We derive an effective Hamiltonian for a particle confined to a thin tube which is gently twisted and curved to form a closed loop. The Hamiltonian describes the effect of both the curvature and the torsion of the loop correctly to the second order. Applied to the case of a tube of circular cross section, it reveals complete analogy with the Aharonov-Bohm effect. We also discuss eigenmodes for Möbius-like rings formed by a tube of non-circular cross section.
We derive an effective Hamiltonian for a particle confined to a thin tube which is gently twisted and curved to form a closed loop. The Hamiltonian describes the effect of both the curvature and the torsion of the loop correctly to the second order. Applied to the case of a tube of circular cross section, it reveals complete analogy with the Aharonov-Bohm effect. We also discuss eigenmodes for Möbius-like rings formed by a tube of non-circular cross section.
We study the quantum dynamics of a particle confined in a twisted tube with a linearly varying cross section. By relating a general linear transformation matrix to the system's Hamiltonian, we use an extended thin-layer method to derive an effective Hamiltonian for tangential motion under mild and general linear transformations. Explicit forms are provided for three fundamental transformations: rotation, scaling, and shearing. Rotation introduces a gauge field coupled to angular momentum, while scaling and shearing produce geometric potentials that lift degeneracies in non-circular cross sections. In square cross sections, these transformations cause energy splittings among formerly degenerate states, whereas circular cross sections retain degeneracy. Through an example combining rotation and squeezing, we analyze state evolution and compute the quantum geometric tensor to quantify geometric response. Our results demonstrate how geometric transformations can tailor quantum states and suggest that circular waveguides are more robust against mode mixing.
Investigating a particle, a charged particle in an electromagnetic field and a particle with spin-orbit interaction confined to a space curve in square and circular cases, respectively, we first find that their effective Hamiltonian can be defined in terms of the ground states of their confined dimensions. And we obtain interesting results: the torsion-induced geometric potential appears in the square case, while in the circular case it vanishes; the geometrically induced gauge potential and Zeeman coupling appear in the circular case; and the geometric gauge potential, curvature and torsion simultaneously emerge in the spin-orbit interaction in the circular case. In the example, a helix nanowire, the previous results are all demonstrated. These results provide a way to generate an artificial gauge field and to manipulate spin transport by designing the geometry of curved nanodevice.
We investigate the effective dynamics of a particle confined near a space curve. In the strict thin-layer reduction of the nondegenerate transverse ground state, torsion does not enter the local effective Hamiltonian, which contains only the curvature-induced scalar geometric potential. In contrast, for a thin guide with finite transverse width, a leading-order adiabatic projection onto the twofold-degenerate first-excited transverse band renders the rotation of the Frenet normal frame dynamically relevant and generates a matrix-valued Abelian gauge potential. Using a projection-based derivation in a co-rotating Frenet-frame basis, we show that this effective gauge potential is directly determined by the local torsion of the curve. The resulting effective Hamiltonian takes a gauge-covariant form and produces two transverse-mode branches whose parabolic dispersions are shifted in opposite directions in momentum space. For closed curves, the associated holonomy is controlled by the integrated torsion and leads to geometric interference. These results provide a direct realization of a Wilczek--Zee-type connection induced purely by spatial geometry in curved quantum waveguides. We further construct a classical-wave analogue using the degenerate bending modes of an isotropic elastic rod, demonstrating that the same torsion-induced gauge structure appears in continuum wave physics.
Twisted cylindrical tubes are important model systems for nanostructures, heterostructures, and curved quantum devices. In this work, we investigate the quantum behavior of an electron confined to a twisted cylindrical surface. By first calculating the strain tensor to obtain the induced surface metric, we employ da Costa's formalism to derive the geometry-induced quantum potential. This potential modifies the Schrödinger equation even in the absence of external forces, allowing us to determine the bound states and energy eigenvalues. This was made in the linear and non-linear torsion regime. Furthermore, we analyze two distinct scattering problems: (i) scattering within an infinite cylinder containing a twisted section, and (ii) scattering of a free particle incident upon a finite twisted cylinder. Our goal is to understand how geometry and strain influence the properties of analogous untwisted systems. It turns out that both the linear and non-linear twists yield to a geometric phase into the wave function, while the da Costa potential is kept unchanged. Consequently, the system supports bound states whose energie spectrum is twist independent. For both scattering problems, we find that the transmission probability is insensitive to torsion, whereas it is significantly affected by the particle angular momentum and the cylinder's radius, exhibiting distinct oscillatory behavior. These findings suggest relevant implications for engineering quantum devices based on materials with controlled curvature and twist.
Abstract The Hamiltonian for a spin zero particle that is confined to a 1D curve embedded in the 3D space is constructed. Confinement is achieved by starting with the particle living in a small tube surrounding the curve, and assuming an infinitely strong normal force that squeezes the thickness of the tube to zero, eventually pinning the particle to the curve. We follow the new approach that we applied to confine a particle to a surface, in that we start with an expression for the 3D momentum operators whose components along and normal to the curve directions are separately Hermitian. The kinetic energy operator expressed in terms of the momentum operator in the normal direction is then a Hermitian operator in this case. When this operator is dropped and the thickness of the tube surrounding the curve is set to zero, one automatically gets the Hermitian curve Hamiltonian that contains the geometric potential term as expected. It is demonstrated that the origin of this potential lies in the ordering or symmetrization of the original 3D momentum operators in order to render them Hermitian. The Hermitian momentum operator for the particle as it is confined to the curve is also constructed and is seen to be similar to what is known as the geometric momentum of a particle confined to a surface in that it has a term proportional to the curvature that is along the normal to the curve. The force operator of the particle on the curve is also derived, and is shown to reduce, for a curve with a constant curvature and torsion, to a -apparently- single component normal to the curve that is a symmetrization of the classical expression plus a quantum term. All the above quantities are then derived for the specific case of a particle confined to a cylindrical helix embedded in 3D space.
We investigate bound states of a non-relativistic scalar particle in a three-dimensional helically twisted (torsional) geometry, considering both the free case and the presence of external radial interactions. The dynamics is described by the Schrödinger equation on a curved spatial background and, when included, by minimal coupling to a magnetic vector potential incorporating an Aharonov--Bohm flux. After separation of variables, the problem reduces to a one-dimensional radial eigenvalue equation governed by an effective potential that combines torsion-induced Coulomb-like and centrifugal-like structures with magnetic/flux-dependent terms and optional model interactions. Because closed-form analytic solutions are not reliable over the parameter ranges required for systematic scans, we compute spectra and eigenfunctions numerically by formulating the radial equation as a self-adjoint Sturm--Liouville problem and solving it with a finite-difference discretization on a truncated radial domain, with explicit convergence control. We analyze four representative scenarios: (i) no external potential, (ii) Cornell-type confinement, (iii) Kratzer-type interaction, and (iv) the small-oscillation regime around the minimum of a Morse potential. We present systematic trends of the low-lying levels as functions of the torsion parameter, magnetic field, and azimuthal sector, and we show that geometric couplings alone can produce effective confinement even in the absence of an external interaction.