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We show that the central finite difference formula for the first and thesecond derivative of a function can be derived, in the context of quantummechanics, as matrix elements of the momentum and kinetic energy operatorsusing, as a basis set, the discrete coordinate eigenkets $\vert x_n\rangle$defined on the uniform grid $x_n=na$. Simple closed form expressions of thematrix elements are obtained starting from integrals involving the canonicalcommutation rule. A detailed analysis of the convergence toward the continuumlimit with respect to both the grid spacing and the approximation order ispresented. It is shown that the convergence from below of the eigenvalues inelectronic structure calculations is an intrinsic feature of the finitedifference method.

Real-space grid representation of momentum and kinetic energy operators for electronic structure calculations