2nd Edition. – Springer Nature, Singapore, Pte Ltd., 2017. — 290 p. — (Springer Tracts in Modern Physics 270) — ISBN 978-981-10-4716-9.Based on a theory of differential equations approach, the first edition presented an analytical one-electron non-spin theory of electronic states in some simple and interesting low-dimensional systems and finite crystals. The essential understanding obtained is that different from that all electronic states are Bloch waves in a crystal with translational invariance as in the conventional solid state physics, a two boundary truncation of the translational invariance may produce two different types of electronic states—boundary dependent or size dependent—in some simple and interesting cases. The size dependent states are stationary Bloch waves due to the finite size of the truncated system, which properties and numbers are determined by the size. The boundary dependent states are a different type of electronic states which properties are determined by the very existence of the boundary. Such an understanding was first learned in one-dimensional crystals and later archived in multi-dimensional crystals as well. Since it was first published more than ten years ago, the existence of such two types of confined states or modes has been confirmed by subsequent investigations of other authors in various low-dimensional systems. The very presence of the boundary dependent confined states is a unique distinction of the quantum confinement of Bloch waves. Whether this is the origin of many unusual properties of low-dimensional systems remains to be understood. Contents Why a Theory of Electronic States in Crystals of Finite Size Is Needed Introduction One-Dimensional Semi-infinite Crystals and Finite Crystals The Periodic Sturm–Liouville Equations Surface States in One-Dimensional Semi-infinite Crystals Electronic States in Ideal One-Dimensional Crystals of Finite Length Low-Dimensional Systems and Finite Crystals Electronic States in Ideal Quantum Electronic States in Ideal Quantum Wires Electronic States in Ideal Finite Crystals or Quantum Dots Epilogue Concluding Remarks Appendixes The Kronig–Penney Model Electronic States in One-Dimensional Symmetric Finite Crystals with a Finite Vout Layered Crystals Analytical Expressions of K One-Dimensional Phononic Crystals One-Dimensional Photonic Crystals Electronic States in Ideal Cavity Structures
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