[en] We discuss the consequences of the introduction of a quantum of time τ₀ in the formalism of non-relativistic quantum mechanics, by referring ourselves, in particular, to the theory of the chronon as proposed by P. Caldirola. Such an interesting ''finite difference'' theory, forwards - at the classical level - a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz's and Dirac's approaches (and even allowing a clear answer to the question whether a free falling charged particle does or does not emit radiation), and - at the quantum level - yields a remarkable mass spectrum for leptons. After having briefly reviewed Caldirola's approach, our first aim is to work out, discuss, and compare to one another the new representations of Quantum Mechanics (QM) resulting from it, in the Schroedinger, Heisenberg and density-operator (Liouville-von Neumann) pictures, respectively. Moreover, for each representation, three (retarded, symmetric and advanced) formulations are possible, which refer either to times t and t-τ₀, or to times t-τ₀/2 and t+τ₀/2, or to times t and t+τ₀, respectively. It is interesting to notice that, when the chronon tends to zero, the ordinary QM is obtained as the limiting case of the ''symmetric'' formulation only; while the ''retarded'' one does naturally appear to describe QM with friction, i.e., to describe dissipative quantum systems (like a particle moving in an absorbing medium). In this sense, discretized QM is much richer than the ordinary one. We also obtain the (retarded) finite-difference Schroedinger equation within the Feynman path integral approach, and study some of its relevant solutions. We then derive the time-evolution operators of this discrete theory, and use them to get the finite-difference Heisenberg equations. When discussing the mutual compatibility of the various pictures listed above, we find that they can be written down in a form such that they result to be equivalent (as it happens in the ''continuous'' case of ordinary QM), even if the Heisenberg picture cannot be derived by ''discretizing'' directly the ordinary Heisenberg representation. Afterwards, some typical applications and examples are studied, as the free particle, the harmonic oscillator and the hydrogen atom; and various cases are pointed out, for which the predictions of discrete QM differ from those expected from ''continuous'' QM. At last, the density matrix formalism is applied to the solution of the measurement problem in QM, with very interesting results, as for instance a natural explication of ''decoherence'', which reveal the power of discretized (in particular, retarded) QM. (author)