The Definability of Fields

We look for a deep connection between mathematics and physics. Our approach is to propose a set theory T which leads to a concise mathematical description of physical fields and to a finite unit of action. The concept of "definability" of fields is then introduced. Definabilty of fields in T is necessary and sufficient for quantization and sufficient to avoid physical antinomies.Comment: 10 Pages. Typo corrected. This paper has been submitted to Chaos, Solitons and Fractal

Supersymmetry Across Nanoscale Heterojunction

We argue that supersymmetric transformation could be applied across the heterojunction formed by joining of two mixed semiconductors. A general framework is described by specifying the structure of ladder operators at the junction for making quantitative estimation of physical quantities. For a particular heterojunction device, we show that an exponential grading inside a nanoscale doped layer is amenable to exact analytical treatment for a class of potentials distorted by the junctions through the solutions of transformed Morse-Type potentials.Comment: 7 pages, 2 figure

A study of the bound states for square potential wells with position-dependent mass

A square potential well with position-dependent mass is studied for bound states. Applying appropriate matching conditions, a transcendental equation is derived for the energy eigenvalues. Numerical results are presented graphically and the variation of the energy of the bound states are calculated as a function of the well-width and mass.Comment: To appear in Phys. Lett. A (Present e-mail of A.G: [email protected]

Exact solvability of potentials with spatially dependent effective masses

We discuss the relationship between exact solvability of the Schroedinger equation, due to a spatially dependent mass, and the ordering ambiguity. Some examples show that, even in this case, one can find exact solutions. Furthermore, it is demonstrated that operators with linear dependence on the momentum are nonambiguous.Comment: 12 page

Exceptional orthogonal polynomials and exactly solvable potentials in position dependent mass Schroedinger Hamiltonians

Some exactly solvable potentials in the position dependent mass background are generated whose bound states are given in terms of Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These potentials are shown to be shape invariant and isospectral to the potentials whose bound state solutions involve classical Laguerre or Jacobi polynomials.Comment: To appear in Physics Letters

A general scheme for the effective-mass Schrodinger equation and the generation of the associated potentials

A systematic procedure to study one-dimensional Schr\"odinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals a new and interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schr\"odinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. This provides a unified approach to many exact results known in the literature, as well as to a lot of new ones.Comment: 16 pages + 1 figure; minor changes + new "free-particle" problem; version published in Mod. Phys. Lett.

Development of an eight-band theory for quantum-dot heterostructures

We derive a nonsymmetrized 8-band effective-mass Hamiltonian for quantum-dot heterostructures (QDHs) in Burt's envelope-function representation. The 8x8 radial Hamiltonian and the boundary conditions for the Schroedinger equation are obtained for spherical QDHs. Boundary conditions for symmetrized and nonsymmetrized radial Hamiltonians are compared with each other and with connection rules that are commonly used to match the wave functions found from the bulk kp Hamiltonians of two adjacent materials. Electron and hole energy spectra in three spherical QDHs: HgS/CdS, InAs/GaAs, and GaAs/AlAs are calculated as a function of the quantum dot radius within the approximate symmetrized and exact nonsymmetrized 8x8 models. The parameters of dissymmetry are shown to influence the energy levels and the wave functions of an electron and a hole and, consequently, the energies of both intraband and interband transitions.Comment: 36 pages, 10 figures, E-mail addresses: [email protected], [email protected]