Extract and approximate expressions for the permeability of potential barriers to light particles

In order to calculate the permeability of an energy barrier to a particle of mass m and energy W it is necessary to solve the Schrödinger equation for the potential energy function V ( x ) representing the barrier. An exact solution has only been obtained for one type of barrier represented by a continuous curve (which we shall call the Eckart barrier), and in this case the resulting expression for the permeability is an inconvenient one for application to problems of reaction velocity. Most treatments of this subject have therefore been based on the following approximate method. If the wave function is written in the form Ψ( x ) = e 8 ( x ) , the one-dimensional Schrödinger equation becomes d 2 s / dx 2 + ( ds / dx ) 2 + 8π 2 m / h 2 {W - V( x )} = 0. (1) This equation can be solved approximately provided that the first term is small compared with the second, i.e. , if | d 2 s / dx 2 / ( ds / dx ) 2 | ≪ 1.