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The '''fractional Schrödinger equation''' is a fundamental equation of [[Physics:Fractional quantum mechanics|fractional quantum mechanics]]. It was discovered by Nick Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term ''fractional Schrödinger equation'' was coined by Nick Laskin.<ref>{{Cite journal |doi = 10.1016/S0375-9601(00)00201-2|title = Fractional quantum mechanics and Lévy path integrals|journal = Physics Letters A|volume = 268|issue = 4–6|pages = 298–305|year = 2000|last1 = Laskin|first1 = Nikolai|arxiv = hep-ph/9910419}}</ref>

==Fundamentals==

The fractional Schrödinger equation in the form originally obtained by Nick Laskin is:<ref>{{cite journal | last=Laskin | first=Nick | title=Fractional Schrödinger equation | journal=Physical Review E | volume=66 | issue=5 | date=18 November 2002 | issn=1063-651X | doi=10.1103/physreve.66.056108 | pmid=12513557 | page=056108|arxiv=quant-ph/0206098}}</ref>

{{Equation box 1
|indent =:
|title=
|equation = <math>i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_\alpha (-\hbar
^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t)</math>
|cellpadding= 5
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

*'''r''' is the 3-dimensional position vector, 
*''ħ'' is the reduced [[Physics:Planck constant|Planck constant]], 
*''ψ''('''r''', ''t'') is the wavefunction, which is the quantum mechanical probability amplitude for the particle to have a given position '''r''' at any given time ''t'', 
*''V''('''r''', ''t'') is a [[Physics:Potential energy|potential energy]], 
*Δ = &part;<sup>2</sup>/&part;'''r'''<sup>2</sup> is the [[Laplace operator]].

Further,
*''D<sub>α</sub>'' is a scale constant with [[Dimensional analysis|physical dimension]] [D<sub>α</sub>] = [energy]<sup>1 − ''α''</sup>·[length]<sup>''α''</sup>[time]<sup>−''α''</sup>, at ''α''&nbsp;=&nbsp;2,  ''D''<sub>2</sub> =1/2''m'', where ''m'' is a particle mass,
*the operator (−''ħ''<sup>2</sup>Δ)<sup>''α''/2</sup> is the 3-dimensional fractional quantum Riesz derivative defined by (see, Ref.[2]);

::<math>
(-\hbar ^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar
)^3}\int d^3pe^{i\frac{\mathbf{pr}}\hbar }|\mathbf{p}|^\alpha \varphi (
\mathbf{p},t),
</math>

Here, the wave functions in the [[Physics:Position and momentum space|position and momentum space]]s; <math>\psi(\mathbf{r},t)</math> and <math> \varphi (\mathbf{p},t)</math> are related each other by the 3-dimensional [[Fourier transform]]s:

:<math>
\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i \mathbf{p}\cdot\mathbf{r}/\hbar}\varphi (\mathbf{p},t),\qquad \varphi (\mathbf{p},t)=\int d^3re^{-i
\mathbf{p}\cdot\mathbf{r}/\hbar }\psi (\mathbf{r},t).
</math>

The index ''α'' in the fractional Schrödinger equation is the Lévy index, 1&nbsp;<&nbsp;''α''&nbsp;≤&nbsp;2. Thus, the fractional Schrödinger equation includes a space [[Derivative|derivative]] of fractional order ''α'' instead of the second order (''α''&nbsp;=&nbsp;2) space derivative in the standard [[Physics:Schrödinger equation|Schrödinger equation]]. Thus, the fractional Schrödinger equation is a fractional differential equation in accordance with modern terminology.<ref>S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional
Integrals and Derivatives, Theory and Applications ~Gordon
and Breach, Amsterdam, 1993</ref> This is the main point of the term ''fractional Schrödinger equation'' or a more general term [[Physics:Fractional quantum mechanics|fractional quantum mechanics]].<ref>{{cite journal | last=Laskin | first=Nick | title=Fractional quantum mechanics | journal=Physical Review E | publisher=American Physical Society (APS) | volume=62 | issue=3 | date=1 August 2000 | issn=1063-651X | doi=10.1103/physreve.62.3135 | pages=3135–3145| pmid=11088808 |arxiv=0811.1769}}</ref> At ''α''&nbsp;=&nbsp;2 fractional Schrödinger equation becomes the well-known [[Physics:Schrödinger equation|Schrödinger equation]].

