2017 # Geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space

**Abstract:** 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…

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“…(10). Unlike the scalar geometrical field κ 2 /4, which results from the action of normal derivatives on the rescale factor [32], the other two additional terms −iτŜ 1 and −iκŜ 3 relate to the parallel transport of the electric field vector along a curve and act as effective gauge terms within the effective dynamics. These two effective gauge terms result in a number of geometry-induced SOI of light.…”

confidence: 99%

“…(10). Unlike the scalar geometrical field κ 2 /4, which results from the action of normal derivatives on the rescale factor [32], the other two additional terms −iτŜ 1 and −iκŜ 3 relate to the parallel transport of the electric field vector along a curve and act as effective gauge terms within the effective dynamics. These two effective gauge terms result in a number of geometry-induced SOI of light.…”

confidence: 99%

“…When the free particle is confined to a curved surface, the effective dynamics would be affected by the geometrical properties 13,18 . In the spirit of the thin-layer quantization scheme, to obtain the effective quantum dynamics describing a particle confined to a curved surface, the Schrödinger equation Eq.…”

confidence: 99%

“…According to the above results, using the formula of the geometric influence 18,35,36 , we can calculate the effective Hamiltonian as…”

confidence: 99%

“…This momentum can be obtained by many different ways including: the hermiticity requirement on derivative part −i ∇ Σ [3], and compatibility of constraint condition n · p + p · n = 0 which means that the motion is perpendicular to the surface normal vector n [4,5], and thin-layer quantization or confining potential formalism which instead considers that particle is confined onto the surface Σ N −1 by means of introduction of a confinement potential along the normal direction of the the surface [6], and dynamical quantum conditions (DQCs) [7], etc. [9,10] It was demonstrated that this momentum (2) satisfies last one of the FQCs (1), when it explicitly takes following simplest form [7],…”

confidence: 99%

“…Experimental justification was performed by comparison of the interference spots formed by the surface plasmon polariton propagating on a cylindrical surface, predicted by the introduction of the geometric momentum or not [8], respectively. Some of previous discussions deal with quite general case [5,7,10], some of them [3,4,6,9] are mainly for a particle on Σ 2 . The geometric momentum (2) suffices to act on state function that has a single component.…”

confidence: 99%