Editing Fractional calculus (section)

===Fractional Schrödinger equation in quantum theory===
The [[fractional Schrödinger equation]], a fundamental equation of [[fractional quantum mechanics]], has the following form:<ref>{{cite journal |last=Laskin |first=N. |year=2002 |title=Fractional Schrodinger equation |journal=Phys. Rev. E |volume=66 |issue=5 |pages=056108 |arxiv=quant-ph/0206098 |citeseerx=10.1.1.252.6732 |doi=10.1103/PhysRevE.66.056108 |pmid=12513557 |bibcode=2002PhRvE..66e6108L |s2cid=7520956}}</ref><ref>{{cite book |doi=10.1142/10541 |title=Fractional Quantum Mechanics |year=2018 |last1=Laskin |first1=Nick |isbn=978-981-322-379-0 |citeseerx=10.1.1.247.5449}}</ref>
<math display="block">i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_{\alpha } \left(-\hbar^2\Delta \right)^{\frac{\alpha}{2}}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t)\,.</math>

where the solution of the equation is the [[wavefunction]] {{math|''ψ''('''r''', ''t'')}} – the quantum mechanical [[probability amplitude]] for the particle to have a given [[position vector]] {{math|'''r'''}} at any given time {{mvar|t}}, and {{mvar|ħ}} is the [[reduced Planck constant]]. The [[potential energy]] function {{math|''V''('''r''', ''t'')}} depends on the system.

Further, <math display="inline">\Delta = \frac{\partial^2}{\partial\mathbf{r}^2}</math> is the [[Laplace operator]], and {{mvar|D<sub>α</sub>}} is a scale constant with physical [[dimensional analysis|dimension]] {{math|1=[''D<sub>α</sub>''] = J<sup>1 − ''α''</sup>·m<sup>''α''</sup>·s<sup>−''α''</sup> = kg<sup>1 − ''α''</sup>·m<sup>2 − ''α''</sup>·s<sup>''α'' − 2</sup>}}, (at {{math|1=''α'' = 2}}, <math display="inline">D_2 = \frac{1}{2m}</math> for a particle of mass {{mvar|m}}), and the operator {{math|(−''ħ''<sup>2</sup>Δ)<sup>''α''/2</sup>}} is the 3-dimensional fractional quantum Riesz derivative defined by
<math display="block">(-\hbar^2\Delta)^\frac{\alpha}{2}\psi (\mathbf{r},t) = \frac 1 {(2\pi \hbar)^3} \int d^3 p e^{\frac{i}{\hbar} \mathbf{p}\cdot\mathbf{r}}|\mathbf{p}|^\alpha \varphi (\mathbf{p},t) \,.</math>

The index {{mvar|α}} in the fractional Schrödinger equation is the Lévy index, {{math|1 < ''α'' ≤ 2}}.

====Variable-order fractional Schrödinger equation====
As a natural generalization of the [[fractional Schrödinger equation]], the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena:<ref>{{cite journal |last1=Bhrawy |first1=A.H. |last2=Zaky |first2=M.A. |year=2017 |title=An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations |journal=Applied Numerical Mathematics |volume=111 |pages=197–218 |doi=10.1016/j.apnum.2016.09.009}}</ref>
<math display="block">i\hbar \frac{\partial \psi^{\alpha(\mathbf{r})} (\mathbf{r},t)}{\partial t^{\alpha(\mathbf{r})} } = \left(-\hbar^2\Delta \right)^{\frac{\beta(t)}{2}}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t),</math>

where <math display="inline">\Delta = \frac{\partial^2}{\partial\mathbf{r}^2}</math> is the [[Laplace operator]] and the operator {{math|(−''ħ''<sup>2</sup>Δ)<sup>''β''(''t'')/2</sup>}} is the variable-order fractional quantum Riesz derivative.