The fractional Schrödinger equation has the following [[Physics:Operator|operator]] form

{{Equation box 1
|indent =:
|title=
|equation = <math>
i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=\hat{H}_\alpha
\psi (\mathbf{r},t)</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

where the fractional Hamilton operator <math>\hat{H}_\alpha </math> is given by

: <math>
\hat{H}_\alpha =D_\alpha \left(-\hbar ^2\Delta \right)^{\alpha /2}+V(\mathbf{r},t). 
</math>

The Hamilton operator, <math>\hat{H}_\alpha </math> corresponds to the [[Physics:Classical mechanics|classical mechanics]] Hamiltonian function introduced by Nick Laskin

: <math>
H_\alpha (\mathbf{p},\mathbf{r})=D_\alpha |\mathbf{p}|^\alpha +V(\mathbf{r},t),
</math>

where '''p''' and '''r''' are the momentum and the position vectors respectively.

===Time-independent fractional Schrödinger equation===

The special case when the Hamiltonian <math>H_\alpha </math> is independent of time
: <math>
H_\alpha = D_\alpha \left(-\hbar ^2\Delta \right)^{\alpha /2} + V(\mathbf{r}), 
</math>
is of great importance for physical applications. 
It is easy to see that in this case there exist the special solution of the fractional Schrödinger equation
: <math>
\psi (\mathbf{r},t)=e^{-(i/\hbar )Et}\phi (\mathbf{r}), 
</math>
where <math>\phi (\mathbf{r})</math> satisfies
:<math>
H_\alpha \phi (\mathbf{r}) = E\phi (\mathbf{r}),  
</math>

or

:<math>
D_\alpha \left(-\hbar ^2\Delta \right)^{\alpha /2} \phi (\mathbf{r}) + V(\mathbf{r})\phi (
\mathbf{r}) = E\phi (\mathbf{r}). 
</math>

This is the '''time-independent fractional Schrödinger equation''' (see, Ref.[2]).

Thus, we see that the [[Wave function|wave function]] <math>\psi (\mathbf{r},t)</math> oscillates with a definite frequency. In [[Physics:Classical physics|classical physics]] the frequency corresponds to the energy. Therefore, the quantum mechanical state has a definite energy ''E''. 
The probability to find a particle at <math>\mathbf{r}</math> is the absolute square of the wave function <math>| \psi (\mathbf{r},t) |^2 .</math>
Because of time-independent fractional Schrödinger equation this is equal to <math>| \phi (\mathbf{r})|^2 </math> and does not depend upon the time. 
That is, the probability of finding the particle at <math>\mathbf{r}</math> is independent of the time. One can say that the system is in a stationary
state. In other words, there is no variation in the probabilities as a function of time.

===Probability current density===
The conservation law of fractional quantum mechanical probability has been discovered for the first time by D.A.Tayurskii and Yu.V. Lysogorski <ref>{{cite journal | last1=Tayurskii | first1=D A | last2=Lysogorskiy | first2=Yu V | title=Superfluid hydrodynamic in fractal dimension space | journal=Journal of Physics: Conference Series | publisher=IOP Publishing | volume=394 | date=29 November 2012 | issn=1742-6588 | doi=10.1088/1742-6596/394/1/012004 | page=012004|arxiv=1108.4666}}</ref>

:<math>
\frac{\partial \rho (\mathbf{r},t)}{\partial t}+\nabla \cdot \mathbf{j}(
\mathbf{r},t)+K(\mathbf{r},t)=0,  
</math>

where <math>\rho (\mathbf{r},t)=\psi ^{\ast }(\mathbf{r},t)\psi (\mathbf{r},t)</math>
is the quantum mechanical probability density and the vector <math>\mathbf{j}(\mathbf{r},t)</math>
can be called by the fractional probability current density vector

:<math>
\mathbf{j}(\mathbf{r},t)=\frac{D_\alpha \hbar }i\left( \psi ^{*}(\mathbf{r}
,t) \left(-\hbar ^2\Delta \right)^{\alpha /2-1}\mathbf{\nabla }\psi (\mathbf{r},t)-\psi (
\mathbf{r},t) \left(-\hbar ^2\Delta \right)^{\alpha /2-1}\mathbf{\nabla }\psi ^{*}(
\mathbf{r},t)\right) ,  
</math>

and

:<math>
\mathit{K}(\mathbf{r},t)=\frac{D_\alpha \hbar }i\left(\mathbf{\nabla }\psi (\mathbf{r}
,t) \left(-\hbar ^2\Delta \right)^{\alpha /2-1}\mathbf{\nabla }\psi ^{*}(\mathbf{r},t)-(\mathbf{\nabla } \psi^{*} (
\mathbf{r},t) \left(-\hbar ^2\Delta \right)^{\alpha /2-1}\mathbf{\nabla }\psi (
\mathbf{r},t)\right) ,  
</math>

here we use the notation (see also [[Matrix calculus#Scope|matrix calculus]]): <math>
\mathbf{\nabla =\partial /\partial r}
</math>.

