The relativistic Dirac–Morse Green’s function

Abstract

Using a recently developed approach for solving the three dimensional Dirac equation with spherical symmetry, we obtain the two-point Green's function of the relativistic Dirac-Morse problem. This is accomplished by setting up the relativistic problem in such a way that makes comparison with the nonrelativistic problem highly transparent and results in a mapping of the latter into the former. The relativistic bound states energy spectrum is easily obtained by locating the energy poles of the Green's function components in a simple and straightforward manner. Introduction: Despite all the work that has been done over the years on the Dirac equation, its exact solutions for local interaction has been limited to a very small set of potentials. Since the original work of Dirac in the early part of last century up until 1989 only the relativistic Coulomb problem was solved exactly. In 1989, the relativistic extension of the oscillator problem (Dirac-Oscillator) was finally formulated and solved by Moshinsky and Szczepaniak [1]. Recently, and in a series of articles [2-6], we presented an effective approach for solving the three dimensional Dirac equation for spherically symmetric potential interaction. The first step in the program started with the realization that the nonrelativistic Coulomb, Oscillator, and S-wave Morse problems belong to the same class of shape invariant potentials which carries a representation of so(2,1) Lie algebra. Therefore, the fact that the relativistic version of the first two problems (Dirac-Coulomb and Dirac-Oscillator) were solved exactly makes the solution of the third, in principle, feasible. Indeed, the relativistic Dirac-Morse problem was formulated and solved in Ref. 2. The bound state energy spectrum and spinor wavefunctions were obtained. Taking the nonrelativistic limit reproduces the familiar Schrödinger-Morse problem. Motivated by these findings, the same approach was applied successfully in obtaining solutions for the relativistic extension of yet another class of shape invariant potentials [3]. These included the Dirac-Scarf, Dirac-RosenMorse I & II, Dirac-Pöschl-Teller, and Dirac-Eckart potentials. Furthermore, using the same formalism quasi exactly solvable systems at rest mass energies were obtained for a large class of power-law relativistic potentials [4]. Quite recently, Guo Jian-You et al succeeded in constructing solutions for the relativistic Dirac-Woods-Saxon and DiracHulthén problems using the same approach [7]. In the fourth and last article of the series in our program of searching for exact solutions to the Dirac equation [5], we found a special graded extension of so(2,1) Lie algebra. Realization of this superalgebra by 2×2 matrices of differential operators acting in the two component spinor space was constructed. The linear span of this graded algebra gives the canonical form of the radial Dirac Hamiltonian. It turned out that the Dirac-Oscillator class, which also includes the Dirac-Coulomb and Dirac-Morse, carries a representation of this supersy...

Solution of the relativistic Dirac–Hulthén problem

“…In fact, the relativistic extension of any known dynamical relationship in the nonrelativistic theory could easily be obtained by this correspondence map. The Green's function, which has prime significance in the calculation of relativistic processes, is such an example [14]. Moreover, the spinor wavefunction is also obtained from the nonrelativistic wavefunction using the same parameter map.…”