Numerical solution of the Dirac equation by a mapped Fourier grid method

Abstract

An efficient numerical method is developed to solve the relativistic hydrogenic Coulomb problem. Combining a pseudospectral method and a change of coordinates, a matrix representation of the relativistic Hamiltonian is constructed in position space. The radial coordinate is mapped from the semiinfinite r-axis to a finite range. The accuracy of the calculation is tested by comparing eigenvalues, sum rules and eigenfunctions with known analytic results. The method can be applied to central field problems for which analytic solutions are unknown.

Sparse grids for the Schrödinger equation

“…We now come back to the electronic Schrödinger equation (1). Note that in general an electronic wave function depends in addition to the positions x i of the electrons also on their associated spin coordinates σ i ∈ {− 1 2 , 1 2 }.…”

“…This result suggests that it is possible to numerically solve (1) up to a prescribed accuracy with an amount of work which does not scale exponentially in the number N of electrons with respect to K. Note that in the case d = 3 the Coulomb potential 1 |x−y|2 is unbounded at the set of coalescence points…”

“…cusp conditions) can be found in [29-31, 50, 52]. In this article we develop and study a generalized sparse grid/hyperbolic cross technique for the electronic Schrödinger equation (1). For reasons of simplicity we restrict ourselves to the setting of a d · N -dimensional product domain Ω = I d·N with I = [0, 2π] and periodic boundary conditions.…”

“…Candidates are other hierarchical global polynomial systems [8,10,12,51,73] or function families with localization properties like wavelets [18], prewavelets [15,42], interpolets [19,22], and related wavelet-like constructs, see [12,16] for a survey. But also multiscale finite element systems and frames [37,40,41,67] or multiscale Gaussians [63] may be used 1,2 .…”

Propagation of a relativistic electron wave packet in the Dirac equation

“…A numerical approach recently has gained in popularity [13,[21][22][23][24][25]. However, neither approach provides a physical insight on the mechanism of the phenomenon.…”