It has been found in Ref.[5] that there are quantum physical conditions when the new term <math>\mathit{K}(\mathbf{r},t)</math> is negligible and we come to the [[Continuity equation#Quantum mechanics|continuity equation]] for quantum probability current and quantum density (see, Ref.[2]):

:<math>
\frac{\partial \rho (\mathbf{r},t)}{\partial t}+\nabla \cdot \mathbf{j}(
\mathbf{r},t)=0.  
</math>

Introducing the [[Physics:Momentum operator|momentum operator]] <math>\hat{\mathbf{p}}=\frac{\hbar }{i}
\frac{\partial }{\partial \mathbf{r}}</math> we can write the vector <math>\mathbf{j}</math> in the form (see, Ref.[2])

:<math>
\mathbf{j}=D_{\alpha }\left( \psi (\hat{\mathbf{p}}^{2})^{\alpha /2-1}
\hat{\mathbf{p}}\psi ^{\ast }+\psi ^{\ast }(\hat{\mathbf{p}}^{\ast
2})^{\alpha /2-1}\hat{\mathbf{p}}^{\ast }\psi \right).  
</math>

This is fractional generalization of the well-known equation for probability current density vector of standard quantum mechanics (see, Ref.[7]).

====Velocity operator====

The quantum mechanical velocity operator <math>\hat{\mathbf{v}}</math> is defined as follows:

:<math>
\hat{\mathbf{v}}=\frac{i}{\hbar }(H_{\alpha }\hat{\mathbf{r}}\mathbf{
-}\hat{\mathbf{r}}H_{\alpha }),
</math>

Straightforward calculation results in (see, Ref.[2])
:<math>
\hat{\mathbf{v}}=\alpha D_{\alpha } \left|\hat{\mathbf{p}}^{2}\right|^{\alpha
/2-1}\hat{\mathbf{p}}\,. 
</math>

Hence,

:<math>
\mathbf{j=}\frac{1}{\alpha }\left( \psi \hat{\mathbf{v}}\psi ^{\ast
}+\psi ^{\ast }\hat{\mathbf{v}}\psi \right) ,\qquad 1<\alpha \leq 2.
</math>

To get the [[Physics:Probability current|probability current]] density equal to 1 (the current when one particle passes through unit area per unit time) the wave function of a free particle has to be normalized as

:<math>
\psi (\mathbf{r},t)=\sqrt{\frac{\alpha }{2\mathrm{v}}}\exp \left[\frac{i}{\hbar }(
\mathbf{p}\cdot\mathbf{r}-Et)\right],\qquad E=D_{\alpha }|\mathbf{p}|^{\alpha
},\qquad 1<\alpha \leq 2,  
</math>

where <math>\mathrm{v}</math> is the particle [[Physics:Velocity|velocity]], <math>\mathrm{v}=\alpha D_{\alpha
}p^{\alpha -1}</math>.

Then we have

:<math>
\mathbf{j}=\frac{\mathbf{v}}{\mathrm{v}},\qquad \mathbf{v}=\alpha D_{\alpha
}\left|\mathbf{p}^{2}\right|^{\frac{\alpha }{2}-1}\mathbf{p,}  
</math>

that is, the vector <math>\mathbf{j}</math> is indeed the [[Unit vector|unit vector]].

== Physical applications ==

===Fractional Bohr atom===


When <math>V(\mathbf{r})</math> is the potential energy of hydrogenlike atom,

: <math>
V(\mathbf{r})=-\frac{Ze^{2}}{|\mathbf{r}|},
</math>

where ''e'' is the [[Physics:Electron charge|electron charge]] and ''Z'' is the [[Physics:Atomic number|atomic number]] of the hydrogenlike atom, (so ''Ze'' is the nuclear charge of the atom), we come to following fractional eigenvalue problem,

: <math>
D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}\phi (\mathbf{r})-\frac{Ze^{2}}{|
\mathbf{r|}}\phi (\mathbf{r})=E\phi (\mathbf{r}).
</math>

This eigenvalue problem has first been introduced and solved by Nick Laskin in.<ref>{{cite journal | last=Laskin | first=Nick | title=Fractals and quantum mechanics | journal=Chaos: An Interdisciplinary Journal of Nonlinear Science | publisher=AIP Publishing | volume=10 | issue=4 | year=2000 | pages=780–790 | issn=1054-1500 | doi=10.1063/1.1050284 | pmid=12779428 }}</ref>

Using the first [[Biography:Niels Bohr|Niels Bohr]] postulate yields
: <math>
 \alpha
D_{\alpha }\left( \frac{n\hbar }{a_{n}}\right) ^{\alpha }=\frac{Ze^{2}}{a_{n}
}, 
</math>

and it gives us the equation for the [[Physics:Bohr radius|Bohr radius]] of the fractional hydrogenlike atom
: <math>
a_{n}=a_{0}n^{\alpha /(\alpha -1)}.  
</math>

