# About the Use of Generalized Forms of Derivatives in the Study of Electromagnetic Problems
## 6 Final Comments
By using a Gaussian as the initial function, we obtain the result reported in Figure 18, where we showed the square modulus of the wave function \(\Psi(\eta,\tau)\) at different times. The relevant evolution is characterized by an asymmetric diffusion, including a shift of the peak and the development of a long tail. We also checked that this behavior is compatible (at short times) with the results obtained with the perturbative solution. The use of pseudo-operators, involving square roots of derivative operators in relativistic Quantum Mechanics is delicate either from the mathematical and physical points of view. However, it is scientifically sound and does not imply any doubtful steps. The operators on both sides of Equation (89) are indeed hermitian, and the associated evolution operator is unitary. This is the minimum requirement when a Quantum Mechanics problem  
Figure 18: \(\left|\,\Psi(\eta,\tau)\,\right|^{2}\) vs. \(\eta\) for different values of \(\tau\), by reference to Equation (89). is posed, since the physical consequences from the associated mathematical treatment is not flawed by intrinsic \"fatal\" errors. Some authors \"generalize\" the Schroedinger equation by directly replacing (time or spatial) derivative with fractional counterparts. The development of a credible formulation of Fractional Quantum Mechanics (FQM), requires the change of the specific paradigms leading to the formulation of the Schroedinger equation itself. For this reason, in Ref. [56], the problem has been afforded starting from a path integral formulation. This study has led to the conclusion that standard Quantum Mechanics follows from the fractality of Brownian motion and FQM deepen its root into the fractality of Levy flights. The formal basis of the (non-relativistic) FQM has been considered with adequate mathematical rigor. The Schroedinger equation derived in [56] reads  
\[ \begin{cases}i\hbar\ \partial_{t}\ \psi=\hat{H}_{a}\ \psi\\ \hat{H}_{a}=-D_{a}(\hbar\ \partial_{x})^{a}+V(x)\end{cases}  \tag{93}\]  
where \(D_{a}\) is a constant, adjusting the associated physical units and defining the link between energy and momentum in FQM [56]. The operator on the r.h.s. of Equation (93) is not Hermitian; this implies that the associated eigenvalues are not real and the corresponding evolution operator is not unitary. An important possibility within this respect is offered by the formalism adopted in dissipative Quantum Mechanics [57] whose mathematical (and physical as well) treatment requires the use of bi-orthogonal states [56; 57] yielding a more appropriate definition of the operator self-adjointedness condition and a correct tool for the evaluation of matrix elements between different quantum states. Regarding the Free Particle evolution, Equation (93) yields  
\[\psi(\xi,\sigma)=e^{i\,T_{a}\,\sigma\,(\partial_{\xi})^{a}}\psi_{0}(\xi), \xi=\frac{x}{\lambda_{c}},\qquad\lambda_{c}=\frac{\hbar}{mc},\qquad\sigma= \frac{ct}{\lambda_{c}},\qquad T_{a}=\frac{D_{a}}{c}\hbar^{a-1} \tag{94}\]  
where we used a more convenient non-dimensional form. The solution of Equation (94) can be obtained using the discussed method. Using the definition of the Riesz fractional derivative [56; 58], we obtain the behavior reported in Figure 19. It is qualitatively similar to that displayed in Figure 18, thus suggesting a non-local nature of the underlying dynamics. In order to complete this discussion, we would like to add as a further example of application of the discussion so far developed to the solution of propagation equations of D'Alembert type, namely \[ \begin{cases}\vec{\
abla}^{2}U-\frac{1}{v^{2}}\partial_{t}^{2}U=0,\qquad\qquad U(x,y, z,0)=f(x,y,z),\\ \partial_{t}U(x,y,z,t)\mid_{t=0}=g(x,y,z),\qquad\qquad\vec{\
abla}^{2}=\partial_{x}^{2}+ \partial_{y}^{2}+\partial_{z}^{2}.\end{cases}  \tag{95}\]  
The use of the generalization of the evolution operator method [59] allows casting of the solution of Equation (95) in the form  
\[