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Journal of Physics A Mathematical and General

Effective potentials of quantum strip waveguides and their dependence upon torsion

Iain John Clark, Anthony J. Bracken

Published Jan 21, 1996• 28 citations
Publisher

Abstract

We investigate the dynamics of a particle constrained to move on a curved quantum strip waveguide embedded in three-space, subject to Dirichlet boundary conditions. By establishing a coordinate system which allows a more general formulation of the dynamics, we demonstrate consistency with previous results and provide evidence that the presence of torsion introduces a further effective potential term. This result has implications for nanostructure device engineering.

DOI
10.1088/0305-4470/29/2/014

Related papers

More on effective potentials of quantum strip waveguides

Iain John Clark

We investigate further the dynamics of a particle constrained to move on a curved quantum strip waveguide embedded in 3-space, subject to Dirichlet boundary conditions. An earlier calculation of the dependence of the effective potential upon torsion and twisting is modified to yield a corrected expression, by use of the `straightening-out' transformation.

Journal of Physics A Mathematical and General•Feb 27, 1998•10 citations

Advances in quantum waveguide dynamics, with applications to physical and biological nanostructures

Iain John Clark

Advances in quantum waveguide dynamics, with applications to physical and biological nanostructuresIAIN JOHN CLARK It is known [1,2] that as semiconductor microstructures get progressively smaller and more free of impurities, it becomes increasingly necessary to use quantum mechanics to describe accurately these systems, which may have features sized between a few and several hundred nanometers.Recent advances in fabrication technology [11] make it feasible to build such nanostructures or mesoscopic systems, as they are known.The term "quantum waveguide" [6] is used to denote a nanostructure which acts as a conducting wire, along which a wavefunction is guided to move by the boundary conditions.Unlike classical wires, curved quantum waveguides have curvature-induced bound states; see [6] for a recent review.This has implications for nanoelectronic circuit design, since the curvature-induced bound states will appear as resonances.However, the effect of torsion (where the waveguide twists out of a plane) and twisting (where the waveguide changes its orientation along its path) upon bound states has attracted little investigation, and is the motivation for the analysis in this work.This thesis develops a unified mathematical approach by which the energy spectrum of a quantum waveguide with curvature and torsion can be analysed, for any crosssectional geometry that remains uniform along the waveguide.Following the introduction and literature review of Chapter 1, a non-relativistic Hamiltonian H for a particle on a quantum waveguide is set up in Chapter 2. A system of curvilinear coordinates is used to express Schrodinger's equation using the Laplace-Beltrami operator V 2 .The metric tensor G is expressed as the product of a curvaturedependent matrix F and the metric tensor G for a "straightened-out" waveguide.The Hamiltonian H is then shown to be unitarily equivalent to the Hamiltonian Hy for a straightened-out waveguide plus interaction terms, Hy -H o + Ky + Vu, where H o is the Hamiltonian for a "straightened-out" waveguide, Ky is a kinetic energy perturbation term and Vu is an effective potential.In Chapter 3, the spectral theory needed to investigate the curvature-induced bound states is developed.The spectrum of a Hamiltonian is partitioned into the

Bulletin of the Australian Mathematical Society•Dec 1, 1997•1 citation

Torsion-induced gauge structure in curved quantum waveguides

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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.

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Quantum mechanics of a particle confined to a space curve embedded in three-dimensional Euclidean space

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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.

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Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides

Jan Stockhofe, Peter Schmelcher

We investigate the quantum mechanics of a single particle constrained to move along an arbitrary smooth reference curve by a confinement that is allowed to vary along the waveguide. The Schr\"odinger equation is evaluated in the adapted coordinate frame and a transverse-mode decomposition is performed, taking into account both curvature and torsion effects and the possibility of a cross-section potential that changes along the curve in an arbitrary way. We discuss the adiabatic structure of the problem, and examine nonadiabatic couplings that arise due to the curved geometry, the varying transverse profile, and their interplay. The exact multimode matrix Hamiltonian is taken as the natural starting point for few-mode approximations. Such approximate equations are provided, and it is worked out how these recover known results for twisting waveguides and can be applied to other types of waveguide designs. The quantum waveguide Hamiltonian is recast into a form that clearly illustrates how it generalizes the Born-Oppenheimer Hamiltonian encountered in molecular physics. In analogy to the latter, we explore the local gauge structure inherent to the quantum waveguide problem and suggest the usefulness of diabatic states, giving an explicit construction of the adiabatic-to-diabatic basis transformation.