Here ''a''<sub>0</sub> is the fractional Bohr radius (the radius of the lowest, ''n'' = 1, Bohr orbit) defined as,

: <math>
a_{0}=\left( \frac{\alpha D_{\alpha }\hbar ^{\alpha }}{Ze^{2}}\right) ^{1/(\alpha -1)}.  
</math>

The energy levels of the fractional hydrogenlike atom are given by

: <math>
E_{n}=(1-\alpha )E_{0}n^{-\alpha/(\alpha -1)},\qquad 1<\alpha \leq 2,
</math>

where ''E''<sub>0</sub> is the [[Physics:Binding energy|binding energy]] of the electron in the lowest Bohr orbit
that is, the energy required to put it in a state with ''E'' = 0 corresponding to ''n'' = ∞,

: <math>
E_{0}=\left( \frac{Ze^{2}}{\alpha D_{\alpha }^{1/\alpha }\hbar }\right) ^{\alpha/(\alpha -1)}.  
</math>

The energy (''α''  − 1)''E''<sub>0</sub> divided by ''ħc'', (''α''  − 1)''E''<sub>0</sub>/''ħc'',  can be considered as fractional generalization of the
[[Physics:Rydberg constant|Rydberg constant]] of standard [[Physics:Quantum mechanics|quantum mechanics]]. For ''α''&nbsp;=&nbsp;2 and ''Z'' = 1  the formula
<math>(\alpha -1)E_{0}/\hbar c</math> is transformed into

: <math>\mathrm{Ry}=me^{4}/2\hbar ^{3}c</math>,
which is the well-known expression for the [[Physics:Rydberg formula|Rydberg formula]].

According to the second [[Biography:Niels Bohr|Niels Bohr]] postulate, the frequency of radiation <math>\omega_{m,n}</math> associated with the transition, say, for example from the orbit ''m'' to the orbit ''n'', is,

: <math>
\omega_{m,n} =\frac{(1-\alpha )E_{0}}{\hbar }\left[ \frac{1}{n^{\frac{\alpha }{
\alpha -1}}}-\frac{1}{m^{\frac{\alpha }{\alpha -1}}}\right]  
</math>.

The above equations are fractional
generalization of the Bohr model. In the special Gaussian case, when (''α''&nbsp;=&nbsp;2) those equations give us the well-known results of the [[Physics:Bohr model|Bohr model]].<ref>{{cite journal | last=Bohr | first=N. | title=XXXVII. On the constitution of atoms and molecules | journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science | publisher=Informa UK Limited | volume=26 | issue=153 | year=1913 | issn=1941-5982 | doi=10.1080/14786441308634993 | pages=476–502| url=https://zenodo.org/record/1430916 }}</ref>

===The infinite potential well===

A particle in a one-dimensional well moves in a potential field <math>V({x})</math>, which is zero for 
<math>-a\leq x\leq a</math> and which is infinite elsewhere,

{{NumBlk|:| <math>
V(x)=\infty ,\qquad x < -a
</math>|{{EquationRef|i}}}}

{{NumBlk|:| <math>
V(x)=0,\quad -a \leq x \leq a
</math>|{{EquationRef|ii}}}}

{{NumBlk|:| <math>
V(x)=\infty ,\qquad \ x>a
</math>|{{EquationRef|iii}}}}

It is evident ''a priori'' that the energy spectrum will be discrete. The solution of the fractional Schrödinger equation for the stationary state with well-defined energy ''E'' is described by a wave function <math>\psi (x)</math>, which can be written as

:<math>
\psi (x,t)=\left(-i\frac{Et}{\hbar }\right)\phi(x)
,</math>

where <math>\phi (x)</math>,  is now time independent. 
In regions ({{EquationNote|i}}) and ({{EquationNote|iii}}),
the fractional Schrödinger equation can be satisfied only if we take <math>\phi (x)=0</math>. In the middle region
({{EquationNote|ii}}), the time-independent fractional Schrödinger equation is (see, Ref.[6]).

