WKB Approximation Scheme

The WKB scheme is typically used in quantum mechanics to find an approximate solution of a time-independent one-dimensional Schrodinger equation of the form  

that we exchanged the interval over which is positive and negative, respectively, compared to [64]. In the limit f2(x) C / Q x), the general solution of (34) yields an approximate solution of (32). 

Using the properties of Bessel functions, i/ (x) can be expressed in terms of Airy functions Ai of negative and positive arguments, respectively [153]. Then, we obtain the uniformly valid Langer solution [48, 64]  

In the next section, we present the results of this approach for polyelectrolyte adsorption onto planar, cylindrical, and spherical surfaces. This is possible because the equation for the Green function reduces, in the corresponding separable coordinates, to a one-dimensional equation comparable to (32) in the ground-state approximation. We confirmed that the WKB applicability condition Q x)/Q x) 1 is satisfied for aU three geometries. The approach applies particularly well above the adsorption-desorption transition, whereas it naturally fails in the proximity of the zero-potential point xq at which Q(xo) = 0. 

---