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arXiv:1610.09663 · European Journal of Physics

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

M S Shikakhwa, N Chair

Published Nov 2, 2016• 10 citations
arXivHTMLPDF

Abstract

Abstract We construct the Hermitian Schrödinger Hamiltonian of spin-less particles and the gauge-covariant Pauli Hamiltonian of spin one-half particles in a magnetic field, which 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 gauge-covariance. 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.

DOI
10.1088/0143-0807/38/1/015402

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Abstract We construct the Hermitian Schrödinger Hamiltonian of spin-less particles and the gauge-covariant Pauli Hamiltonian of spin one-half particles in a magnetic field, which 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 gauge-covariance. 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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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.
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.
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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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.