The Real Solution to Scalar Field Equation in 5D Black String Space

After the nontrivial quantum parameters $\Omega_{n}$ and quantum potentials $V_{n}$ obtained in our previous research, the circumstance of a real scalar wave in the bulk is studied with the similar method of Brevik (2001). The equation of a massless scalar field is solved numerically under the boundary conditions near the inner horizon $r_{e}$ and the outer horizon $r_{c}$. Unlike the usual wave function $\Psi_{\omega l}$ in 4D, quantum number $n$ introduces a new functions $\Psi_{\omega l n}$, whose potentials are higher and wider with bigger n. Using the tangent approximation, a full boundary value problem about the Schr$\ddot{o}$dinger-like equation is solved. With a convenient replacement of the 5D continuous potential by square barrier, the reflection and transmission coefficients are obtained. If extra dimension does exist and is visible at the neighborhood of black holes, the unique wave function $\Psi_{\omega l n}$ may say something to it.