Introduction to Quantum Mechanics 2

Wave-Corpuscle, Quantization and Schrodinger's Equation

Ibrahima Sakho

• Publisher: Wiley-ISTE
• Year: 2020
• Print ISBN: 9781786305015
• eBook ISBN: 9781119694953
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

Quantum mechanics is the foundation of modern technology, due to its innumerable applications in physics, chemistry and even biology.

This second volume studies Schrödingers equation and its applications in the study of wells, steps and potential barriers. It examines the properties of orthonormal bases in the space of square-summable wave functions and Dirac notations in the space of states.

This book has a special focus on the notions of the linear operators, the Hermitian operators, observables, Hermitian conjugation, commutators and the representation of kets, bras and operators in the space of states. The eigenvalue equation, the characteristic equation and the evolution equation of the mean value of an observable are introduced. The book goes on to investigate the study of conservative systems through the time evolution operator and Ehrenfests theorem.

Finally, this second volume is completed by the introduction of the notions of quantum wire, quantum wells of semiconductor materials and quantum dots in the appendices.

1
Schrödinger’s Equation and its Applications

General objective

The general objective is to apply the Schrödinger equation to the study of simple physical systems.

Specific objectives

On completing this chapter, the reader should be able to:

• – know the properties of the square-summable wave functions;
• – know the boundary conditions imposed to any square-summable wave function;
• – distinguish between a physical state in classical mechanics and in quantum mechanics;
• – describe a physical quantity in quantum mechanics;
• – define an operator;
• – define an observable;
• – give examples of operators and observables;
• – know the correspondence principle or rule;
• – define the Hamiltonian of a physical system;
• – express the time-dependent Schrödinger equation;
• – express the stationary Schrödinger equation;
• – know the properties of the Schrödinger equation;
• – integrate the Schrödinger equation for a free particle;
• – integrate the Schrödinger equation for the ground state of the hydrogen atom;
• – apply the Schrödinger equation to the study of quantum wells;
• – apply the Schrödinger equation to the study of quantum dots;
• – apply the Schrödinger equation to the study of potential barriers;
• – apply the Schrödinger equation to the study of potential steps;
• – define the probability current;
• – define the reflection and transmission factors;
• – define the reflection and transmission probabilities;
• – provide an interpretation of the t...

Table of contents

• Cover
• Table of Contents
• Foreword
• Preface
• 1 Schrödinger’s Equation and its Applications
• 2 Hermitian Operator, Dirac’s Notations
• 3 Eigenvalues and Eigenvectors of an Observable
• Appendix 1: Quantum Well of Semiconductor Materials
• Appendix 2: Quantum Dot of Semiconductor Materials
• Appendix 3: Transparency of a Potential Barrier, Resonance
• References
• Index
• End User License Agreement