The linear Schrödinger equation is central to Quantum Chemistry. It is presented within the context of relativistic Quantum Mechanics and analysed both in time-dependent and time-independent forms. The Riccati equation is used to study the one-dimensional Schrödinger equation.
The authors develop the Schrödinger-Riccati equation as an approach to determine solutions of the time-independent, linear Schrödinger equation.

1 Introduction

The Linear Schrödinger Equation
2 Derivation of the Schrödinger Equation
3 The Schrödinger Equation in Position Space
4 The Schrödinger Equation In Momentum Space
5 The Local Schrödinger Equation
6 The Time-Dependent Schrödinger Equation
The Non-Linear Schrödinger Equation
7 The Non-Linear Schrödinger Equation
The Riccati Equation
8 The Riccati Equation and Its Solution
9 Quantum-Mechanical Applications of the Riccati Equation
The Schrödinger-Riccati Equation
10 The Schrödinger-Riccati Equation
11 Numerical Experience with the Schrödinger-Riccati Equation
12 References and Bibliography
Appendix. Matrix Notation.