Band Theory. The Linear Chain of Hydrogen Atoms

4 Band Theory. The Linear Chain of Hydrogen Atoms 

In this way one becomes independent of the number N, as the limits for k become 0 and n/a. Contrary to the numbers k the values for k are not integral numbers. 

Vibrational modes of a chain of N + 1 spheres connected by springs 

The Jk-th wave function of the electrons in a chain of hydrogen atoms results in a similar way. From every atom we obtain a contribution 2 cosnka, i.e. the Is function %n of the n-th atom of the chain takes the place of A0. All atoms have the same function x, referred to the local coordinate system of the atom, and the index n designates the position of the atom in the chain. The k-th wave function is composed of contributions of all atoms  

A wave function composed in this way from the contributions of single atoms is called a Bloch function [in texts on quantum chemistry you will And this function being formulated with exponential functions exp(inka) instead of the cosine functions, since this facilitates the mathematical treatment]. 

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