# Fractional Schrödinger equation

The **fractional Schrödinger equation** is a fundamental equation of 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.^{[1]}

## Fundamentals

The fractional Schrödinger equation in the form originally obtained by Nick Laskin is:^{[2]}

**r**is the 3-dimensional position vector,*ħ*is the reduced 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 potential energy,- Δ = ∂
^{2}/∂**r**^{2}is the Laplace operator.

Further,

*D*is a scale constant with physical dimension [D_{α}_{α}] = [energy]^{1 − α}·[length]^{α}[time]^{−α}, at*α*= 2,*D*_{2}=1/2*m*, where*m*is a particle mass,- the operator (−
*ħ*^{2}Δ)^{α/2}is the 3-dimensional fractional quantum Riesz derivative defined by (see, Ref.[2]);

Here, the wave functions in the position and momentum spaces; and are related each other by the 3-dimensional Fourier transforms:

The index *α* in the fractional Schrödinger equation is the Lévy index, 1 < *α* ≤ 2. Thus, the fractional Schrödinger equation includes a space derivative of fractional order *α* instead of the second order (*α* = 2) space derivative in the standard Schrödinger equation. Thus, the fractional Schrödinger equation is a fractional differential equation in accordance with modern terminology.^{[3]} This is the main point of the term *fractional Schrödinger equation* or a more general term fractional quantum mechanics.^{[4]} At *α* = 2 fractional Schrödinger equation becomes the well-known Schrödinger equation.

The fractional Schrödinger equation has the following operator form

where the fractional Hamilton operator is given by

The Hamilton operator, corresponds to the classical mechanics Hamiltonian function introduced by Nick Laskin

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

### Time-independent fractional Schrödinger equation

The special case when the Hamiltonian is independent of time

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

where satisfies

or

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

Thus, we see that the wave function oscillates with a definite frequency. In 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 is the absolute square of the wave function
Because of time-independent fractional Schrödinger equation this is equal to and does not depend upon the time.
That is, the probability of finding the particle at 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 ^{[5]}

where is the quantum mechanical probability density and the vector can be called by the fractional probability current density vector

and

here we use the notation (see also matrix calculus): .

It has been found in Ref.[5] that there are quantum physical conditions when the new term is negligible and we come to the continuity equation for quantum probability current and quantum density (see, Ref.[2]):

Introducing the momentum operator we can write the vector in the form (see, Ref.[2])

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 is defined as follows:

Straightforward calculation results in (see, Ref.[2])

Hence,

To get the 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

where is the particle velocity, .

Then we have

that is, the vector is indeed the unit vector.

## Physical applications

### Fractional Bohr atom

When is the potential energy of hydrogenlike atom,

where *e* is the electron charge and *Z* is the atomic number of the hydrogenlike atom, (so *Ze* is the nuclear charge of the atom), we come to following fractional eigenvalue problem,

This eigenvalue problem has first been introduced and solved by Nick Laskin in.^{[6]}

Using the first Niels Bohr postulate yields

and it gives us the equation for the Bohr radius of the fractional hydrogenlike atom

Here *a*_{0} is the fractional Bohr radius (the radius of the lowest, *n* = 1, Bohr orbit) defined as,

The energy levels of the fractional hydrogenlike atom are given by

where *E*_{0} is the 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* = ∞,

The energy (*α* − 1)*E*_{0} divided by *ħc*, (*α* − 1)*E*_{0}/*ħc*, can be considered as fractional generalization of the
Rydberg constant of standard quantum mechanics. For *α* = 2 and *Z* = 1 the formula
is transformed into

- ,

which is the well-known expression for the Rydberg formula.

According to the second Niels Bohr postulate, the frequency of radiation associated with the transition, say, for example from the orbit *m* to the orbit *n*, is,

- .

The above equations are fractional
generalization of the Bohr model. In the special Gaussian case, when (*α* = 2) those equations give us the well-known results of the Bohr model.^{[7]}

### The infinite potential well

A particle in a one-dimensional well moves in a potential field , which is zero for and which is infinite elsewhere,

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 , which can be written as

- ,

where , is now time independent. In regions (i) and (iii), the fractional Schrödinger equation can be satisfied only if we take . In the middle region (ii), the time-independent fractional Schrödinger equation is (see, Ref.[6]).

This equation defines the wave functions and the energy spectrum within region (ii), while outside
of the region (ii), x<-a and x>a, the wave functions are zero. The wave function has to be continuous everywhere, thus we impose the boundary conditions for the solutions of the *time-independent fractional Schrödinger equation* (see, Ref.[6]). Then the solution in region (ii) can be written as

To satisfy the boundary conditions we have to choose

and

It follows from the last equation that

Then the even ( under reflection ) solution of the time-independent fractional Schrödinger equation in the infinite potential well is

The odd ( under reflection ) solution of the time-independent fractional Schrödinger equation in the infinite potential well is

The solutions and have the property that

where is the Kronecker symbol and

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

It is obvious that in the Gaussian case (*α* = 2) above equations are ö
transformed into the standard quantum mechanical equations for a particle in a box (for example, see
Eq.(20.7) in ^{[8]})

The state of the lowest energy, the ground state, in the infinite potential
well is represented by the at *n*=1,

and its energy is

### Fractional quantum oscillator

*Fractional quantum oscillator* introduced by Nick Laskin (see, Ref.[2]) is the fractional quantum mechanical model with the Hamiltonian operator defined as

- ,

where *q* is interaction constant.

The fractional Schrödinger equation for the wave function of the fractional quantum oscillator is,

Aiming to search for solution in form

we come to the time-independent fractional Schrödinger equation,

The Hamiltonian is the fractional generalization of the 3D 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 were found in semiclassical approximation (see, Ref.[2]).

