The DKP Equation in Presence of a Woods-Saxon Potential: Transmission Resonances and Bound States

Abstract

In this article, we solve the Duffin-Kemmer-Petiau (DKP) equation in the presence of the Woods-Saxon potential barrier and well for spin-one particles. We derive the scattering solution in terms of the Gaussian hypergeometric function ${}_2{F}_1(a,b;c;z)$, the regularized Gaussian hypergeometric function ${}_2{\tilde{F}}_1(a,b;c;z)$, and the Gamma function $\Gamma (x)$. Our analysis reveals the presence of transmission resonances. To observe these resonances, we calculate and plot the transmission $T$ and reflection $R$ coefficients for various parameters of the Woods-Saxon potential barrier. Our results are compared with those obtained for the square potential barrier and the cusp potential barrier, which represent limiting cases of the Woods-Saxon potential barrier. Furthermore, we investigate the bound state solutions, determining the critical turning point $V_{cr}$ to study particle-antiparticle creation and compute the norm $N$. Finally, we also compare our results with those obtained for the square potential well and the cusp potential well, confirming that pair creation occurs in both potential wells.

40 pages, 20 figures