In the previous chapters we have been dealing mainly with effects involving a single electron, such as the solution of the Schrödinger equation for a hydrogen-like atom. However, the majority of interesting problems in spectroscopy deal with systems that contain many electrons. In this chapter we shall see how to construct many-body wave functions from single-particle wave functions, and how to build a many-body Hamiltonian in matrix form and apply this to the Coulomb interaction for many-electron atoms.

Many-body wave functions

A many-body wave function needs to satisfy several characteristics. First, we need to ensure that the particles in the wave function are identical. This is a clear difference from classical physics, where we can distinguish one object from another. Second, it is with many-body wave functions that the distinction between fermions and bosons comes to the forefront. The construction of many-body basisfunctions starts by choosing a basis of one-particle basisfunctions. Let us denote these by ϕk(r), where k is a generic quantum number describing the quantum states (for example, momentum and spin k → kσ with spin projection σ = ↑, ↓ or k → nlmσ for hydrogen-like atomic orbitals). The basis can consist of eigenfunctions of the one-particle problem. In this case, the one-particle interactions H1 (r1) are already solved and we are only dealing with the two-particle interactions H2(r1, r2). However, this is not essential and we often choose a basis that is convenient to work with. For example, for atoms and ions, convenient basisfunctions are the hydrogen-like atomic orbitals. In the presence of spin-orbit coupling, these states are not eigenfunctions. In this case, both one- and two-particle interactions couple different many-body basisfunctions. In principle, the basis of many-body wave functions has to be complete. In practice, this is impossible since it requires an infinite number of basisfunctions.