Quasilinear and WKB Solutions in Quantum Mechanics

Solutions of the Schrödinger equation by the quasilinearization method (QLM) and by the WKB method are compared. While the latter generates an expansion in powers of ħ, QLM approaches the solution of the equivalent nonlinear Riccati equation by approximating nonlinear terms with a sequence of linear ones. QLM does not rely on the existence of a smallness parameter. It is shown that both energies and wave functions in the first QLM iteration are more accurate than in the WKB approximation. The first QLM iterate is represented by a closed expression, allowing analytical estimates of the effects of parameters on the properties of the quantum systems. Quadratic convergence assures extremely accurate energies and wave functions in a few QLM iterations.