Florian Geyer

Line-of-sight integration: A powerful tool for visualization of three-dimensional scalar fields

An easy conceivable but very powerful method for the visualization of three-dimensional scalar fields is described. The way this method works is illustrated by some examples and the computed pictures are compared with the results of other methods of representation.

Results for Low-Lying States

The results of our extensive numerical calculations are presented in the form of tables and figures in Appendix Al. The energy values (in units of -E ∞) are given for the 38 lowest-lying states of the static Coulomb problem — corresponding to the H atom with infinite proton mass — and for magnetic field strengths in the range 10-4 ≤ β ≤ 103. The tables are ordered according to the azimuthal quantum number m and the z-parity π. The states are labelled by their asymptotic (β = 0 and β → ∞) quantum numbers. Those states that, in the limit β → 0, remain linear combinations of different angular momentum states are additionally marked by a prime, and the angular momentum given indicates the dominant l-component if two angular momentum states are involved. For linear combinations with more than two angular momentum states, this procedure is somewhat arbitrary, as can be seen from the 5g’ 0 and 5d’ 0 states, where the (l = 4)-component is dominant. Therefore, we choose the energy as the ordering parameter. For a given degenerate (n p, m, π)-multiplet, we denote the state with the highest l’ which is lowered most when the magnetic field is switched on (cf. Garstang and Kemic 1974; Garstang 1977). The coefficients of the linear combinations are obtained by diagonalizing the matrices following from the standard perturbation treatment of degenerate states. In Table 4.1 these coefficients are listed for the 38 states considered.

Electromagnetic Transition Probabilities

The calculation of electromagnetic transitions is a standard chapter of quantum mechanics, the essential formula being the famous “golden rule” (see, e.g., Heitler 1953, Messiah 1973). However, the resulting final expressions for field-free transitions or for Zeeman transitions which may be found in the literature cannot simply be applied to atoms in strong magnetic fields. This is because in the derivation of these expressions averaging processes over m are involved, whereas in strong magnetic fields the m-degeneracy is completely destroyed. Let us, therefore, begin by briefly reviewing the considerations, and the formulae, which are valid also for strong magnetic fields.

Stationary Lines and White Dwarf Spectra

One of the most spectacular applications of the calculations presented in the foregoing chapters was the identification of absorption features in the optical spectra of magnetized white dwarf stars, which had defied interpretation for almost 50 years, in terms of stationary components of hydrogen lines in magnetic fields of several 105 T (Angel et al. 1985; Greenstein et al. 1985; Wunner et al. 1985b; Schmidt et al., 1986a,b). By stationary components we mean those transitions whose wavelengths go through maxima or minima as functions of the magnetic field strength. These lines, between 300 nm and 1000 nm, are particularly well recognized if Fig. 4.2a is viewed sideways at flat angles. The fact that these transitions can produce sharp absorption features in white dwarf spectra is obvious when one considers that the magnetic field strength varies (in a dipolar geometry) by a factor of two across the white dwarf and thus all fast moving wavelengths are smeared out.

Methods of Solution for the Magnetized Coulomb Problem

Measuring energies in units of the Rydberg energy E ∞, lengths in units of the Bohr radius a 0, and the magnetic field strength in units of B 0, the Hamiltonian of an electron in a static Coulomb potential and in a uniform magnetic field then reads for spin-down states

Highly Excited States

^{H = - \Delta - \frac{2} {r} + 2\beta {l_z} + {\beta ^2}{\varrho ^2} - 2\beta }

Relativistic Effects, Nuclear Mass Effects, and Landau-Excited States

(3.1) where the magnetic field is assumed to point in the z-direction and ϱ 2 = x 2+y 2. The energies of the corresponding spin-up states are obtained by simply adding 4β. The eigenstates of (3.1) can be classified according to z-parity π and the z-component l z of orbital angular momentum, which are exact symmetries of H, but in general no further separation of the two-dimensional problem is possible.

Helium-Like Atoms in Magnetic Fields of Arbitrary Strengths

Let us consider a system of N charged particles with charges e i and masses m i (i = 1,..., N) in a uniform magnetic field B. The value of the vector potential A at the position r i of particle i is abbreviated by A i = A(r i ). In this section no special gauge is adopted. The Hamiltonian of the system is given by