Fractional Quantum Mechanics Overview

• Fractional Quantum Mechanics is a generalization that replaces the traditional quadratic kinetic energy with a fractional power, capturing nonlocal quantum dynamics.
• It employs the fractional Schrödinger equation and quantum Riesz fractional derivative, enabling the study of anomalous transport and scaling phenomena.
• Applications span disordered systems, mesoscopic physics, and quantum gravity, offering new insights into diffusion, tunneling, and spectral properties.

Fractional quantum mechanics generalizes conventional quantum dynamics by replacing the standard quadratic kinetic energy operator with a non-integer, or "fractional," power of momentum. This framework is motivated by extending the Feynman path integral to Lévy flight trajectories, leading to a fractional Schrödinger equation where the usual Laplacian is replaced by a quantum Riesz fractional derivative. The resulting theory retains core quantum characteristics—such as hermiticity, (generalized) probability current, and parity symmetry—while introducing nonlocal spatial dynamics parameterized by the Lévy index $1 < \alpha \leq 2$. Standard quantum mechanics is recovered for $\alpha = 2$. This theoretical generalization yields new insights and tools for studying anomalous transport, nonlocality, disorder, and scaling phenomena in quantum systems, with applications spanning from mesoscopic physics to quantum gravity phenomenology.

1. Mathematical Foundations

The central object of fractional quantum mechanics is the fractional Schrödinger equation: $$i\hbar\,\frac{\partial\psi(\mathbf{r},t)}{\partial t} = D_{\alpha}\,(-\hbar^2\Delta)^{\alpha/2}\,\psi(\mathbf{r},t) + V(\mathbf{r})\,\psi(\mathbf{r},t)$$ where $D_{\alpha}$ is a generalized scaling coefficient and $(-\hbar^2\Delta)^{\alpha/2}$ is defined via the Fourier transform as: $$(-\hbar^2 \Delta)^{\alpha/2}\psi(x,t) = \frac{1}{2\pi\hbar} \int_{-\infty}^{+\infty} dp\,e^{ipx/\hbar}\,|p|^{\alpha}\,\phi(p,t)$$ This quantum Riesz fractional derivative replaces the standard Laplacian with a nonlocal operator: for $\alpha = 2$ the evolution reduces to that governed by Brownian paths and parabolic dispersion; for $1 < \alpha < 2$ the dynamics are governed by Lévy flights and non-parabolic energy-momentum relations.

The fractional Hamiltonian operator is: $$\hat{H}_{\alpha} = D_{\alpha}|\hat{p}|^{\alpha} + V(\mathbf{r}),\quad \hat{p} = -i\hbar\nabla$$ Self-adjointness holds in the Hilbert space $L^2$, ensuring real energy eigenvalues. The theory admits a generalized continuity equation, with the conserved current: $$j(\mathbf{r}, t) = D_{\alpha}\left[\psi^*(\mathbf{r}, t)(-i\hbar\nabla)^{\alpha-1}\psi(\mathbf{r}, t) + \text{c.c.}\right]$$ and the fractional velocity operator is $v = (1/i\hbar)[\hat{H}_\alpha, \mathbf{r}]$.

Scaling invariance is preserved under: $$t = \lambda t',\quad x = Bx',\quad D_\alpha = B^{\alpha-1} D'_\alpha,\quad \psi(x,t) = B^{-1/2}\psi'(x',t')$$ with normalization preserved.

2. Path Integral and Formal Structure

Fractional quantum mechanics is naturally formulated from a path integral over Lévy flights (as opposed to Gaussian paths in standard quantum mechanics), leading to the configuration space propagator: $$K(x_b, t_b \mid x_a, t_a) = \int[\mathcal{D}x(\tau)\mathcal{D}p(\tau)]\,\exp\left\{\frac{i}{\hbar}\int_{t_a}^{t_b}[p(\tau)\dot{x}(\tau) - H_\alpha(p(\tau), x(\tau))]\,d\tau\right\}$$ Explicit propagators for the free particle can be written in terms of the Fox H-function: $$K^{(0)}(x,t|x_0,0) = \frac{1}{2\pi\hbar} \int dp\,\exp\left\{\frac{i}{\hbar}[p(x-x_0) - D_\alpha |p|^\alpha t]\right\}$$

$$K^{(0)}(x, t) = \frac{1}{\alpha |x|} H^{1,1}_{2,2}\left[\frac{1}{\hbar}\left(\frac{|x|}{D_\alpha t^{1/\alpha}}\right)^{\alpha} \bigg| \cdots\right]$$

