No abstract

We use the “tridiagonal representation approach” to solve the time-independent Schrödinger equation for the bound states of generalized versions of the trigonometric and hyperbolic Pöschl–Teller potentials. These new solvable potentials do not belong to the conventional class of exactly solvable problems. The solutions are finite series of square integrable functions written in terms of the Jacobi polynomial.

Abstract:The aim of this work is to find exact solutions of the one-dimensional Dirac equation that do not belong to the already known conventional class. We write the spinor wavefunction as a bounded infinite sum in a complete basis set, which is chosen such that the matrix representation of the Dirac wave operator becomes tridiagonal and symmetric. This makes the wave equation equivalent to a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation and obtain the relativistic energy spectrum and corresponding state functions. Restriction to the diagonal representation results in the conventional class of solutions. As illustration, we consider the square well with half-sine potential bottom in the vector and scalar coupling under spin symmetry. We obtain the relativistic energy spectrum and show that the nonrelativistic limit agrees with results found elsewhere.

This paper aims at extending our previous work on the solution of the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA). In the approach, we expand the spinor wavefunction in terms of suitable square integrable basis functions that support a tridiagonal matrix representation of the wave operator. This will transform the problem from solving a system of coupled first order differential equations to solving an algebraic three-term recursion relation for the expansion coefficients of the wavefunction. In some cases, solutions to this recursion relation can be related to well-known classes of orthogonal polynomials whereas in other situations solutions represent new class of polynomials. In this work, we will discuss various solvable potentials that obey the tridiagonal representation requirement with special emphasis on simple cases with spin-symmetric and pseudospin-symmetric potential couplings. We conclude by mentioning some potential applications in graphene.

We solve the -dimensional Schrödinger equation with hyperbolic Pöschl-Teller potential plus a generalized ring-shaped potential. After the separation of variable in the hyperspherical coordinate, we used Nikiforov-Uvarov (NU) method to solve the resulting radial equation and obtain explicitly the energy level and the corresponding wave function in closed form. The solutions to the energy eigenvalues and the corresponding wave functions are obtained using the NU method as well.