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  • 1.
    Linnaeus, Staffan
    Dalarna University, School of Technology and Business Studies, Physics.
    Phase-integral solution of the radial Dirac equation2010In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 51, no 3, article id 032304Article in journal (Refereed)
    Abstract [en]

    A phase-integral (WKB) solution of the radial Dirac equation is constructed, retaining perfect symmetry between the two components of the wave function and introducing no singularities except at the classical transition points. The potential is allowed to be the time component of a four-vector, a Lorentz scalar, a pseudoscalar, or any combination of these. The key point in the construction is the transformation from two coupled first-order equations constituting the radial Dirac equation to a single second-order Schroumldinger-type equation. This transformation can be carried out in infinitely many ways, giving rise to different second-order equations but with the same spectrum. A unique transformation is found that produces a particularly simple second-order equation and correspondingly simple and well-behaved phase-integral solutions. The resulting phase-integral formulas are applied to unbound and bound states of the Coulomb potential. For bound states, the exact energy levels are reproduced.

  • 2.
    Linnaeus, Staffan
    Dalarna University, School of Technology and Business Studies, Physics.
    Stokes constants for a singular wave equation2005In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 45, no 5, article id 053505Article in journal (Refereed)
    Abstract [en]

    The Stokes constants for arbitrary-order phase-integral approximations are calculated when the square of the wave number has either two simple zeros close to a second-order pole or one simple zero close to a first-order pole. The treatment is based on uniform approximations. All parameters may assume general complex values.

  • 3. Thylwe, Karl-Erik
    et al.
    Linnaeus, Staffan
    Dalarna University, School of Technology and Business Studies, Physics.
    Semiclassical aspects and supersymmetry of bound Dirac states for central pseudo-scalar potentials2011In: Physica Scripta, ISSN 0031-8949, E-ISSN 1402-4896, Vol. 84, no 2, article id 025006Article in journal (Refereed)
    Abstract [en]

    Relativistic bound states for a linear, radial pseudo-scalar potential model are discussed. The two radial Dirac components are known to have a close connection to partner states in super-symmetric quantum mechanical theory. The pseudo-scalar potential plays the role of the 'super potential'. Hence, the Dirac components satisfy decoupled Schrodinger-type equations with isospectral, so-called, 'partner potentials' except possibly for a single state; the ground state corresponding to one of the partner potentials. The energy spectrum of a confining linear radial potential is discussed in some detail. Accurate amplitude-phase computations and a novel semiclassical (phase-integral) approach are presented.

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