Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Chinese Physics Letters

Schrödinger Equation of a Particle on a Rotating Curved Surface

Long Du, Yong-Long Wang, Guo-Hua Liang, Guang-Zhen Kang, Hong-Shi Zong

Published Mar 1, 2016• 4 citations
Publisher

Abstract

We derive the Schrödinger equation of a particle constrained to move on a rotating curved surface S. Using the thin-layer quantization scheme to confine the particle on S, and with a proper choice of gauge transformation for the wave function, we obtain the well-known geometric potential V g and an additive Coriolis-induced geometric potential in the co-rotational curvilinear coordinates. This novel effective potential, which is included in the surface Schrödinger equation and is coupled with the mean curvature of S, contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian. We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.

DOI
10.1088/0256-307x/33/3/030301

Related papers

Schrödinger Equation for a Particle on a Curved Surface in an Electric and Magnetic Field

Giulio Ferrari, Giampaolo Cuoghi

We derive the Schrödinger equation for a spinless charged particle constrained to move on a curved surface in the presence of an electric and magnetic field. The particle is confined on the surface using a thin-layer procedure, which gives rise to the well-known geometric potential. The electric and magnetic fields are included via the four potential. We find that there is no coupling between the fields and the surface curvature and that, with a proper choice of the gauge, the surface and transverse dynamics are exactly separable. Finally, we derive an analytic form of the Hamiltonian for spherical, cylindrical, and toroidal surfaces.

arXiv:0803.2670•Physical Review Letters•Jun 12, 2008•202 citations

An application of Heun functions in the quantum mechanics of a constrained particle

Alexandre G. M. Schmidt, Matheus E. Pereira

Using the thin-layer quantization, we formulate the problem of a Schrödinger particle constrained to move along a coordinate surface of the bi-spherical coordinate system. In three-dimensional space, the free Schrödinger equation is not separable in this coordinate system. However, when we consider the equation for a particle constrained to a given surface, there are only two degrees of freedom. One has to introduce a geometrical potential to attach the particle to the surface. This well-known potential has two contributions: one from Gauss’ curvature and the other from the mean curvature. The Schrödinger equation leads to a general Heun equation. We solve it exactly and present the eigenfunctions and plots of the probability densities, and, as an application of this methodology, we study the problem of an electric charge propagating along these coordinate surfaces in the presence of a uniform magnetic field.

Journal of Mathematical Physics•Apr 1, 2023•5 citations

Geometric Potential and Dirac Quantization

Dingkun Lian, Liangdong Hu, Quanhui Liu

Abstract A fundamental problem regarding the Dirac quantization of a free particle on an ( ) curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature‐induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin‐layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator‐ordering‐free section is identified and is then found sufficient to lead to the expected form of geometric potential.

arXiv:1703.06388•Annalen der Physik•Apr 3, 2018•16 citations

Pauli equation for a charged spin particle on a curved surface in an electric and magnetic field

Yong-Long Wang, Long Du, Chang-Tan Xu +2 more

We derive the Pauli equation for a charged spin particle confined to move on a spatially curved surface $\mathcal{S}$ in an electromagnetic field. Using the thin-layer quantization scheme to constrain the particle on $\mathcal{S}$, and in the transformed spinor representations, we obtain the well-known geometric potential ${V}_{g}$ 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 $\mathcal{S}$. 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.

arXiv:1411.1169•Physical Review A•Oct 22, 2014•57 citations

Quantum dynamics within curved thin layers with deviation

Run Cheng, Hao Zhao, Cui-Bai Luo +4 more

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.

Chinese Physics B•Apr 12, 2024•0 citations

Quantum particle on a surface: Catenary surface and paraboloid of revolution

S. Habib Mazharimousavi

Abstract We revisit the Schrödinger equation of a quantum particle that is confined on a curved surface. Inspired by the seminal work of da Costa (1980 Phys. Rev. A 23 1982) we find the field equation in a more convenient notation. The contribution of the principal curvatures in the effective binding potential on the surface is emphasized. Furthermore, using the so-called Monge-Gauge we construct the approximate Schrödinger equation for a flat surface with small fluctuations. Finally, the resulting Schrödinger equation is solved for some specific surfaces. In particular, we give exact solutions for a particle confined on a Catenary surface and a paraboloid of revolution.

arXiv:2107.05621•Physica Scripta•Oct 22, 2021•5 citations
We derive the Schrödinger equation of a particle constrained to move on a rotating curved surface S. Using the thin-layer quantization scheme to confine the particle on S, and with a proper choice of gauge transformation for the wave function, we obtain the well-known geometric potential V g and an additive Coriolis-induced geometric potential in the co-rotational curvilinear coordinates. This novel effective potential, which is included in the surface Schrödinger equation and is coupled with the mean curvature of S, contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian. We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.
We derive the Schrödinger equation for a spinless charged particle constrained to move on a curved surface in the presence of an electric and magnetic field. The particle is confined on the surface using a thin-layer procedure, which gives rise to the well-known geometric potential. The electric and magnetic fields are included via the four potential. We find that there is no coupling between the fields and the surface curvature and that, with a proper choice of the gauge, the surface and transverse dynamics are exactly separable. Finally, we derive an analytic form of the Hamiltonian for spherical, cylindrical, and toroidal surfaces.
Using the thin-layer quantization, we formulate the problem of a Schrödinger particle constrained to move along a coordinate surface of the bi-spherical coordinate system. In three-dimensional space, the free Schrödinger equation is not separable in this coordinate system. However, when we consider the equation for a particle constrained to a given surface, there are only two degrees of freedom. One has to introduce a geometrical potential to attach the particle to the surface. This well-known potential has two contributions: one from Gauss’ curvature and the other from the mean curvature. The Schrödinger equation leads to a general Heun equation. We solve it exactly and present the eigenfunctions and plots of the probability densities, and, as an application of this methodology, we study the problem of an electric charge propagating along these coordinate surfaces in the presence of a uniform magnetic field.
Abstract A fundamental problem regarding the Dirac quantization of a free particle on an ( ) curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature‐induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin‐layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator‐ordering‐free section is identified and is then found sufficient to lead to the expected form of geometric potential.
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.
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.
Abstract We revisit the Schrödinger equation of a quantum particle that is confined on a curved surface. Inspired by the seminal work of da Costa (1980 Phys. Rev. A 23 1982) we find the field equation in a more convenient notation. The contribution of the principal curvatures in the effective binding potential on the surface is emphasized. Furthermore, using the so-called Monge-Gauge we construct the approximate Schrödinger equation for a flat surface with small fluctuations. Finally, the resulting Schrödinger equation is solved for some specific surfaces. In particular, we give exact solutions for a particle confined on a Catenary surface and a paraboloid of revolution.