: <math>
D_\alpha (\hbar \nabla )^\alpha \phi (x)=E\phi (x).  
</math>

This equation defines the wave functions and the energy spectrum within region ({{EquationNote|ii}}), while outside of the region (ii), ''x''&nbsp;<&nbsp;−''a'' and ''x''&nbsp;>&nbsp;''a'', the wave functions are zero. The wave function <math>\phi (x)</math> has to be continuous everywhere, thus we impose the boundary conditions <math>\phi (-a)=\phi (a)=0</math> for the solutions of the ''time-independent fractional Schrödinger equation'' (see, Ref.[6]). Then the solution in region ({{EquationNote|ii}}) can be written as

:<math>
\phi (x)=A\exp(ikx) +B\exp(-ikx).
</math>

To satisfy the boundary conditions we have to choose

:<math>
A = -B\exp(-i2ka),
</math>

and
:<math>
\sin(2ka) =0.
</math>

It follows from the last equation that

:<math>
2ka = n\pi.
</math>

Then the even (<math>\phi _n^{\mathrm{even}}(-x) = \phi _n^{\mathrm{even}}(x)</math> under reflection <math>x\rightarrow -x</math>) solution of the time-independent fractional Schrödinger equation <math>\phi ^{\mathrm{even}}(x)</math> in the infinite potential well is

:<math>
\phi _n^{\mathrm{even}}(x)=\frac 1{\sqrt{a}}\cos \left[ \frac{
n\pi x}{2a}\right] , \quad n = 1, 3, 5, \ldots
</math>

The odd (<math>\phi _n^{\mathrm{odd}}(-x) = -\phi _n^{\mathrm{odd}}(x)</math> under reflection <math>x\rightarrow -x</math>) solution of the time-independent fractional Schrödinger equation <math>\phi ^{\mathrm{even}}(x)</math> in the infinite potential well is

:<math>
\phi _n^{\mathrm{odd}}(x)=\frac 1{\sqrt{a}}\sin \left[ \frac{
n\pi x}{2a}\right] , \quad n = 2, 4, 6, \ldots
</math>

The solutions <math>\phi ^{\mathrm{even}}(x)</math> and <math>\phi ^{\mathrm{odd}}(x)</math> have the
property that

:<math>
\int\limits_{-a}^{a}dx\phi _{m}^{\mathrm{even}}(x)\phi _{n}^{\mathrm{even}}(x)
= \int\limits_{-a}^{a}dx\phi _{m}^{\mathrm{odd}}(x)\phi _{n}^{\mathrm{odd}}(x)
= \delta _{mn},
</math>

where <math>\delta _{mn}</math> is the [[Kronecker symbol]] and

:<math>
\int\limits_{-a}^{a}dx\phi _{m}^{\mathrm{even}}(x)\phi _{n}^{\mathrm{odd}
}(x)=0.
</math>

The eigenvalues of the particle in an infinite potential well are (see, Ref.[6])

: <math>
E_n=D_\alpha \left( \frac{\pi \hbar }{2a}\right) ^\alpha n^\alpha ,\qquad
\qquad n=1,2,3, \ldots, \qquad 1<\alpha \leq 2.  
</math>

It is obvious that in the Gaussian case (''α''&nbsp;=&nbsp;2) above equations are ö
transformed into the standard quantum mechanical equations for a [[Physics:Particle in a box|particle in a box]] (for example, see
Eq.(20.7) in <ref>L.D. Landau and E.M. Lifshitz, Quantum mechanics (Non-relativistic Theory), Vol.3, Third Edition, Course of Theoretical Physics, Butterworth-Heinemann, Oxford, 2003</ref>)

The state of the lowest energy, the [[Physics:Ground state|ground state]], in the infinite potential
well is represented by the <math>\phi _n^{\mathrm{even}}(x)</math> at ''n''=1,
: <math>
\phi _{\mathrm{ground}}(x)\equiv \phi _1^{\mathrm{even}}(x)=\frac 1{\sqrt{a}
}\cos \left(\frac{\pi x}{2a}\right),
</math>

and its energy is

: <math>
E_{\mathrm{ground}}=D_{\alpha }\left( \frac{\pi \hbar }{2a}\right) ^{\alpha}.
</math>
===Fractional quantum oscillator===

''Fractional quantum oscillator'' introduced by Nick Laskin (see, Ref.[2]) is the fractional quantum  mechanical model with the [[Physics:Hamiltonian operator|Hamiltonian operator]] <math>H_{\alpha ,\beta }</math> defined as

:<math>
H_{\alpha,\beta}=D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}+q^{2}|\mathbf{
r}|^{\beta },\quad 1<\alpha \leq 2,\quad 1<\beta \leq 2,  
</math>

where ''q'' is interaction constant.

The fractional Schrödinger equation for the wave
function <math>\psi (\mathbf{r},t)</math> of the fractional quantum oscillator is,

:<math>
i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_{\alpha }(-\hbar
^{2}\Delta )^{\alpha /2}\psi (\mathbf{r},t)+q^{2}|\mathbf{r}|^{\beta }\psi (
\mathbf{r},t)  
</math>

Aiming to search for solution in form

:<math>
\psi (\mathbf{r},t)=e^{-iEt/\hbar }\phi (\mathbf{r}),
</math>

we come to the time-independent fractional Schrödinger equation,

:<math>
D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}\phi (\mathbf{r},t)+q^{2}|\mathbf{
r}|^{\beta }\phi (\mathbf{r},t)=E\phi (\mathbf{r},t).  
</math>

The Hamiltonian <math>H_{\alpha,\beta}</math> is the fractional
generalization of the 3D [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]] Hamiltonian of standard quantum
mechanics.

