Special Topics In One-Dimensional Quantum Mechanics: Selected Exercises In Spatial and Momentum Translations

The present eBook is a – hopefully successful – attempt to present, in the framework of quantum mechanics, the spatial and momentum translation operators in one dimension, through a series of specifically designed exercises and applications. The eBook is divided into two parts. In the first two sections, the translation operators are defined and their basic properties are proved. It follows a series of problems with detailed solutions, covering a number of aspects of translations applied to various one-dimensional quantum systems, such as the harmonic oscillator, the infinite well, the attractive delta potential, and the attractive Coulomb potential, with particular emphasis on the harmonic oscillator, which serves as a model system for both spatial and momentum translations. The reader is assumed to have a basic knowledge of the postulates and the mathematical formalism of quantum mechanics, including the Dirac notation and the ladder operator method of the harmonic oscillator.