Relativistic Shifted- l Expansion Technique for Dirac and Klein-Gordon Equations

Abstract

The shifted -l expansion technique (SLET) is extended to solve for Dirac particle trapped in spherically symmetric scalar and/or 4-vector potentials. A parameter λ = 0, 1 is introduced in such a way that one can obtain the Klein-Gordon (KG) bound states from Dirac bound states. The 4-vector Coulomb, the scalar linear, and the equally mixed scalar and 4-vector power-law potentials are used in KG and Dirac equations. Exact numerical results are obtained for the 4-vector Coulomb potential in both KG and Dirac equations. Highly accurate and fast converging results are obtained for the scalar linear and the equally mixed scalar and 4-vector power-law potentials.

The Asymptotic Iteration Method for Dirac and Klein–gordon Equations With a Linear Scalar Potential

“…10 and 11. 11 The values in square parentheses are those given by Gunion Comparing our results, with the numerically predicted ones 10 in Table 2, we have scored a minimum accuracy of 99.61% and a maximum accuracy of 99.76%. In Table 3 the minimum accuracy is noted to be 99.05% and the maximum is 100.00%.…”

“…(28) with S(r) = Ar, A = 0.137 GeV 2 , m = 1.12 GeV, V (r) = 0 and κ = . The values in round parentheses are those of shifted expansion method 11. The values in square parentheses are those given by Gunion and Li.…”

Perturbed Coulomb potentials in the Klein-Gordon equation via the shifted-lexpansion technique

“…Very recently we have introduced a shifted -l expansion technique ( SLET ) to solve for the Schrödinger [8], and the Dirac equations for some model potentials [9]. SLET is a reformation to the existing shifted -N expansion technique ( SLNT ) [1,10-12, and references therein].…”

“…(ii) V (r) = 0 and S(r) = −A 2 /r, which has no experimental evidence, to the best of our knowledge, thus our calculations are only of academic interest. (iii) V (r) = S(r) = −A/r which represents not only a KG -particle in an equally mixed Lorentz scalar and Lorentz vector potential but also a Dirac particle in the same potential mixture, where l = j + 1/2 and the radial KG wave function represents the radial large -component of the Dirac -spinor [4,9]. (iv) V (r) = −A 1 /r + kr and S(r) = 0 representing a π − meson in a Coulomb potential perturbed by a linear Lorentz vector interaction kr.…”

“…where β is a suitable shift to be determined and is mainly introduced to avoid the trivial case when l ′ = 0. To start the systematic 1/ l expansion [8,9] we define…”