Quantum mechanics in one band of a solid

K.G. Nichols, E. V. Vernon

Paolo Francisco, Tadan Cobb, Shawn A. Hilbert +1 more

When many individual atoms come together to form a solid, their interaction splits their electronic energy levels to form continuous bands separated by forbidden energy ranges known as band gaps. Introducing defects in a solid results in new electron energy levels that may lie inside the bandgaps. The presence of these defect levels is the heart of the semiconductor-based devices that play a significant role in the modern world. Quantum mechanics provides the best description of interacting atoms. However, band formation is not unique to small-scale atomic interactions but rather is a result of the wave-nature of Schrödinger's equation, which governs quantum mechanics. Using oscillations in a mass-spring system, we present a table-top, classical analog to the quantum system illustrating how defects in a one-dimensional lattice produce changes to the band structure. A pair of masses connected by a spring plays the role of a single atom. Interactions between “atoms” are introduced with weak coupling springs producing two distinct frequency bands from the translational and fundamental modes. Defects are introduced by altering an oscillator pair's total mass or internal spring constant. We provide the theoretical groundwork and experimental verification of the model along with a discussion of the value and limitations of the model as a macroscopic tool to visualize the microscopic world.

Heimo G. Latal

ISBN: 0 471 96364 X Although the author claims to have written a `novel introduction' to the quantum theory of matter, this book is certainly just one of a wealth of books on that subject appearing on the market every year. What is really novel about the presentation is that the more `formal aspects' of quantum mechanics, especially a discussion of the Schrodinger equation, are relegated to an appendix, while in the text only specific solutions to that fundamental equation are discussed. On the other hand, the book certainly provides an extensive outline of various applications of quantum mechanics to atoms, molecules and solids. On the atomic level even the emission of light, spin and many electron atoms are treated; similarly on the molecular scale the most important aspects are presented. Almost one half of the book is devoted to the quantum theory of solids, and here the range of topics includes, of course, the band model and its various implications, but also surfaces and non-crystalline solids. Due to the aim of the book the author had to present various complex situations in an abbreviated form, thereby sometimes losing the rigour of the necessary proof of the statements. But, taken altogether, the book will provide the reader with an overall view, and sometimes an in-depth discussion, of the quantum world of atoms, molecules and solids. Somewhat distracting might be the large number of footnotes collected at the end of each chapter; of advantage, however, are the selections of problems there.

I. M. Lifshit︠s︡

Fabian H. L. Eßler, Holger Frahm, Frank Göhmann +2 more

The description of solids at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons. It is impossible to solve the corresponding many-body problems analytically or numerically, although much insight can be gained from the analysis of simplified models. An important example is the Hubbard model, which describes interacting electrons in narrow energy bands, and which has been applied to problems as diverse as high-Tc superconductivity, band magnetism, and the metal-insulator transition. This 2005 book presents a coherent, self-contained account of the exact solution of the Hubbard model in one dimension. The early chapters will be accessible to beginning graduate students with a basic knowledge of quantum mechanics and statistical mechanics. The later chapters address more advanced topics, and are intended as a guide for researchers to some of the more topical results in the field of integrable models.