödinger equation. Firstly, the notion of energy quantization is introduced by semi-analytical means. A particle is confined within a rectangular well. It is seen, eventually, that the corresponding solutions to the time-independent Schrödinger equation can only exist for a few, specific energies. This phenomenon is also seen for other physical quantities, such as angular momentum and charge. Next, the situation in which the potential experienced by a quantum particle is periodic is given particular attention. It may be studied numerically by minor adjustments of the framework already developed. The case of a periodic potential is important as, to a large extent, it forms the basis for understanding solid state physics. Since the Hamiltonian is Hermitian, its eigenvectors form an orthonormal set in which any state or wave function may be expanded. This is exploited in the last part of the chapter, which is dedicated to determining ground states – the energy eigenstate of minimal energy. This is done in two ways – by using what is called the variational principle and by so-called propagation in imaginary time. These methods are implemented for several examples and compared to full solutions.

The theory of chemical bond formation in molecules and extended crystals is outlined. We start from the Born–Oppenheimer approximation, which associates the forces experienced by nuclei to the quantum electronic state. The Schrödinger equation for diatomic molecules reveals the formation of stable molecules when electrons are occupying “bonding” molecular orbitals. These are linear combinations of atomic orbitals (LCAO), in which the nuclei “share” electrons that effectively mask the electrostatic repulsion between them. The formation of effective LCAOs relies on compatibility in symmetry and energy of the underlying atomic orbitals. This is ubiquitously found in covalently bounded molecules, including conjugated polyatomic molecules. In the absence of effective LCAO, ionic bonds can be formed by charge transfer between atomic orbitals. In periodic lattices, effective LCAOs result in broad energy bands, which increase electrical conductivity. Conductor-to-insulator transition in response to the type of LCAO in the underlying material is demonstrated for a model system.