## 4 Physical Numerical Applications  
The fractional quantum mechanics based on the Levy path integral generalizes the standard quantum mechanics based on the well-known Feynman path integral. It has been realized that the understanding of complex quantum and classical physics phenomena requires the implementation of the Levy fights random process instead of Brownian motion. The motivation behind the involvement of Levy fights into consideration is that if the path integral over Brownian trajectories leads to the well-known Schrodinger equation, then the path integral over Levy trajectories leads to the fractional Schrodinger equation. This section is intended for displaying some numerical applications to show the efficiency, reliability, and simplicity of the proposed method in finding conformable series coefficients for both linear and nonlinear FSEs associated with trapping potential. Wolfram's Mathematica software package was used in all computational processes. **Example 1**. : _Consider the following linear fractional Schrodinger equation with zero trapping potential [42; 43]:_  
\[iT_{t}^{a}u(x,t)-u_{xx}(x,t)=0,\ t\geq 0,\ 0<a\leq 1, \tag{20}\]  
_subject to the initial condition_  
\[u(x,0)=1+\cosh(2x),x\in\mathbb{R}. \tag{21}\]This model is a special case of FSE that arises in the modeling of free particle state, where the particle moves without any potential energy. In particular, the exact solution of this model at \(\alpha=1\) is \(u(x,t)\ =1+\cosh(2x)e^{-4 it}\). To achieve our goal, let \(u(x,t)=v(x,t)+ive(x,t)\) so that \(u(x,0)=v_{0}(x)+iw_{0}(x)\). Then, the corresponding coupled system is given by  
\[ \begin{array}{l}T_{t}^{\alpha}v(x,t)-w_{xx}(x,t)=0,\\ T_{t}^{\alpha}w(x,t)+v_{xx}(x,t)=0,\end{array}  \tag{22}\]  
subject to the following initial conditions  
\[v(x,0)=1+\cosh(2x)\ \text{and}\ w(x,0)=0. \tag{23}\]  
According to CRPS algorithm, the \(n\)th-truncated series solutions are given by  
\[ \begin{array}{l}v_{n}(x,t)=1+\cosh(2x)+\sum\limits_{k=1}^{n}a_{k}(x)\frac{t ^{\alpha}}{a^{k}k!},\\ w_{n}(x,t)=\sum\limits_{k=1}^{n}b_{k}(x)\frac{t^{\alpha}}{a^{k}k!},\end{array}  \tag{24}\]  
while the \(n\)th-residual functions can be defined by  
\[ \begin{array}{l}Res_{v}^{n}(x,t)=T_{t}^{\alpha}v_{n}(x,t)-\left(w_{n} \right)_{xx}(x,t),\\ Res_{w}^{n}(x,t)=T_{t}^{\alpha}w_{n}(x,t)+(v_{n})_{xx}(x,t).\end{array}  \tag{25}\]  
For \(n=1\), substitute the first truncated series \(v_{1}(x,t)=1+\cosh(2x)+a_{1}(x)\frac{t^{\alpha}}{\alpha}\) and \(w_{1}(x,t)=b_{1}(x)\frac{t^{\alpha}}{\alpha}\) into the first truncated residual functions \(Res_{v}^{1}(x,t)\) and \(Res_{w}^{1}(x,t)\) such that  
\[ \begin{array}{l}Res_{v}^{1}(x,t)=a_{1}(x)-\frac{b_{1}^{n}(x)t^{\alpha}}{ \alpha},\\ Res_{w}^{1}(x,t)=b_{1}(x)+\left(4\cosh(2x)+\frac{a_{1}^{n}(x)t^{\alpha}}{ \alpha}\right).\end{array}  \tag{26}\]  
By using \(Res_{v}^{1}(x,0)=0\) and \(Res_{w}^{1}(x,0)=0\), one can get  
\[a_{1}(x)=0\ \text{and}\ b_{1}(x)=-4\cosh(2x).\]  
Therefore, the first CRPS approximations of Equations (22) and (23) are  
\[v_{1}(x,t)=1+\cosh(2x)\ \text{and}\ w_{1}(x,t)=-4\cosh(2x)\frac{t^{\alpha}}{ \alpha}.\]  
For \(n=2\), substitute the second truncated series solutions \(v_{2}(x,t)=1+\cosh(2x)+a_{2}(x)\frac{t^{2\alpha}}{2\alpha^{2}}\) and \(w_{2}(x,t)=-4\cosh(2x)\frac{t^{\alpha}}{\alpha}+b_{2}(x)\frac{t^{2\alpha}}{ 2\alpha^{2}}\) into the second truncated residual functions \(Res_{v}^{2}(x,t)\) and \(Res_{w}^{2}(x,t)\) such that  
\[ \begin{array}{l}Res_{v}^{2}(x,t)=\frac{a_{2}(x)t^{\alpha}}{\alpha}+\left( \frac{16\cosh(2x)t^{\alpha}}{\alpha}-\frac{b_{2}^{n}(x)t^{2\alpha}}{2\alpha^{ 2}}\right),\\ Res_{w}^{2}(x,t)=\frac{b_{2}(x)t^{\alpha}}{\alpha}+\frac{a_{2}^{n}(x)t^{2 \alpha}}{2\alpha^{2}}.\end{array}