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.
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