On the linear potential hill

By explicitly solving the Schrödinger equation for a particle encountering a linear potential hill (V = 0, x < 0; V = V ₀ x/a, 0 < x < a; V = V ₀, x ≳ a), we obtain corrections to the quantum (a → 0) and classical (a → ∞) limits for the transmission coefficient, demonstrating the influence of the range a over which the potential rises.