Quantization on curved surfaces

Mladenov, Iväılo M.

doi:10.1002/qua.10292

Cited by 6 publications

(6 citation statements)

References 25 publications

(25 reference statements)

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“…The same result can be reached from great circle geometry but the derivation is longer. Returning now to Cartesian coordinates for the further motion in the plane, the point of exit from the sphere and reentry to the plane is x y reentry = R sin α cos φ exit sin φ exit (12) and therefore the final part of the trajectory which continues further into the plane and has the same energy constant Ω, hence the same speed v, must be x(t) y(t) out = R sin α cos φ exit sin φ exit +v(t−t exit ) cos θ sin θ (13) for some scattering angle θ.…”

Section: Scattering Trajectoriesmentioning

confidence: 99%

“…Explicit solutions for quantum particle motion on a curved surface have only been provided in a few simple cases, however, including spheres [5], ellipsoids [1,12], and tori [6]. Here we provide explicit solutions for a significantly different kind of curved surface, namely one formed by joining a portion of a sphere to an infinite plane, to make a bump or bubble that bulges out of the plane, which is otherwise flat.…”

Section: Introductionmentioning

confidence: 99%

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Sphere on a plane: Two-dimensional scattering from a finite curved region

Anglin¹,

Wamba²

2022

Preprint

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Non-relativistic particles that are effectively confined to two dimensions can in general move on curved surfaces, allowing dynamical phenomena beyond what can be described with scalar potentials or even vector gauge fields. Here we consider a simple case of piecewise uniform curvature: a particle moves on a plane with a spherical extrusion. Depending on the latitude at which the sphere joins the plane, the extrusion can range from an infinitesimal bump to a nearly full sphere that just touches the plane. Free classical motion on this surface of piecewise uniform curvature follows geodesics that are independent of velocity, while quantum mechanical scattering depends on energy. We compare classical, semi-classical, and fully quantum problems, which are all exactly solvable, and show how semi-classical analysis explains the complex quantum differential cross section in terms of interference between two classical trajectories: the sphere on a plane acts as a kind of double slit.

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Section: Scattering Trajectoriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sphere on a plane: Two-dimensional scattering from a finite curved region

Anglin¹,

Wamba²

2022

Preprint

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“…However, we should emphasize that the general question of identifying what are the canonical pairs in curved space is an involved one, and the standard approach to the problem is the Dirac quantization scheme. This issue is quite intricate and we refer the reader to some exemplary works for further reading [20][21][22][23]. Now, having identified the Hermitian canonical (and thus the physical) radial momentum operator, we can express the kinetic energy in terms of it.…”

Section: Hamiltonian For a Spin-less Particle On The Surface Of A Cyl...mentioning

confidence: 99%

Hermitian and gauge-covariant Hamiltonians for a particle in a magnetic field on cylindrical and spherical surfaces

Shikakhwa

2016

Eur. J. Phys.

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We construct the Hermitian Schrödinger Hamiltonian of spin-less as well as the gauge-covariant Pauli Hamiltonian of spin one-half particles in a magnetic field that are confined to cylindrical and spherical surfaces. The approach does not require the use of involved differential-geometrical methods and is intuitive and physical, relying on the general requirements of Hermicity and gaugecovariance. The surfaces are embedded in the full three-dimensional space and confinement to the surfaces is achieved by strong radial potentials. We identify the Hermitian and gauge-covariant (in the presence of a magnetic field) physical radial momentum in each case and set it to zero upon confinement to the surfaces . The resulting surface Hamiltonians are seen to be automatically Hermitian and gauge-covariant. The well-known geometrical kinetic energy also emerges naturally.

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“…A free particle moving on a one-sheeted hyperboloid was discussed both at the classical and quantum levels and novel equations of motion were found in [7]. Quantization on a prolate and on a oblate ellipsoid, considering applications to multiple-shell fullerenes, can be found in [8]. A special surface on which many theoretical investigations are realized is the cone.…”

Section: Introductionmentioning

confidence: 99%

Landau levels, self-adjoint extensions and Hall conductivity on a cone

et al. 2014

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In this work we obtain the Landau levels and the Hall conductivity at zero temperature of a twodimensional electron gas on a conical surface. We investigate the integer quantum Hall effect considering two different approaches. The first one is an extrinsic approach which employs an effective scalar potential that contains both the Gaussian and the mean curvature of the surface. The second one, an intrinsic approach where the Gaussian curvature is the sole term in the scalar curvature potential. From a theoretical point of view, the singular Gaussian curvature of the cone may affect the wave functions and the respective Landau levels. Since this problem requests self-adjoint extensions, we investigate how the conical tip could influence the integer quantum Hall effect, comparing with the case were the coupling between the wave functions and the conical tip is ignored. This last case corresponds to the so-called Friedrichs extension. In all cases, the Hall conductivity is enhanced by the conical geometry depending on the opening angle. There are a considerable number of theoretical papers concerned with the self-adjoint extensions on a cone and now we hope the work addressed here inspires experimental investigation on these questions about quantum dynamics on a cone.PACS. PACS-key discribing text of that key -PACS-key discribing text of that key

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Quantization on curved surfaces

Cited by 6 publications

References 25 publications

Sphere on a plane: Two-dimensional scattering from a finite curved region

Sphere on a plane: Two-dimensional scattering from a finite curved region

Hermitian and gauge-covariant Hamiltonians for a particle in a magnetic field on cylindrical and spherical surfaces

Landau levels, self-adjoint extensions and Hall conductivity on a cone

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