The Penetration of a Potential Barrier by Electrons

TLDR

A potential barrier studied by Fowler and others can be represented analytically by the function V (Eq. (1)). The Schrödinger equation for this potential is solvable using hypergeometric functions, yielding a calculable reflection coefficient for electrons with energy W (Eq. (15)).

Abstract

A potential barrier of the kind studied by Fowler and others may be represented by the analytic function $V$ (Eq. (1)). The Schr\"odinger 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, $1\ensuremath{-}\ensuremath{\rho}=\mathrm{exp}{\ensuremath{-}\ensuremath{\int}\frac{4\ensuremath{\pi}}{h}{(2m(V\ensuremath{-}W))}^{\frac{1}{2}}\mathrm{dx}}$ 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 $W<{V}_{max}$.