2.6M - 1. Institut für Theoretische Physik - Universität Stuttgart

Summary
Because of the huge magnetic fields B ∼ 10 8 T, the Landau excitation energy is of the
magnitude ∼ 10 keV. For the atomic ground state it is therefore adequate to restrict
the wavefunction only to the lowest Landau level n = 0. At this level all spins are
aligned anti-parallel to the external magnetic field. The single-particle wavefunctions
Pνm(z) are the results of solving the Hartree-Fock equations in adiabatic approximation
(see equation (3.15)), formulated as an equivalent variational problem (see equation
(3.22)) [26]. In doing so B-spline interpolation [30] and the finite element method [14] is
used. In listing 3.1 an example for an input file for calculating the wavefunction Pνm(z)
is given. Figure 3.4 shows the single-particle wavefunctions in adiabatic approximation
for helium, and figures 3.6 and 3.7 those for iron. Finally it is not only possible to
calculate the wavefunction Pνm(z) at any position, but it is also possible to calculate
their derivatives which are required for the quantum force (see equation (3.44), (3.48)
and (3.49)) and the local energy (see equation (3.59) and (3.60)). The Landau states
Φ0m(ρ, ϕ) = N e imϕ ρ |m| β
−
e 2 ρ2
are given analytically. The evaluation of the spin function is simple because of the
alignment of all spins leading to a lowering in energy ES = � N
i=1 βˆσzi = −N β. This
thesis considers both the electron-electron as well as the electron-core interaction within
a Jastrow factor (Ψ JF = e −u , with the Padé-Jastrow function [33] u). The cusp condition
[13] configures the parameter a JF , an estimate (see equation (3.69) and (3.70)) for the
parameter b JF is given by the comparison of the diamagnetic energy part (see equation
(3.38)) and the energy of the lowest Landau level (see equation (3.55)). The Padé-
Jastrow function for a many-electron system is given by
u = − 1
4
N�
i,j =1
i