Ramazan Koc

A systematic study on the exact solution of the position dependent mass Schrödinger equation

An algebraic method of constructing potentials for which the Schrodinger equation with position dependent mass can be solved exactly is presented. A general form of the generators of su(1,1) algebra has been employed with a unified approach to the problem. Our systematic approach reproduces a number of earlier results and also leads to some novelties. We show that the solutions of the Schrodinger equation with position dependent mass are free from the choice of parameters for position dependent mass. Two classes of potentials are constructed that include almost all exactly solvable potentials.

Scattering in abrupt heterostructures using a position dependent mass Hamiltonian

Abstract. Transmission probabilities of the scattering problem with a position dependent mass are studied. After sketching the basis of the theory, within the context of the Schrödinger equation for spatially varying effective mass, the simplest problem, namely, transmission through a square well potential with a position dependent mass barrier is studied and its novel properties are obtained. The solutions presented here may be advantageous in the design of semiconductor devices.

A new class of quasi-exactly solvable potentials with a position-dependent mass

A new class of quasi-exactly solvable potentials with a variable mass in the Schrodinger equation is presented. We have derived a general expression for the potentials, including Natanzon confluent potentials. The general solution of the Schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials.

Noncrystallographic Coxeter group H4 in E8

The E8 lattice is constructed in terms of icosians by matching two sets of F4 lattices described by quaternions. Embedding the noncrystallographic group H4 into the Weyl group W(E8) has been described using matrix generators with an emphasis on the relevant Coxeter elements. The conjugacy classes of H4 in terms of quaternions and the characters of the two four-dimensional irreducible representations are explicitly calculated.

Exact solution of position dependent mass Schrödinger equation by supersymmetric quantum mechanics

A supersymmetric technique for the solution of the effective mass Schrodinger equation is proposed. Exact solutions of the Schrodinger equation corresponding to a number of potentials are obtained. The potentials are fully isospectral with the original potentials. The conditions for the shape invariance of the potentials are discussed.

Polyhedra obtained from Coxeter groups and quaternions

We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtained as the orbit of the Coxeter-Weyl group acting on the highest weight representing a selected irreducible representation of the Lie group. This paper, in particular, deals with the determination of the vertices of the Platonic and Archimedean solids from the Coxeter diagrams A3, B3, and H3 in the context of the quaternionic representations of the root systems and the Coxeter-Weyl groups. We use Lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral, octahedral, and icosahedral symmetries are related to the Coxeter-Weyl groups representing the symmetries of the diagrams of A3, B3, and H3, respectively. This technique leads to the determination of the vertices of all Platonic and Archimedean solids except two chiral polyhedra, snubcuboctahedr...

Quaternionic root systems and subgroups of the Aut(F4)

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions, and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4),SP(2)≈SO(5),SO(8),SO(9),F4 and E8 in terms of the discrete elements of the division algebras. The roots themselves display the groups structures besides the octonionic roots of E8 which form a closed octonion algebra. The automorphism group Aut(F4) of the Dynkin diagram of F4 of order 2304, the largest crystallographic group in four-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F4. The Weyl groups of many Lie algebras, such as, G2,SO(7),SO(8),SO(9),SU(3)XSU(3), and SP(3)×SU(2) have been constructed as the subgroups of Aut(F4). We have also class...

The exact solution of the s-wave Klein–Gordon equation for the generalized Hulthén potential by the asymptotic iteration method

The bound state solution of the (1+1)-dimensional Klein?Gordon (KG) equation for the generalized Hulth?n potential has been studied in the framework of the asymptotic iteration method (AIM). The energy values and the corresponding eigenfunctions are obtained for mixed forms of the Hulth?n vector potential and scalar potential.

Quaternionic roots of SO(8), SO(9), F4 and the related Weyl groups

The root systems of SO(8), SO(9) and F4 are constructed by quaternions. Triality manifests itself as permutations of pure quaternion units e1, e2 and e3. It is shown that the automorphism groups of the associated root systems are the finite subgroups of O(4) generated by left-right actions of unit quaternions on the root systems. The relevant finite groups of quaternions, the binary tetrahedral and binary octahedral groups, play essential roles in the construction of the Weyl groups and their conjugacy classes. The relations between the Dynkin indices, standard orthogonal vector and the quaternionic weights are obtained.

Catalan solids derived from three-dimensional-root systems and quaternions

Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter–Dynkin diagrams A3, B3, and H3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A3), W(B3), and W(H3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result from the orbits derived from fundamental weights. The Platonic solids are dual to each other; however, the duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), and (011) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011), and (111), which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A3), W(B3), and W(H3) by the quaternions sim...