Determination of the shape of a potential barrier from the tunneling transmission coefficient

Given the shape of a potential barrier $V(x)$, there is an explicit expression in the WKB approximation for the transmission coefficient for penetration of the barrier. The problem considered in this paper is to find the inverse relation, determining $V(x)$ from the transmission coefficient. It is shown that the width of the barrier at energy $E$ can be determined if the transmission coefficient is known for energies between $E$ and ${V}_{max}$, the barrier maximum. Other properties of the turning points can be deduced for potentials which can be written in terms of a parameter $\ensuremath{\lambda}$ as $V(x)={V}_{0}(x)\ensuremath{-}\ensuremath{\lambda}\ensuremath{\varphi}(x)$, if the dependence of transmission coefficient on $\ensuremath{\lambda}$ is known. For the specific case of field emission, one can find the potential $V(x)$ from the energy and field dependence of the transmission coefficient.