These propagators retain the Gaussian form only for $\alpha = 2$; for fractional $\alpha$ the dynamics exhibit heavy-tailed statistics and anomalous diffusion.

3. Applications and Exactly Solvable Models

3.1 Free Particle and Kernel

The momentum-space solution for a free particle is: $$\psi(p, t) = \exp\left[ -\frac{i}{\hbar} D_\alpha |p|^\alpha t \right]\psi(p, 0)$$ Fourier transformation yields the fundamental kernel.

3.2 Infinite Potential Well

For $V(x) = 0$ in $|x| \leq a$, $V(x) = \infty$ outside, the spectrum is determined by boundary condition $ψ(|x|\leq a) = 0$ at $|x|=a$: $$-D_\alpha(-\hbar^2 d^2/dx^2)^{\alpha/2}\psi(x) = E\psi(x)$$ Eigenfunctions are sine/cosine with wave numbers $k_n$ determined by boundary conditions, and energies $E_n = D_\alpha (\hbar k_n)^\alpha$.

3.3 Delta and Linear Potentials

For the attractive delta potential $V(x) = -\gamma\delta(x)$: $$E = -\left[\frac{\gamma B(1/\alpha, 1-1/\alpha) \hbar^\alpha D_\alpha}{\alpha - 1}\right]^{\alpha/(\alpha-1)}$$ Bound state wave functions are expressible via Fox H-functions.

For a linear potential $V(x) = Fx$, solutions yield quantized energies reducing to Airy function results at $\alpha = 2$.

3.4 Fractional Bohr Atom

For $V(r) = -Ze^2/r$,

$$a_n = a_0 n^{\alpha-1},\quad a_0 = \left(\frac{\alpha D_\alpha \hbar^\alpha}{Ze^2}\right)^{1/(\alpha - 1)},\quad E_n = (1 - \alpha) E_0 n^{-\alpha/(\alpha-1)}$$

3.5 Fractional Oscillator

The fractional oscillator Hamiltonian $$H_{\alpha,\beta} = D_\alpha(-\hbar^2\Delta)^{\alpha/2} + q^2|r|^\beta$$ yields energy spectrum via semiclassical quantization: $$E_n \propto \left[q^{2/\beta}(D_\alpha^{1/\alpha})\right]^{\frac{\alpha\beta}{\alpha + \beta}}(n + \text{const.})^{\frac{\alpha\beta}{\alpha + \beta}}$$ Only standard $(\alpha, \beta) = (2,2)$ yields equidistant level spacing; otherwise, spectral statistics become nontrivial, a property exploited in models of quark confinement.

4. Statistical Mechanics and Many-Body Generalizations

The extension to statistical mechanics involves defining the density operator via the path integral over the "fractional Euclidean action". For many-body systems, the reduced density matrix formalism leads to Thomas–Fermi models with generalized kinetic energy functional: $$T_\text{TF}[\rho] = C_F \int \rho(\vec{r})^{\alpha/3+1} d\vec{r},\quad C_F = \frac{3}{\alpha + 3} D_\alpha \hbar^\alpha (3\pi^2)^{\alpha/3}$$ The electron degeneracy pressure in a free gas becomes: $$p = \frac{\alpha}{\alpha + 3}(3\pi^2)^{\alpha/3} D_\alpha \hbar^\alpha \rho^{\alpha/3+1}$$ Such generalizations allow for the construction of fractional density functional theory (DFT), building on the fact that the Hohenberg–Kohn theorems hold for Hermitian fractional Hamiltonians.

5. Nonlocality, Scaling, and Anomalous Effects

The nonlocality of the Riesz fractional derivative significantly alters quantum transport and spectral properties.

