36 research outputs found
Calculation of the two-photon decay rates of hydrogen-like ions by using B-polynomials
A new approach is laid out to investigate the two photon atomic transitions.
It is based on application of the finite basis solutions constructed from the
Bernstein Polynomial (B-Polynomial) sets. We show that such an approach
provides a very promising route for the relativistic second- (and even
higher-order) calculations since it allows for analytical evaluation of the
involved matrices elements. In order to illustrate possible applications of the
method and to verify its accuracy, detailed calculations are performed for the
2s_{1/2}-1s_{1/2} transition in neutral hydrogen and hydrogen-like ions, and
are compared with the theoretical predictions based on the well-established
B-spline-basis-set approach
Mathematical surprises and Dirac's formalism in quantum mechanics
By a series of simple examples, we illustrate how the lack of mathematical
concern can readily lead to surprising mathematical contradictions in wave
mechanics. The basic mathematical notions allowing for a precise formulation of
the theory are then summarized and it is shown how they lead to an elucidation
and deeper understanding of the aforementioned problems. After stressing the
equivalence between wave mechanics and the other formulations of quantum
mechanics, i.e. matrix mechanics and Dirac's abstract Hilbert space
formulation, we devote the second part of our paper to the latter approach: we
discuss the problems and shortcomings of this formalism as well as those of the
bra and ket notation introduced by Dirac in this context. In conclusion, we
indicate how all of these problems can be solved or at least avoided.Comment: Largely extended and reorganized version, with new title and abstract
and with 2 figures added (published version), 54 page