This book presents an analytical theory of the electronic states in ideal low dimensional systems and finite crystals based on a differential equation theory approach. It provides precise and fundamental understandings on the electronic states in ideal low-dimensional systems and finite crystals, and offers new insights into some of the basic problems in low-dimensional systems, such as the surface states and quantum confinement effects, etc., some of which are quite different from what is traditionally believed in the solid state physics community. Many previous predictions have been confirmed in subsequent investigations by other authors on various relevant problems. In this new edition, the theory is further extended to one-dimensional photonic crystals and phononic crystals, and a general theoretical formalism for investigating the existence and properties of surface states/modes in semi-infinite one-dimensional crystals is developed. In addition, there are various revisions and improvements, including using the Kronig-Penney model to illustrate the analytical theory and make it easier to understand. This book is a valuable resource for solid-state physicists and material scientists.

The theory of electronic states in crystals is the very basis of modern solid state physics. In traditional solid state physics - based on the Bloch theorem - the theory of electronic states in crystals is essentially a theory of electronic states in crystals of in?nite size. However, that any real crystal always has a ?nite size is a physical reality one has to face. The di?erence between the electronic structure of a real crystal of ?nite size and the electronic structure obtained based on the Bloch theorem becomes more signi?cant as the crystal size decreases. A clear understanding of the properties of electronic states in real crystals of ?nite size has both theoretical and practical signi?cance. Many years ago when the author was a student learning solid state physics at Peking University, he was bothered by a feeling that the general use of the periodic boundary conditions seemed unconvincing. At least the e?ects of such a signi?cant simpli?cation should be clearly understood. Afterward, he learned that many of his school mates had the same feeling. Among many solid state physics books, the author found that only in the classic book Dynamic Theory of Crystal Lattices by Born and Huang was there a more detailed discussion on the e?ects of such a simpli?cation in an Appendix.

This book presents an analytical theory of the electronic states in ideal low dimensional systems and finite crystals based on a differential equation theory approach. It provides precise and fundamental understandings on the electronic states in ideal low-dimensional systems and finite crystals, and offers new insights into some of the basic problems in low-dimensional systems, such as the surface states and quantum confinement effects, etc., some of which are quite different from what is traditionally believed in the solid state physics community. Many previous predictions have been confirmed in subsequent investigations by other authors on various relevant problems. In this new edition, the theory is further extended to one-dimensional photonic crystals and phononic crystals, and a general theoretical formalism for investigating the existence and properties of surface states/modes in semi-infinite one-dimensional crystals is developed. In addition, there are various revisions and improvements, including using the Kronig-Penney model to illustrate the analytical theory and make it easier to understand. This book is a valuable resource for solid-state physicists and material scientists.