• Tunneling: For a delta barrier, the transmission at zero energy is nonzero for $\alpha < 2$:

$T_0 = \cos^2 (\pi/\alpha)$

implying finite zero-energy tunneling due to nonlocal propagation.

• Scaling Behavior: Under appropriate scale transformations, the fractional Schrödinger equation exhibits invariance, facilitating the paper of scaling limits and critical behavior in complex systems.
• Anomalous Diffusion: Fractional quantum mechanics models describe anomalous diffusion, beyond the standard quantum Brownian motion paradigm, by incorporating Lévy statistics directly into quantum propagation.

6. Physical Relevance and Outlook

Fractional quantum mechanics has implications in:

• Disordered and Mesoscopic Systems: Models with $\alpha < 2$ can simulate the effects of strong disorder or long-range hopping, with specific relevance to quantum wells, heterostructures, perovskite semiconductors, and quantum wires where the kinetic energy may deviate from the parabolic law due to structural or environmental factors.
• High-Energy and Condensed Matter Physics: Fractional oscillators are applied in quarkonium spectroscopy, while many-body extensions open the path to studying nonlocal interacting fermion systems in a DFT framework.
• Quantum Gravity Phenomenology: Corrections to the Schrödinger equation involving fractional Laplacians have been linked to deep infrared effects and fractal geometry in quantum spacetime (Varão et al., 22 May 2024).

7. Summary Table: Key Equations and Applications

Concept	Key Equation or Structure	Physical Domain/Application
Fractional Schrödinger Eq.	$$i\hbar \partial_t \psi = D_\alpha(-\hbar^2\Delta)^{\alpha/2} \psi + V \psi$$	Universal Quantum Dynamics
Riesz Fractional Derivative	Fourier: $(–\hbar^2 d^2/dx^2)^{\alpha/2}$ in momentum space	Nonlocality, Anomalous Diffusion
Free Particle Kernel	$K^{(0)}(x,t) = (1/\alpha |x|) H^{1,1}_{2,2}[... ]$	Propagation, Scattering
Infinite Potential Well	$E_n = D_\alpha (\hbar k_n)^\alpha$	Quantum Wells, Bound States
Delta Potential (Bound State)	$$E = -\left[ \gamma B(... ) \hbar^\alpha D_\alpha/(\alpha-1)\right]^{\alpha/(\alpha-1)}$$	Resonant/Discrete Structure
Fractional Bohr Atom	$E_n \propto n^{-\alpha/(\alpha-1)}$, $a_n \propto n^{\alpha-1}$	Atomic, Excitonic Systems
Fractional Oscillator	$$E_n \propto (n + \text{const.})^{\frac{\alpha\beta}{\alpha + \beta}}$$	Spectral Anomaly, Quarkonium

• Principles and foundations: "Principles of Fractional Quantum Mechanics" (Laskin, 2010)
• Tunneling and nonlocal transport: "Tunneling in Fractional Quantum Mechanics" (Oliveira et al., 2010)
• Statistical mechanics and DFT: "Applications of density matrix in the fractional quantum mechanics" (Dong, 2010)
• Scattering theory: "Scattering problems in the fractional quantum mechanics governed by the 2D space fractional Schrodinger equation" (Jianping, 2013), "Fractional Green's function for the time-dependent scattering problem in the Space-time-fractional quantum mechanics" (Jianping, 2013)
• Physical applications and advanced topics: "Fractional quantum mechanics meets quantum gravity phenomenology" (Varão et al., 22 May 2024), "Quantum Well in Fractional Quantum Mechanics" (Laskin, 29 Aug 2025), "On the discreet spectrum of fractional quantum hydrogen atom in two dimensions" (Stephanovich, 2019)

Fractional quantum mechanics thus provides a rigorously constructed and flexible extension of quantum theory, allowing for systematic exploration of nonlocal and anomalous quantum phenomena. Its consistent mathematical underpinnings, together with its compatibility with fundamental quantum principles, make it a robust framework for ongoing theoretical and applied research.