7. Conclusion

We proceed with a brief discussion of the relationship between the derivative matrix D<sup>N</sup> and an important concept in quantum mechanics; the concept of self-adjoint operators [8, 9]. In particular, we focus on the momentum operator, whose continuous coordinate representation (operation) is given by �i d=dq, i.e., a derivative times �i, in the case of infinite-dimensional Hilbert space.

In the finite-dimensional complex vectorial space (where each vector define a sequence gi � �<sup>N</sup> i¼1 of complex numbers such that P <sup>i</sup> gi � � � � 2 < ∞). A transformation A is usually called Hermitian, when its entries ai,j are such that ai,j <sup>¼</sup> <sup>a</sup><sup>∗</sup> j,i ( <sup>∗</sup> denote the complex conjugate). Our matrix D<sup>N</sup> is related to an approximation of the derivative (see Section 3) which uses second order finite differences. Therefore, we can ask if the matrix �iD<sup>N</sup> is also Hermitian.

Let P<sup>N</sup> ¼ �iD<sup>N</sup> and v ¼ ix be the eigenvalue of DN, where x ∈ ℝ is a free parameter, the corresponding eigenvalue of �iD<sup>N</sup> is indeed the real value x; which is one of the properties of a Hermitian matrix, as is also the case of infinite-dimensional space (for the Hilbert space on a finite interval, these values are discrete, and for the Hilbert space on the real line, these values conform the continuous spectrum, instead of discrete eigenvalues). Other characteristic of �iD<sup>N</sup> is that the eigenvector corresponding to x is the same exponential function which is the eigenfunction of �i d=dx (see Section 2).

Furthermore, let P† <sup>N</sup> denote the adjoint of PN. Thus, if we restrict our attention to the offdiagonal entries ð Þ P<sup>N</sup> i,j ¼ �ið Þ D<sup>N</sup> i,j , it is fulfilled that P† N i,j ¼ �idj,i <sup>∗</sup> ¼ �idi,j <sup>¼</sup> ð Þ <sup>P</sup><sup>N</sup> i,j (noticing that, with v ¼ ix then χð Þ¼ x;Δ sinð Þ x; Δ =x ∈ ℝ). Even more, if we do not care about the two entries di,i for i ¼ 1, N, we will have a Hermitian matrix. Finally, as it was seen in Section 4, we can say that P<sup>N</sup> can be considered as a suitable approximation to the conjugate matrix to the coordinate matrix.

In conclusion, we have introduced a matrix with the properties that a Hermitian matrix should comply with, except for two of its entries. Besides, our partition provides congruency between discrete, continuous, and matrix treatments of the exponential function and of its properties.
