Variations on the Planar Landau Problem: Canonical Transformations, A Purely Linear Potential and the Half-Plane

Journal of Physics A Mathematical and Theoretical (Impact Factor: 1.77). 09/2009; DOI: 10.1088/1751-8113/42/48/485209
Source: arXiv

ABSTRACT The ordinary Landau problem of a charged particle in a plane subjected to a perpendicular homogeneous and static magnetic field is reconsidered from different points of view. The role of phase space canonical transformations and their relation to a choice of gauge in the solution of the problem is addressed. The Landau problem is then extended to different contexts, in particular the singular situation of a purely linear potential term being added as an interaction, for which a complete purely algebraic solution is presented. This solution is then exploited to solve this same singular Landau problem in the half-plane, with as motivation the potential relevance of such a geometry for quantum Hall measurements in the presence of an electric field or a gravitational quantum well.

  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: For the example of the infinitely deep well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self-adjoint operator. We then describe the self-adjoint extensions and their spectra for the momentum and the Hamiltonian operators in different physical situations. Some consequences are worked out, which could lead to experimental checks. Comment: 25 pages, Latex file, extended version of Am. J. Phys. 69 (2001) 322
    American Journal of Physics 03/2001; · 0.78 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We demonstrate how a one parameter family of interacting noncommuting Hamiltonians, which are physically equivalent, can be constructed in noncommutative quantum mechanics. This construction is carried out exactly (to all orders in the noncommutative parameter) and analytically in two dimensions for a free particle and a harmonic oscillator moving in a constant magnetic field. We discuss the significance of the Seiberg-Witten map in this context. It is shown for the harmonic oscillator potential that an approximate duality, valid in the low-energy sector, can be constructed between the interacting commutative and a noninteracting noncommutative Hamiltonian. This approximation holds to order 1/B and is therefore valid in the case of strong magnetic fields and weak Landau-level mixing.
    Physical review D: Particles and fields 04/2005; 71(8).
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular we discuss the infinite and finite non-commutative spherical well in two dimensions. Using this, bound-states and scattering can be discussed unambiguously. Here we focus on the infinite well and solve for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored.
    Journal of Physics A Mathematical and Theoretical 10/2007; · 1.77 Impact Factor

Full-text (2 Sources)

Available from
May 22, 2014