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arXiv:1703.06388 · Annalen der Physik

Geometric Potential and Dirac Quantization

Dingkun Lian, Liangdong Hu, Quanhui Liu

Published Apr 3, 2018• 16 citations
arXivHTMLPDF

Abstract

Abstract A fundamental problem regarding the Dirac quantization of a free particle on an ( ) curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature‐induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin‐layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator‐ordering‐free section is identified and is then found sufficient to lead to the expected form of geometric potential.

DOI
10.1002/andp.201700415

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Abstract A fundamental problem regarding the Dirac quantization of a free particle on an ( ) curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature‐induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin‐layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator‐ordering‐free section is identified and is then found sufficient to lead to the expected form of geometric potential.
Abstract Geometric momentum is the appropriate momentum for a particle constrained to move on a curved surface, which depends on the extrinsic curvature and leads to observable effects, and curvature-induced quantum potentials appear for a nonrelativistic free particle on the surface. In the context of multi-component quantum states, the geometric momentum should be rewritten as a generally covariant geometric momentum, which contains an additional term defined as the gauge potential. For a Dirac fermion constrained on a two-dimensional hypersurface, we derive the generally covariant geometric momentum, and demonstrate that no curvature-induced geometric potentials arise on a pseudosphere or a helical surface. The dynamical quantization conditions are verified to be effective in dealing with constrained systems on hypersurfaces, enabling the derivation of both the generally convariant geometric momentum and the geometric potential for a spin particle constrained on parametrically defined surfaces.
For a free particle that moves non-relativistically on a curved surface, there are curvature-induced quantum potentials that significantly influence the surface quantum states. However, the experimental results in topological insulators, whether curved or not, indicate no evidence of such a potential for the relativistic particles constrained on the curved surface. Within the framework of Dirac quantization scheme, we demonstrate a general result that for a Dirac fermion on a two-dimensional curved surface of revolution, no curvature-induced quantum potential is permissible.
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 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.
In this work, Schrödinger and Dirac equations will be examined in geometries that confine the particles to hypersurfaces. For this purpose, two methods will be considered. The first method is the thin layer method which relies on explicit use of geometrical relations and the squeezing of a certain coordinate of space (or spacetime). The second is Dirac’s quantization procedure involving the modification of canonical quantization making use of the geometrical constraints. For the Dirac equation, only the first method will be considered. Lastly, the results of the two methods will be compared and some notes on the differences between the results will be included.
Abstract Combining the deviation between thin layers’ adjacent surfaces with the confining potential method applied to the quantum curved systems, we derive the effective Schrödinger equation describing the particle constrained within a curved layer, accompanied by a general geometric potential V gq composed of a compression-corrected geometric potential V gq * and a novel potential V gq * * brought by the deviation. Applying this analysis to the cylindrical layer emerges two types of deviation-induced geometric potential, resulting from the the cases of slant deviation and tangent deviation, respectively, which strongly renormalizes the purely geometric potential and contribute to the energy spectrum based on a very substantial deepening of bound states they offer.