Some numerical problems in quantum theory

This thesis presents the results of an exercise in practical numerical analysis whose aim is the estimation of realistic 'a posteriori' error bounds for the solutions of atomic structure calculations computed by finite-difference methods. In performing these calculations the basic problem is to determine the eigenstates of a particle in a central force field by solving the one-dimensional radial form of the Schrodinger (non-relativistic) or Dirac (relativistic) equation, subject to homogeneous boundary conditions. The system represented in either case is essentially a second-order ordinary linear differential system of the type arising in many branches of atomic, molecular, and nuclear physics. In particular, the Hartree-Fock approach ta the problem of calculating atomic structures involves the solution of one or more of these equations by an iterative scheme. Chapter 1 describes the general aims of the thesis: these are to produce non-pessimistic error estimates for the solutions computed on any particular grid by obtaining first-order bounds on the solutions of the finite-difference systems satisfied by the errors themselves. The intention is to gain a realistic assessment of the order of magnitude of the error at every point over the range of tabulation of each of the computed solutions, thus indicating the number of figures worth quoting in the results. The sources of error taken into account include the errors arising from replacement of the derivatives by truncated finite-difference expansions (''truncation errors') and those arising from uncertainties in any initial data that may be used in performing the calculation ('physical errors'). Compared with these, the effect of rounding errors is assumed sufficiently small to be neglected. Chapter 2 contains a description of a standard numerical method of determining the eigenstates of a particle in a central field in the non-relativistic approximation by solving the radial form of the Schrodinger equation, employing for this purpose a combination of initial-value and boundary-value techniques. An extension of the method is described for dealing with the situation which arises for example in atomic structure theory in which an inhomogeneous term is present in the equation. The problem is not then strictly of eigenvalue type but a solution satisfying a specified normalization condition exists only for certain discrete values of the energy parameter. The results which form the basis of the error analysis are presented in Chapter 3. The exact differential system is regarded as a perturbation of the finite-difference system actually solved, and the errors themselves are regarded as forming the solution of a finite-difference system. Taking into account each source of error and working to the first order in small quantities, this system is 'solved' by finding a first-order estimate of the error in the computed eigenvalue and a similar estimate of the error in each element of the vector representing the eigenfunction. Attention is paid to the problem of ensuring that the error estimate reflects as closely as possible the form of the actual error in the computed eigenfunction over the range of integration. In Chapter 4 the formulae resulting from this analysis are rewritten in terms of the notation used in Chapter 2. Several sets of results are presented for the special case in which the central force field is a Coulomb field. In this case the analytic solutions of the exact differential system are known and the computed error estimates may be compared with the actual errors in the computed solutions. In making this comparison, separate consideration is given to the effect on the results of truncation error and simulated physical errors in the cases of both homogeneous, and inhomogeneous systems. Chapter 5 is concerned with the numerical calculation of the eigenstates of a particle in a central field by solving the radial form of the wave equation in Dirac's relativistic quantum theory. The radial wave functions now have two components satisfying a pair of first-order differential equations, but the techniques employed for computing the solutions are similar to those described in Chapter 2. In presenting the method, the question of the stability of the techniques: is discussed. An error analysis is developed along similar lines to the treatment presented in Chapter 3 and the results of the analysis are tested in the same way as in Chapter 4 by making use of the known solutions for the Coulomb field. The subject of Chapter 6 is the application of the error analysis: to the problem of calculating atomic structures in the non-relativistic approximation by solving the Hartree-Fock equations by the iterative self-consistent field technique. A general description of the technique is presented, and a method is proposed for determining 'a posteriori' error bounds on the solutions by incorporating the error analysis into an additional iteration of the self-consistent field procedure. The bounds are derived by taking into account the effect of the truncation errors associated with the replacement of the exact differential system by the approximating finite-difference system over a given grid, and these bounds are independent of the number of iterations performed to achieve self-consistency. Results are presented for the two simplest atomic systems, representing the ground states of atoms or ions of two electrons and four electrons respectively, and suggestions are made for modifying the methods of analysis so as to effect improvements in the results.