Approximate analytic solutions of the Schrödinger equation for the generalized anharmonic oscillator

Abstract

High precision approximate analytic expressions for the ground state energies and wavefunctions of the generalized anharmonic oscillator with the potential U(x)=g2x2/2+λ|x|p are obtained by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The suitable initial guess for the first iteration ensures the high accuracy of the resulting expressions in a wide range of the parameters g⩾0, λ⩾0 and p⩾2, the latter need not be an integer. It is found that the precision of the obtained wavefunctions is close to 0.1% that is more accurate than Wentzel–Kramers–Brillouin (WKB) solutions by two to three orders of magnitude. In addition, the SWKB excited state energies have been calculated using the obtained ground state wavefunctions. The latter results are in good agreement with the exact ones. All of the above mentioned enables us to make accurate analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the generalized anharmonic oscillators.