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In this paper a systematic formalism for dealing with non-relativistic timedependent quantum Hamiltonians is presented. The starting point is the well-known Lewis and Riesenfeld idea which involves the construction of an invariant operator I(x, t) which defines both the dynamics of the physical system and the canonical formalism that has to be used in order to obtain a consistent theoretical framework. In order to exhibit the full power of the formalism we discuss two examples: the generalized harmonic oscillator and the infinite square well with a moving boundary. As the first example has already been analyzed by the present authors from other different points of view, we are able to compare the results of the canonical formalism with these other approaches and, as it was expected, we obtain identical descriptions of this physical system. After this, we turn to the case of the square well with a moving boundary. The main surprise is that in order to obtain consistency with the formalism an effective interaction appears which seems to be due to the time dependence of the boundary. Also consistency with the principle of minimal coupling and gauge invariance is obtained just by using this canonical operator formalism. Finally some interesting physical applications are suggested and discussed.

Here, the quantization of Dirac fermions in lithographically defined graphene nanoconstrictions is studied. Quantized conductance is observed in single nanoconstrictions fabricated on top of a thin hexamethyldisilazane layer over a Si/SiO2 wafer. This nanofabrication method allows to obtain well defined edges in the nanoconstrictions, thus reducing the effects of edge roughness on the conductance. The occurrence of ballistic transport is proved and several size quantization plateaus are identified in the conductance at low temperature. Experimental data and numerical simulations show good agreement, demonstrating that the smoothening of the plateaus is not related to edge roughness but to quantum interference effects.

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