Schrödinger–Riccati equation: Feasibility study for the He‐isoelectronic series

Abstract

ABSTRACT:The Schrö dinger-Riccati equation has been applied to the study of some systems of the He-isoelectronic series. The statistical analysis of the local ratios of the predicted function over an accurate function yields, with a sample mean of 1.000, the following confidence intervals (at a 99% level of confidence) for the true population means: He I, 0.0013 (for 24,896 points); Li II, 0.0015 (for 29,354 points); Be III, 0.0018 (for 31,290 points); C V, 0.0024 (for 28,639 points); and O VII, 0.0058 (for 23,784 points).

Few sketches on connections between the Riccati and Ermakov–Milne–Pinney equations

“…The Schrödinger equation that describes a quantum particle of mass m moving in the one‐dimensional potential U ( x ) is the second‐order linear differential equation 15 The Riccati differential equation, either (1) or (2), is widely used in many areas of quantum mechanics (see e.g., Refs. 16–24 for current works): viz., in the WKB approximation 19, supersymmetric (SUSY) quantum mechanics 20, 21, and quantum chemistry (see Refs. 22–24 and references therein).…”

“…16–24 for current works): viz., in the WKB approximation 19, supersymmetric (SUSY) quantum mechanics 20, 21, and quantum chemistry (see Refs. 22–24 and references therein). In particular, the SUSY quantum mechanics is applied to explicitly solve the Schrödinger equation (4) with variety of potentials which is widely used in the quantum chemistry 21.…”

“…In particular, the SUSY quantum mechanics is applied to explicitly solve the Schrödinger equation (4) with variety of potentials which is widely used in the quantum chemistry 21. The Schrödinger–Riccati approach proposed in 23 allows to accurately estimate the ground state of atoms.…”