A practice problem for Schrödinger transmission

We present the case of quantal transmission through a smooth, single-piece exponential potential, V (x) = −V0e x/a, in contrast to the piece-wise continuous potentials, as a pedagogic model to demonstrate the analytic extraction of transmission (reflection) co-efficient. The Schrödinger transmission through a one-dimensional potential well/barrier constitutes an essential topic in the courses and text-books [1,2] on quantum mechanics. The main interest is to find the probability of transmission, T (E), and that of reflection, R(E), when a particle is incident with an energy E at the potential from one side. For a real potential these two probanilities are such that T (E) +R(E) = 1, ensuring the conservation of the incident flux of the particles. Due to simplicity of treatment, usually a rectangilar well/barrier or a sharp step-potentials are included as examples. These potentials are both piece-wise constant and piece-wise continuous, the former is a three-piece and the latter is a two-piece function. For the case of three-piece potentials : V (|x| ≥ a) = 0 and V (−a ≤ x ≤ a) = f(x), where the Schrödinger equation does not possess analytic form, the numerical computation of T (E) is performed by letting Ψ(x ≤ −a) = Ae + Be, Ψ(−a ≤ x ≤ a) = Cu(x) + Dv(x) and Ψ(x ≥ a) = Fe [3]. The functions u(x) and v(x) denote two linearly indedependent solutions of the Schrödinger equation with initial values as u(0) = 1, u(0) = 0; v(0) = 0, v(0) = 1, for numerical integration of the Schrödinger from x = 0 upto x = ±a. Here the prime denotes differentiation with respect to x. For the case of a rectangular well/barrier of depth/height, V0, these functions are cosκx and sinκx κ , where κ = √