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

TL;DR

The paper develops a semiclassical framework based on momentous quantum mechanics to study quantum particles constrained to curved surfaces, applying it to a particle on a sphere with both free and non-central Makarov potentials. By integrating Dirac-constraint quantization and geometrical momentum, it derives a quantum-corrected Hamiltonian $H_Q$ and coupled equations for classical variables and quantum moments, revealing quantum back-reaction on trajectories. For the free sphere, quantum fluctuations induce azimuthal phase shifts of about $8$--$12\%$ and broaden the wave packet (e.g., $\sqrt{G^{2,0,0,0}}$ grows by roughly a factor of 3), while with the Makarov potential (strong asymmetry, $\gamma=-1.9$), semiclassical dynamics push trajectories toward the southern hemisphere about $40\%$ faster and increase trajectory densities by a factor of $3$--$4$, in qualitative agreement with full quantum predictions. These results validate the momentous approach for curved geometries and point to observable signatures in STM, ultrafast spectroscopy, and quantum transport in curved nanostructures and ring-like molecules.

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 ($γ$ = -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.

Figures (5)

• Figure 1: a) Single classical (orange solid line) and semiclassical (blue dashed line) trajectories on the sphere surface for a free quantum particle with initial condition $a = P_{\theta_0} = 1$. The semiclassical trajectory shows measurable deviation from the classical geodesic due to quantum back-reaction through moments. The figure shows the time evolution of b) the polar angle, $\theta (t)$, and c) the azimuthal angle, $\phi(t)$. The gray shaded region shows the uncertainty belts $\pm\sqrt{G^{2,0,0,0}(t)}$ and $\pm\sqrt{G^{0,0,2,0}(t)}$, respectively, with dashed and dashed-dotted gray lines indicating the upper and lower limits of the uncertainty belt. The Heisenberg uncertainty relations follow in d)$\Delta G_\theta(t) = G^{2,0,0,0} G^{0,2,0,0} - (G^{1,1,0,0})^2 \geq \hbar^2/4$ and d)$\Delta G_\phi(t) = G^{0,0,2,0} G^{0,0,0,2} - (G^{0,0,1,1})^2 \geq \hbar^2/4$ for each angle. In both cases, the relations are satisfied, confirming physical consistency.
• Figure 2: a) Classical and b) semiclassical trajectories with varying polar momentum $0\leq a \leq 10$. The temporal evolution of the polar and azimuthal angles, $\theta$ and $\phi$, is shown in c) and d) for the classical case and in e) and f) for the semiclassical case, respectively. The classical trajectories form a symmetric pattern, while semiclassical trajectories show broader distribution and slight asymmetry due to the quantum corrections as shown in g) for $\theta$, $\sqrt{G^{2,0,0,0}}(a,t)$ and h) for $\phi$, $\sqrt{G^{0,0,2,0}}(a,t)$. i) Heisenberg uncertainty relations for the polar angle, $\Delta G_\theta(t)\geq\hbar/4$ and j) azimuthal angle, $\Delta G_\phi(t)\geq\hbar/4$. As shown, the relations hold for both variables.
• Figure 3: Single trajectory comparison for Makarov potentials with $\beta=2$, $\gamma=-0.2,$, $-1.0$ and $-1.9$ and $a=1$. The polar and azimuthal angles, $\theta(t)$ and $\phi(t)$, classical (dashed blue lines) and semiclassical (orange solid lines), are compared respectively in a) and b) for different values of $\gamma$ indicated in the inset. The gray areas correspond to the uncertainty belt delimited by its upper and lower limits, shown with dashed and dashed-dotted lines. For weak asymmetry ($\gamma=-0.2$), the semiclassical case shows enhanced oscillations and a broader uncertainty belt. Likewise, $\phi(t)$ shows a phase difference that develops faster than the free particle case. For moderate asymmetry ($\gamma=-1.0$), the trajectory moves into the southern hemisphere ($\theta>\pi/2$) by $t\approx3$ and $\phi(t)$ shows a strong phase deviation ($>15\%$). For strong asymmetry time evolution ($\gamma=-1.9$), the semiclassical $\theta(t)$ rapidly descends to $\theta \approx 2.3$ rad ($132^\circ$) by $t \approx 0.8$, consistent with quantum probability density peak Oliveira2019. Its classical value reaches a similar value only at $t \approx 1.2$ (40% longer). For strong asymmetry, large phase differences in $\phi(t)$ emerge immediately. As the particle approaches the southern pole, the uncertainty relation is no longer satisfied, and the numerical solution terminates earlier than for weaker Makarov potentials.
• Figure 4: Evolution of trajectory ensembles on sphere surface for different values of the Makarov potential $\beta=2$, a)$\gamma=-0.2$, b)$\gamma=-1.0$, and c)$\gamma=-1.9$ for $0\leq t \leq 1.34$ and $-8\leq a\leq 8$. On the left, the classical trajectories are shown, and on the right, the semiclassical ones. a) For weak asymmetry $\gamma=-0.2$, classical trajectories remain nearly symmetric about the equator and semiclassical trajectories exhibit an earlier onset of southern preference and broader spatial distribution. b) For moderate asymmetry $\gamma=-1.0$, the semiclassical trajectories have a more pronounced clustering in the southern hemisphere, demonstrating amplification of asymmetry by quantum corrections. c) For strong asymmetry $\gamma=-1.9$, the semiclassical trajectories present a distinct concentration in the southern hemisphere showing rapid evolution towards the southern pole, with trajectory density ratio $N(\theta>\pi/2)/N(\theta<\pi/2) \approx 3.8$ by $t=10$ (here only $t\leq 1.34$ is shown), matching quantum mechanical prediction factor $\sim 4$Oliveira2019.
• Figure 5: Time evolution of a) the polar ($\theta$) and d) the azimuthal ($\phi$) angles and their quantum corrections, namely b)$\sqrt{G^{2,0,0,0}}$ and e)$\sqrt{G^{0,0,2,0}}$, as functions of $-8\leq a\leq 8$, for different values of the Makarov potential: $\beta=2$, $\gamma=-0.2$ (left), $-1.0$ (center) and $-1.9$ (right). Here, to emphasize the asymmetry induced by the Makarov potential, each variable is shown as a filled contour plot. As seen in a), the semiclassical trajectories deviate toward the southern hemisphere, $\theta>\pi/2$. When $\gamma$ decreases, the trajectories migrate towards the south in shorter time frames. As seen in d), as time increases, the precession rates of $\phi$ get strongly modified by quantum back-reaction. All the trajectories remain physical, as shown by the Heisenberg uncertainty relations, $\Delta G_{\theta,\phi}$, in e) and f).