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

whence

- .

At the turning points . Hence, the classical motion is possible in the range .

A routine use of the Bohr-Sommerfeld quantization rule yields

where the notation means the integral over one complete period of the classical motion and is the turning point of classical motion.

To evaluate the integral in the right hand we introduce a new variable . Then we have

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

Therefore,

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

This equation is generalization of the well-known energy levels equation of the
standard quantum harmonic oscillator (see, Ref.[7]) and is transformed into it at *α* = 2 and *β* = 2.
It follows from this equation that at the energy levels are equidistant. When and the equidistant energy levels can be for *α* = 2 and *β* = 2 only. It means that the only standard quantum harmonic oscillator has an 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 .

## See also

- Schrödinger equation
- Path integral formulation
- Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
- Fractional calculus
- Quantum harmonic oscillator

## References

- ↑ N. Laskin, (2000), Fractional Quantum Mechanics and Lévy Path Integrals.
*Physics Letters*268A, 298-304. - ↑ N. Laskin, (2002), Fractional Schrödinger equation,
*Physical Review*E66, 056108 7 pages.*(also available online: http://arxiv.org/abs/quant-ph/0206098)* - ↑ S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications ~Gordon and Breach, Amsterdam, 1993
- ↑ N. Laskin, (2000), Fractional Quantum Mechanics,
*Physical Review*E62, 3135-3145.*(also available online: http://arxiv.org/abs/0811.1769)* - ↑ D.A.Tayurskii and Yu.V. Lysogorskiy, (2011), http://arxiv.org/abs/1108.4666, see also, D.A.Tayurskii and Yu.V. Lysogorskiy, (2012), Super fluid hydrodynamic in fractal dimension space,
*Journal of Physics: Conference Series*394 (2012) 012004 5 pages,*(available online: http://iopscience.iop.org/1742-6596/394/1/012004/pdf/1742-6596_394_1_012004.pdf)* - ↑ N. Laskin, (2000), Fractals and quantum mechanics. Chaos 10, 780-790
- ↑ N. Bohr, (1913), Phil. Mag. 26, 1, 476, 857
- ↑ L.D. Landau and E.M. Lifshitz, Quantum mechanics (Non-relativistic Theory), Vol.3, Third Edition, Course of Theoretical Physics, Butterworth-Heinemann, Oxford, 2003

- R. Herrmann (2011). "9".
*Fractional Calculus, An Introduction for Physicists*. World Scientific. ISBN 981 4340 24 3. - J. Klafter; S.C. Lim; R. Metzler (2012).
*Fractional Dynamics: Recent Advances*. World Scientific. p. 426. ISBN 981-434-059-6. - V.E. Tarasov (2010). "19".
*Fractional dynamics*. Nonlinear physical science.**0**. Springer. ISBN 3-642-140-033. - J. Sabatier, O.P.Agrawal, J.A.T.Machado (2007).
*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*. Springer. ISBN 1-402-060-424. - D. Baleanu; J.A.T. Machado; A.C.J. Luo (2012). "17".
*Fractional Dynamics and Control*. Springer. ISBN 1-461-404-576. - F. Pinsker, W. Bao, Y. Zhang, H. Ohadi, A. Dreismann, J. J. Baumberg (2015), http://arxiv.org/abs/1508.03621

## Further reading

- N. Laskin, Fractals and quantum mechanics. Chaos 10 (2000) 780-790
- M. Naber, Time fractional Schrodinger equation. J. Math. Phys. 45 (2004) 3339-3352. arXiv:math-ph/0410028
- Xiaoyi Guo and Mingyu Xu, Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47, (2006) 082104.
- S. Wang, M. Xu, Generalized fractional Schrödinger equation with space-time fractional derivatives J. Math. Phys. 48 (2007) 043502
- J. Dong, M. Xu, Some solutions to the space fractional Schrödinger equation using momentum representation method , J. Math. Phys. 48, 072105 (2007).
- S. Wang, M. Xu, Generalized fractional Schrödinger equation with space-time fractional derivatives J. Math. Phys. 48 (2007) 043502
- J. Dong, M. Xu, Space–time fractional Schrödinger equation with time-independent potentials, J. Math.l Anal. Applic. 344 (2008) 1005–1017.
- V.E. Tarasov, Fractional Heisenberg equation. Phys. Lett. A 372 (2008) 2984-2988. arXiv:0804.0586v1
- V.E. Tarasov, Weyl quantization of fractional derivatives. J. Math. Phys. 49 (2008) 102112. arXiv:0907.2699
- A. Iomin, Fractional-time quantum dynamics. Phys. Rev. E 80, (2009) 022103.
- V.E. Tarasov, Fractional Dynamics of Open Quantum Systems. in Fractional Dynamics, 2010, pp. 467-490. DOI: 10.1007/978-3-642-14003-7_20
- V.E. Tarasov, Fractional Dynamics of Hamiltonian Quantum Systems. in Fractional Dynamics, 2010, pp. 457-466. DOI: 10.1007/978-3-642-14003-7_20
- E.C. Oliveira, F.S. Costa, J. Vaz (Jr.), The fractional Schrödinger equation for delta potentials, J. Math. Phys. 51, 123517 (2010).
- E.C. Oliveira, J. Vaz Jr, "Tunneling in Fractional Quantum Mechanics" J. Phys. A 44 (2011) 185303.
- S.S. Bayın, On the consistency of the solutions of the space fractional Schrödinger equation. J. Math. Phys. 53 (2012) 042105 arXiv:1203.4556