The time independent Schrödinger equation stems from quantum theory axioms as a partial
differential equation. This work aims at providing a novel discrete geometric formulation
of this equation in terms of integral variables associated with precise geometric
elements of a pair of three-dimensional interlocked grids, one of them based on tetrahedra.
We will deduce, in a purely geometric way, a computationally efficient discrete
counterpart of the time independent Schrödinger equation in terms of a standard
symmetric eigenvalue problem. Moreover boundary and interface conditions together with
non homogeneity and anisotropy of the media involved are accounted for in a straightforward
manner.
This approach yields to a sensible computational advantage with respect to the finite element
method, where a generalized eigenvalue problem has to be solved instead. Such a
modeling tool can be used for analyzing a number of quantum phenomena in modern
nano-structured devices, where the accounting of the real 3D geometry is a crucial issue.
