We derive here the many-body electronic Hamiltonian with full account of the spin in the non-relativistic limit. Starting from the single-electron Dirac equation and the notion of the spinor, we derive the Pauli equation for an electron in the electromagnetic field and its non-relativistic variant. Then the general many-electron case is studied to show how the spinor idea is generalised by building the spinor of the whole system as a direct poroduct of spinors of every electron. Corespondingly, the matrix Hamiltonian of such a system is built, which acts on such spinors. Next, a complete expansion of a general many-body wave function in terms of elementary many-body spinors (Slater determinants) is constructed, and corresponding matrix elements of one- and two-body operators are worked out that appear to be identical to the Slater rules. Then the second quantisation is developed for this fully relativistic case. Finally, the non-relativistic limit is considered in detail for the general many-fermion case and the quantum problem is reformulated in terms of the large components of the spinor only. The derived Hamiltonian contains the spin–orbit based on the total electric field.

Chapter 14 is entirely devoted to the electron spin and an introduction to quantum entanglement. The first part deals with the groundbreaking discovery of spin, its introduction into quantum formalism, and some of its most important effects on atomic spectra, notably the anomalous Zeeman effect. Historically, spin has been considered as an angular momentum that particles can have by the mere fact of their existence, which is called "intrinsic" and does not require any explanation. To address this shortcoming, Section 4 presents a possible explanation for the origin of electron spin as a result of its interaction with the vacuum field. Section 5 introduces the entangled system of two particles with spin, which provides an opportunity to discuss, necessarily schematically, the Schrödinger cat and the Einstein-Podolsky-Rosen thought experiment, as well as the Bell inequalities.

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).