====Energy levels of the 1D fractional quantum oscillator in semiclassical approximation====

The energy levels of 1D fractional quantum oscillator with the Hamiltonian function <math>
H_{\alpha}=D_{\alpha }|p|^{\alpha }+q^{2}|x|^{\beta }</math> were found in semiclassical approximation (see, Ref.[2]).

We set the total energy equal to ''E'', so that

:<math>
E=D_{\alpha }|p|^{\alpha }+q^{2}|x|^{\beta },  
</math>

whence

:<math>
|p|=\left( \frac{1}{D_{\alpha }} \left(E-q^{2}|x|^{\beta }\right)\right)
^{1/\alpha }.
</math>

At the turning points <math>p=0</math>. Hence, the classical motion is possible in the range <math>|x|\leq \left(E/q^{2}\right)^{1/\beta }</math>.

A routine use of the Bohr-Sommerfeld quantization rule yields

:<math>
2\pi \hbar \left(n + \frac{1}{2}\right) = \oint p dx = 4\int\limits_{0}^{x_{m}}pdx=4\int\limits_{0}^{x_{m}}D_{\alpha
}^{-1/\alpha }\left(E-q^{2}|x|^{\beta }\right)^{1/\alpha }dx, 
</math>

where the notation <math>\oint  </math> means the integral over one complete period of
the classical motion and <math>x_{m}=\left(E/q^{2}\right)^{1/\beta }</math> is the turning point of classical motion.

To evaluate the integral in the right hand we introduce a new variable <math>y=x\left(E/q^{2}\right)^{-1/\beta }</math>. Then we have

:<math>
\int\limits_0^{x_m}D_\alpha ^{-1/\alpha } \left(E-q^2|x|^\beta \right)^{1/\alpha }dx=\frac 1{D_\alpha ^{1/\alpha }q^{2/\beta }}E^{\frac 1\alpha +\frac 1\beta
}\int\limits_0^1 dy \left(1 - y^\beta \right)^{1/\alpha }.
</math>

The integral over ''dy'' can be expressed in terms of the Beta-function,

:<math>
\int\limits_{0}^{1}dy \left(1-y^{\beta } \right)^{1/\alpha }=\frac{1}{\beta }
\int\limits_{0}^{1}dzz^{\frac{1}{\beta }-1}(1-z)^{\frac{1}{\alpha }}=\frac{1
}{\beta }\Beta \left(\frac{1}{\beta },\frac{1}{\alpha }+1\right).  
</math>

Therefore,

:<math>
2\pi \hbar \left(n + \frac 1 2 \right) = \frac 4{D_\alpha ^{1/\alpha }q^{2/\beta }}E^{\frac
1\alpha +\frac 1\beta }\frac 1\beta \Beta \left(\frac 1\beta ,\frac 1\alpha +1\right).
</math>

The above equation gives the energy levels of stationary states for the 1D fractional quantum oscillator (see, Ref.[2]),

:<math>
E_{n}=\left( \frac{\pi \hbar \beta D_{\alpha }^{1/\alpha }q^{2/\beta }}{2\Beta(
\frac{1}{\beta },\frac{1}{\alpha }+1)}\right) ^{\frac{\alpha \beta }{\alpha
+\beta }}\left(n+\frac{1}{2}\right)^{\frac{\alpha \beta }{\alpha +\beta }}.
</math>

This equation is generalization of the well-known energy levels equation of the
standard [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]] (see, Ref.[7]) and is transformed into it at ''α''&nbsp;=&nbsp;2 and ''β''&nbsp;=&nbsp;2.
It follows from this equation that at <math>\frac{1}{\alpha }+\frac{1}{\beta }=1</math> the energy levels are equidistant. When <math>1<\alpha \leq 2 </math> and <math>1<\beta \leq 2</math> the equidistant energy levels can be for ''α''&nbsp;=&nbsp;2 and ''β''&nbsp;=&nbsp;2 only. It means that the only standard quantum harmonic oscillator has an [[Equidistant|equidistant]] energy spectrum.

=== Fractional quantum mechanics in solid state systems ===
The effective mass of states in solid state systems can depend on the wave vector k, i.e. formally one considers m=m(k). Polariton Bose-Einstein condensate modes are examples of states in solid state systems with mass sensitive to variations and locally in k fractional quantum mechanics is experimentally feasible [https://arxiv.org/abs/1508.03621].

