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 formulatio n of the theory are then summarized and it is shown how they lead to an elu cidation and deeper understanding of the aforementioned problems. After str essing the equivalence between wave mechanics and the other formulations of quantum mechanics, i.e. matrix mechanics and Dirac's abstract Hilbert spac e formulation, we devote the second part of our paper to the latter approac h: we discuss the problems and shortcomings of this formalism as well as th ose of the bra and ket notation introduced by Dirac in this context. In con clusion, we indicate how all of these problems can be solved or at least av oided.