Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Background
- 3 Tensor algebra
- 4 Group theory
- 5 Many-body effects I: Coulomb interactions
- 6 The scattering amplitude
- 7 Many-body effects II: Solid-state effects
- 8 X-ray absorption and resonant X-ray scattering
- 9 Nonresonant and resonant inelastic X-ray scattering
- Appendix A Tensors
- References
- Index
5 - Many-body effects I: Coulomb interactions
Published online by Cambridge University Press: 05 January 2015
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Background
- 3 Tensor algebra
- 4 Group theory
- 5 Many-body effects I: Coulomb interactions
- 6 The scattering amplitude
- 7 Many-body effects II: Solid-state effects
- 8 X-ray absorption and resonant X-ray scattering
- 9 Nonresonant and resonant inelastic X-ray scattering
- Appendix A Tensors
- References
- Index
Summary
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.
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- Theory of Inelastic Scattering and Absorption of X-rays , pp. 76 - 103Publisher: Cambridge University PressPrint publication year: 2015