=== Self-accelerating beams ===
Self-accelerating beams, such as the [[Physics:Airy beam|Airy beam]], are known solutions of the conventional free Schrödinger equation (with <math> \alpha=2 </math> and without a potential term). Equivalent solutions exist in the free fractional Schrödinger equation. 
The time-dependent fractional Schrödinger equation in momentum space (assuming <math> \hbar=1 </math> and with one spatial coordinate) is:
:<math>
i \frac{\partial \varphi (p,t)}{\partial t}=D_\alpha |p|^\alpha
\varphi (p,t)</math>.

In position space, an Airy beam is typically expressed using the special Airy function, although it possesses a more transparent expression in momentum space:
:<math> \psi(x,t=0)= \mathrm{Ai}(b x)\exp(ax), \qquad \varphi(p,t=0)=\frac{1}{2 b \sqrt{2 \pi}}\exp\left(\frac{(a-ip)^3}{3 b^3}\right) .</math>
Here, the exponential function ensures the square-integrability of the wave function, i.e. that the beam possesses a finite energy, in order to be a physical solution. The parameter <math>a</math> controls the exponential cut-off on the beam’s tail, while the parameter <math>b</math> controls the width of the peaks in position space. 
The Airy beam solution for the fractional Schrödinger equation in momentum space is obtained from simple integration of the above equation and initial condition: 
:<math>
\varphi(p,t) =\frac{1}{2 b \sqrt{2 \pi}}\exp\left(-i D_\alpha |p|^\alpha t + \frac{(a - i p)^3}{3 b^3}\right) .</math>

This solution self-accelerates at a rate proportional to <math> t^{\frac{2}{3-\alpha}}</math>.<ref>{{Cite journal |doi = 10.1103/PhysRevResearch.2.033274|title = Self-accelerating beam dynamics in the space fractional Schrödinger equation |journal = Physical Review Research |volume = 2|pages = 033274 |year = 2020|last1 = Colas|first1 = David|arxiv =2006.12743}}</ref> When taking <math> \alpha=2 </math> for the conventional Schrödinger equation, one recovers the original Airy beam solution with a parabolic acceleration (<math> \propto t^2 </math>).

== See also ==
*[[Physics:Schrödinger equation|Schrödinger equation]]
*[[Physics:Path integral formulation|Path integral formulation]]
*Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
*[[Fractional calculus]]
*[[Physics:Quantum harmonic oscillator|Quantum harmonic oscillator]]
*[[Physics:Variable-order fractional Schrödinger equation|Variable-order fractional Schrödinger equation]]

== References ==

<references/>
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*{{cite book|title=Fractional Calculus, An Introduction for Physicists|author=R. Herrmann|year=2011|publisher=World Scientific|chapter=9|isbn=978-981-4340-24-3|chapter-url=https://books.google.com/books?id=mPXzp1f7ycMC&pg=PA97&dq=fractional+schrodinger+equation#v=onepage}}
*{{cite book|title=Fractional Dynamics: Recent Advances|author1=J. Klafter |author2=S.C. Lim |author3=R. Metzler |year=2012|publisher=World Scientific|page=426|isbn=978-981-434-059-5|url=https://books.google.com/books?id=wwfqHij4V2MC&dq=fractional+schrodinger+equation}}
*{{cite book|title=Fractional dynamics|author=V.E. Tarasov|year=2010|volume=0|series=Nonlinear physical science|publisher=Springer|chapter=19|isbn=978-3-642-140-037|chapter-url=https://books.google.com/books?id=Mc193Swt_JQC&pg=PA465&dq=Fractional+quantum+mechanics+N+Laskin#v=onepage}}
*{{cite book|title=Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering |author=J. Sabatier, O.P.Agrawal, J.A.T.Machado|year=2007|publisher=Springer|isbn=978-1-402-060-427|url=https://books.google.com/books?id=8lBgSDVXULwC&pg=PA126&dq=Fractional+quantum+mechanics+N+Laskin#v=onepage}}
*{{cite book|title=Fractional Dynamics and Control|author1=D. Baleanu |author2=J.A.T. Machado |author3=A.C.J. Luo |year=2012|publisher=Springer|chapter=17|isbn=978-1-461-404-576|chapter-url=https://books.google.com/books?id=8tpLwiLONDkC&pg=PA210&dq=Fractional+quantum+mechanics+N+Laskin#v=onepage}}
*{{cite journal | last1=Pinsker | first1=F. | last2=Bao | first2=W. | last3=Zhang | first3=Y. | last4=Ohadi | first4=H. | last5=Dreismann | first5=A. | last6=Baumberg | first6=J. J. | title=Fractional quantum mechanics in polariton condensates with velocity-dependent mass | journal=Physical Review B | volume=92 | issue=19 | date=25 November 2015 | issn=1098-0121 | doi=10.1103/physrevb.92.195310 | page=195310| arxiv=1508.03621 }}

