On the linear potential hill

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