A comparison of WKB (Wentzel–Kramers–Brillouin) and SWKB eigenfunctions of the Schrödinger equation for potentials in the class encountered in supersymmetric quantum mechanics is presented. The potentials that are studied are those that result from the transformation of a Fokker–Planck eigenvalue problem to a Schrödinger equation. Linear Fokker–Planck equations of the type considered in this paper give the probability distribution function for a large number of physical situations. The time-dependent solutions can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker–Planck operator. The specific Fokker–Planck operator considered is the one that describes the thermalization of electrons in the inert gases. The WKB and SWKB semiclassical approximations are compared with exact numerical results. Although the eigenvalues can be very close to the exact values, we find significant departures for the eigenfunctions.
Time-dependent Darboux transformation and supersymmetric hierarchy of Fokker–Planck equations