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Quantum mechanics of a particle on a torus knot: Curvature and torsion effects

D. Biswas, S. Ghosh

Abstract In this paper, we study the subtle effect of constraints on the quantum dynamics of a point particle moving on a non-trivial torus knot. The particle is kept on the knot by the constraints, generated by curvature and torsion. In the Geometry-Induced Potential (GIP) approach, the Schrödinger equation for the system yields new results in particle energy eigenvalues and eigenfunctions, in contrast with existing results that ignored curvature and torsion effects. Our results depend on Γ, parameter that characterizes the global features of both the embedding torus and, more interestingly, the knottedness of the path.

arXiv:1908.06423•Europhysics Letters (EPL)•Oct 1, 2020•12 citations
We investigate the dynamics of a particle constrained to move on a curved quantum strip waveguide embedded in three-space, subject to Dirichlet boundary conditions. By establishing a coordinate system which allows a more general formulation of the dynamics, we demonstrate consistency with previous results and provide evidence that the presence of torsion introduces a further effective potential term. This result has implications for nanostructure device engineering.
Advances in quantum waveguide dynamics, with applications to physical and biological nanostructuresIAIN JOHN CLARK It is known [1,2] that as semiconductor microstructures get progressively smaller and more free of impurities, it becomes increasingly necessary to use quantum mechanics to describe accurately these systems, which may have features sized between a few and several hundred nanometers.Recent advances in fabrication technology [11] make it feasible to build such nanostructures or mesoscopic systems, as they are known.The term "quantum waveguide" [6] is used to denote a nanostructure which acts as a conducting wire, along which a wavefunction is guided to move by the boundary conditions.Unlike classical wires, curved quantum waveguides have curvature-induced bound states; see [6] for a recent review.This has implications for nanoelectronic circuit design, since the curvature-induced bound states will appear as resonances.However, the effect of torsion (where the waveguide twists out of a plane) and twisting (where the waveguide changes its orientation along its path) upon bound states has attracted little investigation, and is the motivation for the analysis in this work.This thesis develops a unified mathematical approach by which the energy spectrum of a quantum waveguide with curvature and torsion can be analysed, for any crosssectional geometry that remains uniform along the waveguide.Following the introduction and literature review of Chapter 1, a non-relativistic Hamiltonian H for a particle on a quantum waveguide is set up in Chapter 2. A system of curvilinear coordinates is used to express Schrodinger's equation using the Laplace-Beltrami operator V 2 .The metric tensor G is expressed as the product of a curvaturedependent matrix F and the metric tensor G for a "straightened-out" waveguide.The Hamiltonian H is then shown to be unitarily equivalent to the Hamiltonian Hy for a straightened-out waveguide plus interaction terms, Hy -H o + Ky + Vu, where H o is the Hamiltonian for a "straightened-out" waveguide, Ky is a kinetic energy perturbation term and Vu is an effective potential.In Chapter 3, the spectral theory needed to investigate the curvature-induced bound states is developed.The spectrum of a Hamiltonian is partitioned into the
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.
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 quantum mechanics of a single particle constrained to move along an arbitrary smooth reference curve by a confinement that is allowed to vary along the waveguide. The Schr\"odinger equation is evaluated in the adapted coordinate frame and a transverse-mode decomposition is performed, taking into account both curvature and torsion effects and the possibility of a cross-section potential that changes along the curve in an arbitrary way. We discuss the adiabatic structure of the problem, and examine nonadiabatic couplings that arise due to the curved geometry, the varying transverse profile, and their interplay. The exact multimode matrix Hamiltonian is taken as the natural starting point for few-mode approximations. Such approximate equations are provided, and it is worked out how these recover known results for twisting waveguides and can be applied to other types of waveguide designs. The quantum waveguide Hamiltonian is recast into a form that clearly illustrates how it generalizes the Born-Oppenheimer Hamiltonian encountered in molecular physics. In analogy to the latter, we explore the local gauge structure inherent to the quantum waveguide problem and suggest the usefulness of diabatic states, giving an explicit construction of the adiabatic-to-diabatic basis transformation.
Abstract In this paper, we study the subtle effect of constraints on the quantum dynamics of a point particle moving on a non-trivial torus knot. The particle is kept on the knot by the constraints, generated by curvature and torsion. In the Geometry-Induced Potential (GIP) approach, the Schrödinger equation for the system yields new results in particle energy eigenvalues and eigenfunctions, in contrast with existing results that ignored curvature and torsion effects. Our results depend on Γ, parameter that characterizes the global features of both the embedding torus and, more interestingly, the knottedness of the path.