Analytical Solutions of the 1D Dirac Equation Using the Tridiagonal Representation Approach

Abstract

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.

The eigenvalue problem of one-dimensional Dirac operator

“…From the quantum-chemical perspective the most important are applications of the onedimensional Dirac equation to the description of quantum systems whose spectrum has an energy gap and whose properties may be conveniently simulated by a properly taylored Dirac equation. To this category belong works on the edge states, on the Landau levels, on the theoretical modeling of graphene [12][13][14][15], also including external electromagnetic fields [16], and advanced studies on radially twisted carbon nanotubes [17,18], specific calculations relevant to the theoretical simulation of one-dimensional graphene structures performed, among others, in [19,20]. One-dimensional Dirac equation has also been used to the description of the -electron systems in conjugated molecules [21].…”

“…In case of numerical solutions, an iterative loop in which the two numbers are adjusted to each other is necessary. In any case, at most one eigenvalue may be equal to a fixed parameter.According to(20) If qr > 0 , i.e. if the scalar part dominates in the total potential, then the eigenvalue problem (29) is Hermitian.…”

Exact solvability of two new 3D and 1D non-relativistic potentials within the TRA framework

“…are functions of the energy and the potential parameters which to be determined later [15][16][17][18]. Here we are assuming a discrete energy spectrum, in case of the presence of an additional continuous spectrum the wave function will be expanded in both discrete and continuous Fourier components keeping in mind that discrete and continuous energy spectra do not overlap.…”