The Schrödinger and Riccati equations
type: Item
instance of: scholarly article
title: The Schrödinger and Riccati equations
zbMATH Open document ID: 0947.34071
author: Yanyan Li
published in: Lecture Notes in Chemistry
publication date: 2000-03-21T00:00:00Z
review text: The major topics of the book are: (i) Linear Schrödinger equation, (ii) The nonlinear Schrödinger equation, (iii) The Riccati equation and (iv) The Schrödinger-Riccati equation. The authors introduce the Schrödinger equation as the reduction of the generalized Dirac-Breit equation for a many-particle system in the nonrelativistic form, (though the title of the chapter is ``Derivation of the Schrödinger equation, in the text it is mentioned that the equation cannot be derived). In the next chapter, local and global properties, in particular the viral theorem, are discussed in the configuration space. This is followed by observations on the momentum representation of both the equation and the wave function by Fourier transformation. This is illustrated with some representative numerical calculations. In the chapter: The local Schrödinger equation, the topics discussed are alternatives for the computational solution of the equation (local energy methods) and a survey of alternative numerical methods for global solution. The following chapter, The time-dependent Schrödinger equation, presents the three representations namely (a) Schrödinger, (b) Heisenberg and (c) interaction representation (Dirac), time evolution in interaction representation and transition probabilities are also given. In the brief part, The nonlinear Schrödinger equation, the discussion is confined only on conservation laws and conserved quantities, and numerical solution to the (one-dimensional) equation. The only parts of the book concomitant to the title of the book are the last parts. The brief penultimate part, The Riccati equation, consists of the well-known reduction of the linear homogeneous second-order equation to the Riccati equation, followed by a discussion on some of the properties of its solution. The last part namely, the Schrödinger-Riccati equation is an essentially quantum mechanical application of the Riccati equation. This deals with an alternative to the perturbation theory whic…
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 81-02
zbMATH DE Number: 1419150
zbMATH Keywords: generalized Dirac-Breit equation
zbMATH Keywords: many-particle system
zbMATH Keywords: interaction representation
zbMATH Keywords: transition probabilities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1419150