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


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Shikakhwa M. S. , Chair N.

EUROPEAN JOURNAL OF PHYSICS, vol.38, no.1, 2017 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 1
  • Publication Date: 2017
  • Doi Number: 10.1088/0143-0807/38/1/015402
  • Journal Name: EUROPEAN JOURNAL OF PHYSICS
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Keywords: quantum mechanics on spheres and cylinders, Schrodinger equation on a sphere and a cylinder, gauge-covariant Hamiltonian on a curved surfaced, QUANTUM-MECHANICS

Abstract

We construct the Hermitian Schrodinger 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.