Mechanical motions of atoms near their equilibrium geometry are approximated by a universal mapping on a collection of independent harmonic oscillators. The respective frequencies are characteristic to the material, for example, lattice phonons or molecular vibrations. The quantum description of atomic motions involves the solution of the Schrödinger equation for the harmonic oscillator in the space of proper wave functions. The energy levels obtained are shown to be equally spaced where the level spacing is proportional to the oscillator frequency. The stationary solutions are identified in terms of Hermite polynomials, demonstrating remarkable differences from the classical harmonic oscillator. The importance of the zero-point energy is emphasized in the context of the stability of chemical bonds, where the Harmonic approximation is shown to be reasonably valid for typical interatomic bonds in standard thermal conditions. This explains the relevance of the harmonic approximation for analyzing the absorption spectrum of infrared radiation by molecules.

The Schrödinger equation is reformulated as a universal continuity equation, which connects between changes in the particles probability density distribution to probability current densities (fluxes). The formulation of particle conservation in terms of stationary fluxes enables one to associate stationary wave functions also to open quantum systems characterized by stationary particle currents. These functions are (improper) solutions of the stationary Schrödinger equation, obtained under scattering boundary conditions. These boundary conditions can be fulfilled for any positive asymptotic kinetic energy, hence, the energy spectrum of the scattering states is continuous. We demonstrate flux calculations in scattering through a one-dimensional potential energy well/barrier, focusing on transmission and reflection probabilities. Nonclassical phenomena such as transmission at energies below a potential energy barrier (quantum tunneling), or reflections at energies above a potential energy well are analyzed. The phenomenon of full transmission through a double barrier structure (resonant tunneling) is introduced in the context of nanoscale transport.

The foundations for understanding the electronic structure of many-electron atoms are introduced. We start from the discovery of the spin and introduce spin operators. The spin existence is shown to “upgrade” the state of single particles into a product space with the spin subspace, and to impose constraints on states of identical particles, which must be symmetric (bosons) or antisymmetric (fermions) under particle transpositions. The many-electron state in the atom is therefore approximated as an antisymmetrized products (Slater determinant) of single-electron states (spin-orbitals). The variationally optimal orbitals are shown to be solutions to the Hartree–Fock equations, and the assignment of electrons to these orbitals in the atomic ground state reflects the Pauli exclusion and Aufbau principles, thus explaining the trends in the periodic table of the elements in terms of their electronic configurations. Special attention is given to two-electron systems, demonstrating the exchange stabilization of triplet versus singlet states (Hund’s rule).