# Physics:Fractional quantum mechanics

In physics, fractional quantum mechanics is a generalization of standard quantum mechanics, which naturally comes out when the Brownian-like quantum paths substitute with the Lévy-like ones in the Feynman path integral. This concept was discovered by Nick Laskin who coined the term fractional quantum mechanics.[1]

## Fundamentals

Standard quantum mechanics can be approached in three different ways: the matrix mechanics, the Schrödinger equation and the Feynman path integral.

The Feynman path integral[2] is the path integral over Brownian-like quantum-mechanical paths. Fractional quantum mechanics has been discovered by Nick Laskin (1999) as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. A path integral over the Lévy-like quantum-mechanical paths results in a generalization of quantum mechanics.[3] If the Feynman path integral leads to the well known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation.[4] The Lévy process is characterized by the Lévy index α, 0 < α ≤ 2. At the special case when α = 2 the Lévy process becomes the process of Brownian motion. 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.[5] This is the key point to launch the term fractional Schrödinger equation and more general term fractional quantum mechanics. As mentioned above, at α = 2 the Lévy motion becomes Brownian motion. Thus, fractional quantum mechanics includes standard quantum mechanics as a particular case at α = 2. The quantum-mechanical path integral over the Lévy paths at α = 2 becomes the well-known Feynman path integral and the fractional Schrödinger equation becomes the well-known Schrödinger equation.

### Fractional Schrödinger equation

The fractional Schrödinger equation discovered by Nick Laskin has the following form (see, Refs.[1,3,4])

$\displaystyle{ i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_\alpha (-\hbar ^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t)\, }$

using the standard definitions:

• r is the 3-dimensional position vector,
• ħ is the reduced Planck constant,
• ψ(r, t) is the wavefunction, which is the quantum mechanical function that determines the 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/∂r2 is the Laplace operator.

Further,

• Dα is a scale constant with physical dimension [Dα] = [energy]1 − α·[length]α[time]α, at α = 2, D2 =1/2m, where m is a particle mass,
• the operator (−ħ2Δ)α/2 is the 3-dimensional fractional quantum Riesz derivative defined by (see, Refs.[3, 4]);
$\displaystyle{ (-\hbar ^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i \mathbf{p}\cdot \mathbf{r}/\hbar}|\mathbf{p}|^\alpha \varphi ( \mathbf{p},t), }$

Here, the wave functions in the position and momentum spaces; $\displaystyle{ \psi(\mathbf{r},t) }$ and $\displaystyle{ \varphi (\mathbf{p},t) }$ are related each other by the 3-dimensional Fourier transforms:

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

The index α in the fractional Schrödinger equation is the Lévy index, 1 < α ≤ 2.

### 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.

## References

1. Laskin, Nikolai (2000). "Fractional quantum mechanics and Lévy path integrals". Physics Letters A 268 (4–6): 298–305. doi:10.1016/S0375-9601(00)00201-2.
2. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals ~McGraw-Hill, New York, 1965
3. Laskin, Nick (1 August 2000). "Fractional quantum mechanics". Physical Review E (American Physical Society (APS)) 62 (3): 3135–3145. doi:10.1103/physreve.62.3135. ISSN 1063-651X. Bibcode2000PhRvE..62.3135L.
4. Laskin, Nick (18 November 2002). "Fractional Schrödinger equation". Physical Review E (American Physical Society (APS)) 66 (5): 056108. doi:10.1103/physreve.66.056108. ISSN 1063-651X. PMID 12513557. Bibcode2002PhRvE..66e6108L.
5. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications ~Gordon and Breach, Amsterdam, 1993
• Samko, S.; Kilbas, A.A.; Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books. ISBN 978-2-88124-864-1.
• Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 978-0-444-51832-3.
• Pinsker, F.; Bao, W.; Zhang, Y.; Ohadi, H.; Dreismann, A.; Baumberg, J. J. (25 November 2015). "Fractional quantum mechanics in polariton condensates with velocity-dependent mass". Physical Review B (American Physical Society (APS)) 92 (19): 195310. doi:10.1103/physrevb.92.195310. ISSN 1098-0121.