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arXiv:2111.14058 · Physical Review A

Quantum mechanics of a fermion confined to a curved surface in Foldy-Wouthuysen representation

Hao Zhao, Yong-Long Wang, Cheng-Zhi Ye, Run Cheng, Guo-Hua Liang, Hui Liu

Published May 25, 2022• 8 citations
arXivHTMLPDF

Abstract

In Foldy-Wouthuysen representation, this paper deduces the effective quantum mechanics for a relativistic particle confined to a curved surface in an external field by using the thin-layer quantization scheme. It is found that the quantum effects caused by introducing the confining potential can be taken as the results of relativistic correction in the nonrelativistic limit. Meanwhile, the spin connection of the curved surface can produce a Zeeman-like gap through the relativistic correction term. In addition, the confining potential can provide a curvature-independent energy shift resulting from the Zitterbewegung effect. As an example, this paper applies the effective Hamiltonian to a torus surface, in which the spin effects related to the introduced confining potential are obtained. These results demonstrate the scaling of the noncommutation of the nonrelativistic limit and the thin-layer quantization procedure.

DOI
10.1103/physreva.105.052220

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By considering the spin connection, we deduce the effective equation for a spin-1/2 particle confined to a curved surface with the nonrelativistic limit and in the thin-layer quantization formalism. We obtain a pseudo-magnetic-field and an effective spin-orbit interaction generated by the spin connection. Geometrically, the pseudo-magnetic field is proportional to the Gaussian curvature and the effective spin-orbit interaction is determined by the Weingarten curvature tensor. In particular, we find that the pseudo-magnetic-field and the effective spin-orbit interaction can be employed to separate the electrons with different spin orientations. All these results are demonstrated in two examples: a straight cylindrical surface and a bent one.
We derive the Pauli equation for a charged spin particle confined to move on a spatially curved surface S\mathcal{S}S in an electromagnetic field. Using the thin-layer quantization scheme to constrain the particle on S\mathcal{S}S, and in the transformed spinor representations, we obtain the well-known geometric potential Vg{V}_{g}Vg​ and the presence of ${e}^{\ensuremath{-}i\ensuremath{\varphi}}$, which can generate additive spin connection geometric potentials by the curvilinear coordinates derivatives, and we find that the two fundamental evidences in the literature [Giulio Ferrari and Giampaolo Cuoghi, Phys. Rev. Lett. 100, 230403 (2008)] are still valid in the present system without source current perpendicular to S\mathcal{S}S. Finally, we apply the surface Pauli equation to spherical, cylindrical, and toroidal surfaces, in which we obtain expectantly the geometric potentials and new spin connection geometric potentials, and find that only the normal Pauli matrix appears in these equations.
Investigating the geometric effects resulting from the detailed behaviors of the confining potential, we consider square and circular confinements to constrain a particle to a space curve. We find a torsion-induced geometric potential and a curvature-induced geometric momentum just in the square case, while a geometric gauge potential solely in the circular case. In the presence of electromagnetic field, a geometrically induced magnetic moment couples with magnetic field as an induced Zeeman coupling only for the circular confinement also. As spin-orbit interaction is considered, we find some additional terms for the spin-orbit coupling, which are induced not only by torsion, but also curvature. Moreover, in the circular case, the spin also couples with an intrinsic angular momentum, which describes the azimuthal motions mapped on the space curve. As an important conclusion for the thin-layer quantization approach, some substantial geometric effects result from the confinement boundaries. Finally, these results are proved on a helical wire.
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In the spirit of the thin-layer quantization approach, we give the formula of the geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space. The geometric contributions can result from the reduced commutation relation between the acted function depending on normal variable and the normal derivative. According to the formula, we obtain the geometric potential, geometric momentum, geometric orbital angular momentum, geometric linear Rashba and cubic Dresselhaus spin-orbit couplings. As an example, a truncated cone surface is considered. We find that the geometric orbital angular momentum can provide an azimuthal polarization for spin, and the sign of the geometric Dresselhaus spin-orbit coupling can be flipped through the inclination angle of generatrix.
Abstract Combining the deviation between thin layers’ adjacent surfaces with the confining potential method applied to the quantum curved systems, we derive the effective Schrödinger equation describing the particle constrained within a curved layer, accompanied by a general geometric potential V gq composed of a compression-corrected geometric potential V gq * and a novel potential V gq * * brought by the deviation. Applying this analysis to the cylindrical layer emerges two types of deviation-induced geometric potential, resulting from the the cases of slant deviation and tangent deviation, respectively, which strongly renormalizes the purely geometric potential and contribute to the energy spectrum based on a very substantial deepening of bound states they offer.