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.