The Penetration of a Potential Barrier by Electrons

Abstract

A potential barrier of the kind studied by Fowler and others may be represented by the analytic function $V$ (Eq. (1)). The Schrödinger equation associated to this potential is soluble in terms of hypergeometric functions, and the coefficient of reflection for electrons approaching the barrier with energy $W$ is calculable (Eq. (15)). The approximate formula, $1-\rho =\exp \{-\int \frac{4\pi }{h}{\mathrm{}\left(2m\right(V-W\left)\right)}^{\frac{1}{2}}\mathrm{dx}\}$ is shown to agree very well with the exact formula when the width of the barrier is great compared to the de Broglie wave-length of the incident electron, and $W<\mathrm{}{V}_{max}$.