1. Introduction

2. Approaches to quantum transport

3. Wigner functions

4. Effective potentials

5. Numerical solutions

6. Particle approaches

7. Collisions and the Wigner function

8. Entanglement

9. Quantum chemistry

10. Semi-classical communications

11. Quantum optics

12. Quantum physics

The Wigner function is almost as old as quantum mechanics itself. Wigner's development was not aimed precisely at a phase-space representation, but by the desire to examine how thermodynamic equilibrium would incorporate any additional potentials that arose from the quantum description. In this pursuit, he developed his own form of a "quantum" potential. What Wigner triggered by his work was not only the recognition that phase-space descriptions could exist in quantum mechanics, but that there were many types of additional potentials that arose in these quantum descriptions. What has drawn considerable interest to the Wigner function, and its cousins, is an enhanced interest in non-equilibrium statistical mechanics. Wigner's phase-space formulation facilitates the study of quantum-classical transitions and clearly identifies behavior that is quantum mechanical in nature. The Wigner function has found many applications due to its ability to clearly designate non-classical behavior through its non-Gaussian shapes and its negative excursions. This allows it to clearly illustrate the presence and importance of entanglement, a pure quantum property.

This book is designed to give a background to the origins and development of Wigner functions, as well as its mathematical underpinnings. Along the way the authors emphasise the connections, and differences, from the more popular non-equilibrium Green's function approaches. But, the key importance lies in inclusion of applications of the Wigner function to various fields of science, including quantum information, coherent optics, and superconducting qubits. These disciplines approach it differently, and the goal here is to give a unified background and highlight how it is utilized in the different disciplines.

David Ferry is Regents' Professor Emeritus in the School of Electrical, Computer, and Energy Engineering at Arizona State University. He was also graduate faculty in the Department of Physics and the Materials Science and Engineering program at ASU, as well as Visiting Professor at Chiba University in Japan. In the distant past, he received his doctorate from the University of Texas, Austin, and spent a postdoctoral period at the University of Vienna, Austria. He enjoys teaching (which he refers to as 'warping young minds') and research.

Mihail (Mixi) Nedjalkov received his PhD in Physics in 1990 and his DSc in Mathematics in 2011 at the Bulgarian Academy of Sciences (BAS). He is Associate Professor with the Institute of Information and Communication Technologies, BAS. His research interests include physics and modeling of classical and quantum carrier transport in semiconductor materials, devices and nanostructures, collective phenomena, theory and application of Monte Carlo methods.