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| This paper analyzes the relativistic bound states with the direct coupling of vector and scalar type-II exponential potentials, for particular under the manner of equal coupling, viz $(potential)$ for s-wave states. To solve the Klein-Gordon equation(KGE) under such circumstances is the crucial work, and manifold variable-transformations leads to an object equation whose solutions distinguish with different V_1. For V_1>0, one proceeds to obtain a parabolic cylinder equation(PCE) of the first type, which has an even and an odd parabolic cylinder series as two linearly independent solutions. Normalization requirements exclude the odd function and break the even function off to a polynomial, which gives rise to an elegant structure of the energy spectrum that explicitly increase along the radii. The case V_1<0, which leads to PCE of the second type, is in serious correspondence with that of V_1>0. As to the wave function in both cases, besides the canonical polynomial arising from parabolic cylinder series which can be complex, another terminated Taylor expansions are introduced on R. Subsequently, bound states of Dirac equation(DE) based on the complete set [H,kappa,J2,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. |
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| Keywords:Relativistic Wave Equation;Type-II Exponential Potential;Bound State;Parabolic Cylinder Equation |
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