== Further reading ==
* {{cite book
  |title=Fractional Quantum Mechanics
  |first=N.|last=Laskin
  |publisher=World Scientific
  |year=2018
|doi=10.1142/10541
  |isbn = 978-981-322-379-0|citeseerx = 10.1.1.247.5449}}
* {{cite journal | last=Naber | first=Mark | title=Time fractional Schrödinger equation | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=45 | issue=8 | year=2004 | issn=0022-2488 | doi=10.1063/1.1769611 | pages=3339–3352|arxiv=math-ph/0410028}}
* {{cite journal | last1=Guo | first1=Xiaoyi | last2=Xu | first2=Mingyu | title=Some physical applications of fractional Schrödinger equation | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=47 | issue=8 | year=2006 | issn=0022-2488 | doi=10.1063/1.2235026 | page=082104}}
* {{cite journal | last1=Wang | first1=Shaowei | last2=Xu | first2=Mingyu | title=Generalized fractional Schrödinger equation with space-time fractional derivatives | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=48 | issue=4 | year=2007 | issn=0022-2488 | doi=10.1063/1.2716203 | page=043502}}
* {{cite journal | last1=Dong | first1=Jianping | last2=Xu | first2=Mingyu | title=Some solutions to the space fractional Schrödinger equation using momentum representation method | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=48 | issue=7 | year=2007 | issn=0022-2488 | doi=10.1063/1.2749172 | page=072105}}
* {{cite journal | last1=Dong | first1=Jianping | last2=Xu | first2=Mingyu | title=Space–time fractional Schrödinger equation with time-independent potentials | journal=Journal of Mathematical Analysis and Applications | publisher=Elsevier BV | volume=344 | issue=2 | year=2008 | issn=0022-247X | doi=10.1016/j.jmaa.2008.03.061 | pages=1005–1017}}
* {{cite journal | last=Tarasov | first=Vasily E. | title=Fractional Heisenberg equation | journal=Physics Letters A | volume=372 | issue=17 | year=2008 | issn=0375-9601 | doi=10.1016/j.physleta.2008.01.037 | pages=2984–2988|arxiv=0804.0586}}
* {{cite journal | last=Tarasov | first=Vasily E. | title=Weyl quantization of fractional derivatives | journal=Journal of Mathematical Physics | volume=49 | issue=10 | year=2008 | issn=0022-2488 | doi=10.1063/1.3009533 | page=102112|arxiv=0907.2699}}
* {{cite journal | last=Iomin | first=Alexander | title=Fractional-time quantum dynamics | journal=Physical Review E | volume=80 | issue=2 | date=28 August 2009 | issn=1539-3755 | doi=10.1103/physreve.80.022103 | page=022103| pmid=19792181 | arxiv=0909.1183 }}
* {{cite book | last=Tarasov | first=Vasily E. | title=Nonlinear Physical Science | chapter=Fractional Dynamics of Open Quantum Systems | publisher=Springer Berlin Heidelberg | location=Berlin, Heidelberg | year=2010 | isbn=978-3-642-14002-0 | issn=1867-8440 | doi=10.1007/978-3-642-14003-7_20 | pages=467–490}}
* {{cite book | last=Tarasov | first=Vasily E. | title=Nonlinear Physical Science | chapter=Fractional Dynamics of Hamiltonian Quantum Systems | publisher=Springer Berlin Heidelberg | location=Berlin, Heidelberg | year=2010 | isbn=978-3-642-14002-0 | issn=1867-8440 | doi=10.1007/978-3-642-14003-7_19 | pages=457–466}}
* {{cite journal | last1=de Oliveira | first1=Edmundo Capelas | last2=Costa | first2=Felix Silva | last3=Vaz | first3=Jayme | title=The fractional Schrödinger equation for delta potentials | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=51 | issue=12 | year=2010 | issn=0022-2488 | doi=10.1063/1.3525976 | page=123517| url=http://www.repositorio.unicamp.br/handle/REPOSIP/73668 }}
* {{cite journal | last1=de Oliveira | first1=E Capelas | last2=Vaz | first2=Jayme | title=Tunneling in fractional quantum mechanics | journal=Journal of Physics A: Mathematical and Theoretical | publisher=IOP Publishing | volume=44 | issue=18 | date=5 April 2011 | issn=1751-8113 | doi=10.1088/1751-8113/44/18/185303 | page=185303| arxiv=1011.1948 }}
* {{cite journal | last=Bayın | first=Selçuk Ş. | title=On the consistency of the solutions of the space fractional Schrödinger equation | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=53 | issue=4 | year=2012 | issn=0022-2488 | doi=10.1063/1.4705268 | page=042105|arxiv=1203.4556}}

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