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| This paper analyzes the relativistic bound states with the direct coupling of a vector and a scalar hyperbolic tangent or cotangent potential, viz V_0 tanh(lambda r) and V_0 (lambda r), for particular under the coupling of V(r)=S(r) for s-wave states. The two kinds of potentials are calculated separately as two cases, the comparison of which reflects elegant correspondence throughout the process. To solve the Klein-Gordon equation(KGE) under such circumstances is the crucial work, which lays the foundation to solve Dirac equation(DE), and manifold variable-transformations lead to an object equation which has the familiar structure of hypergeometric equation. The replacements should be carefully selected, among which the technique of flexible parameter is employed for the crucial simplification. The normalization requirement excludes the other induced function of hypergeometric type as a component of the eigenfunction, and breaks off the regular Gauss function to a polynomial, which gives rise to the energy spectrum. Yet, since its the solution of a sextic algebraic equation, an implicit formalism is employed for simplicity. The case of hyperbolic tangent potential and the hyperbolic cotangent potential share the same structure of the object dynamical equation and energy spectrum, yet their eigenfunctions slightly differ on the induced independent variable via -xi and xi. Whereafter, bound states of Dirac equation(DE) based on the complete set [hatH,hatkappa,hatmathbf J 2,hat J_z] is calculated. The specific coupling manner and $s$-wave condition lead to the identical radial DE with that of KGE, which yields the solution for the up component straightforwardly, and subsequently induces the down component.
( Comments: 11 pages, no figure) |
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| Keywords:Relativistic Wave Equation;Hyperbolic Tangent and Cotangent Potential;Bound State;Hypergeometric Equation |
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