Beiträge zur numerischen Behandlung der Schrödinger-Gleichung im Falle des Yukawa-Potentials

Abstract

Earlier calculations byL. Hulthen andK. V. Laurikainen concerning approximate solutions of the eigenvalue problem (1) and (2) in the casel=0 [1], have been generalized for other values ofl. Asymptotic expansions, valid for all integer values ofl, have been deduced for large values ofk [3], the expansion for the lowest eigen value being (5). As possible generalizations of the function (3)-which has turned out to be the “best fit” in the casel=0-for the casel=1, trial functions (9), (10), and (11) have been studied, (11) giving the best results (Table 1). In the casel=2, the trial function (12) was then chosen and was found to lead to a very satisfactory convergence. These trial functions can be generalized for all values ofl, no doubt leading to a good convergence in the variational procedure; the calculations, however, become very laborious for higher values ofl. Some results are given in Table 2. For calculation of the lowest eigenvalueb 1 and the corresponding parametersh v there are also given interpolation formulas (13) for the casel=0, takingn=4 in the trial function (3) and normalizing the approximative eigenfunction to unity.-A more detailed paper will be published in theAnnales Universitatis Turkuensis [5].