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.