Book contents
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- 1 Introduction
- 2 Quantum scattering with a spherically symmetric potential
- 3 The variational method for the Schrödinger equation
- 4 The Hartree–Fock method
- 5 Density functional theory
- 6 Solving the Schrödinger equation in periodic solids
- 7 Classical equilibrium statistical mechanics
- 8 Molecular dynamics simulations
- 9 Quantum molecular dynamics
- 10 The Monte Carlo method
- 11 Transfer matrix and diagonalisation of spin chains
- 12 Quantum Monte Carlo methods
- 13 The finite element method for partial differential equations
- 14 The lattice Boltzmann method for fluid dynamics
- 15 Computational methods for lattice field theories
- 16 High performance computing and parallelism
- Appendix A Numerical methods
- Appendix B Random number generators
- Index
2 - Quantum scattering with a spherically symmetric potential
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- 1 Introduction
- 2 Quantum scattering with a spherically symmetric potential
- 3 The variational method for the Schrödinger equation
- 4 The Hartree–Fock method
- 5 Density functional theory
- 6 Solving the Schrödinger equation in periodic solids
- 7 Classical equilibrium statistical mechanics
- 8 Molecular dynamics simulations
- 9 Quantum molecular dynamics
- 10 The Monte Carlo method
- 11 Transfer matrix and diagonalisation of spin chains
- 12 Quantum Monte Carlo methods
- 13 The finite element method for partial differential equations
- 14 The lattice Boltzmann method for fluid dynamics
- 15 Computational methods for lattice field theories
- 16 High performance computing and parallelism
- Appendix A Numerical methods
- Appendix B Random number generators
- Index
Summary
Introduction
In this chapter, we shall discuss quantum scattering with a spherically symmetric potential as a typical example of the problems studied in computational physics [1, 2]. Scattering experiments are perhaps the most important tool for obtaining detailed information on the structure of matter, in particular the interaction between particles. Examples of scattering techniques include neutron and X-ray scattering for liquids, atoms scattering from crystal surfaces and elementary particle collisions in accelerators. In most of these scattering experiments, a beam of incident particles hits a target which also consists of many particles. The distribution of scattered particles over the different directions is then measured, for different energies of the incident particles. This distribution is the result of many individual scattering events. Quantum mechanics enables us, in principle, to evaluate for an individual event the probabilities for the incident particles to be scattered off in different directions; and this probability is identified with the measured distribution.
Suppose we have an idea of what the potential between the particles involved in the scattering process might look like, for example from quantum mechanical energy calculations (programs for this purpose will be discussed in the next few chapters). We can then parametrise the interaction potential, i.e. we write it as an analytic expression involving a set of constants: the parameters. If we evaluate the scattering probability as a function of the scattering angle for different values of these parameters, and compare the results with experimental scattering data, we can find those parameter values for which the agreement between theory and experiment is optimal.
- Type
- Chapter
- Information
- Computational Physics , pp. 14 - 28Publisher: Cambridge University PressPrint publication year: 2007