Anomalous quantum transport via the fractional Schrödinger equation

We investigate the free‑particle fractional Schrödinger equation (FSE) as a minimal model for anomalous quantum transport in fractal‑like media. We replacing the standard Laplacian with a noninteger‑order operator and this uncover three hallmark effects: (i) heavy‑tailed probability densities that deviate markedly from Gaussian spreading; (ii) anomalous scaling of the second moment, in excellent agreement with theoretical predictions; and (iii) temporal coherence “echoes” in the fidelity signaling intrinsic memory of past configurations. Through high‑precision split‑step Fourier simulations on a periodic domain, we observed these phenomena for alpha=2.0,;1.8,;1.5. Our results delineate clear, experimentally accessible signatures—tail‑exponent measurements in photonic lattices, dispersion‑law extraction in ultracold‑atom interferometry, and echo protocols in Bose–Einstein condensates—that can directly validate or constrain the fractional‑order parameter alpha. This study thus establishes a concise, testable framework for fractional quantum dynamics laying groundwork for more deep exploration of nonlocality, memory effects, and their potential role in unifying quantum mechanics with emergent gravitational phenomena.