Semiclassical effective description of a quantum particle on a sphere with non-central potential

TL;DR

This paper introduces a semiclassical framework for quantum particles on curved surfaces, revealing significant quantum corrections to classical dynamics, especially in non-central potentials, with implications for nanostructures and molecular systems.

Contribution

It develops a quantum-corrected semiclassical approach on curved surfaces, incorporating quantum back-reaction effects and demonstrating their impact on particle trajectories and distributions.

Findings

Quantum fluctuations cause measurable phase shifts in azimuthal precession.

Quantum corrections amplify asymmetry in non-central potentials.

Trajectories are driven toward specific regions faster than classical predictions.

Abstract

We develop a semiclassical framework for studying quantum particles constrained to curved surfaces using the momentous quantum mechanics formalism, which extends classical phase-space to include quantum fluctuation variables (moments). In a spherical geometry, we derive quantum-corrected Hamiltonians and trajectories that incorporate quantum back-reaction effects absent in classical descriptions. For the free particle, quantum fluctuations induce measurable phase shifts in azimuthal precession of approximately 8-12%, with uncertainty growth rates proportional to initial moment correlations. When a non-central Makarov potential is introduced, quantum corrections dramatically amplify its asymmetry. For strong coupling ($\gamma$ = -1.9), the quantum-corrected force drives trajectories preferentially toward the southern hemisphere on timescales 40% shorter than classical predictions, with trajectory densities exhibiting up to 3-fold enhancement in the preferred region. Throughout evolution, the solutions rigorously satisfy Heisenberg uncertainty relations, validating the truncation scheme. These results demonstrate that quantum effects fundamentally alter semiclassical dynamics in curved constrained systems, with direct implications for charge transport in carbon nanostructures, exciton dynamics in curved quantum wells, and reaction pathways in cyclic molecules.