2004/09/21 by Victor Mandelzweig, Mandelzweig, Victor
Physics and Astronomy · #FOS: Physical sciences #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0409137
23 pages
arxiv created 2004/09/21 · arxiv updated 2009/12/01
It is shown that the quasilinearization method (QLM) sums the WKB series. The method approaches solution of the Riccati equation (obtained by casting the Schrödinger equation in a nonlinear form) by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of a smallness parameter. Each p-th QLM iterate is expressible in a closed integral form. Its expansion in powers of ℏ reproduces the structure of the WKB series generating an infinite number of the WKB terms. Coefficients of the first 2p terms of the expansion are exact while coefficients of a similar number of the next terms are approximate. The quantization condition in any QLM iteration, including the first, leads to exact energies for many well known physical potentials such as the Coulomb, harmonic oscillator, Pöschl-Teller, Hulthen, Hyleraas, Morse, Eckart etc.