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573lec39

# 573lec39 - 5.73 Lecture 39 39 1 One Dimensional Lattice...

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39 - 1 5.73 Lecture 39 updated September 19, 2003 One Dimensional Lattice: Weak Coupling Limit See Baym “Lectures on Quantum Mechanics” pages 237-241. Each atom in lattice represented as a 1-D V(x) that could bind an unspecified number of electronic states. Lattice could consist of two or more different types of atoms. Periodic structure: repeated for each “unit cell”, of length . Consider a finite lattice (N atoms) but impose periodic (head-to-tail) boundary condition. L = N Each unit cell, eq: A + A B × V i (x) i-th unit cell This is an infinitely repeated finite interval: Fourier Series V x e V K n iKnx n ( ) = = π =−∞ 2 “reciprocal lattice vector”

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39 - 2 5.73 Lecture 39 updated September 19, 2003 V n is the (possibly complex) Fourier coefficient of the part of V(x) that looks like a free particle state with wave-vector Kn (momentum Kn). Note that Kn is larger than the largest k (shortest λ ) free particle state that can be supported by a lattice of spacing . Kn n k = π π π 2 , first Brillouin Zone for k We will see that the lattice is able to exchange momentum in quanta of n K with the free particle. In 3-D, is a vector. To solve for the effect of V(x) on a free particle, we use perturbation theory. r K 1. Define basis set. H p ( ) ( ) ( ) / ( ) tan 0 2 2 2 2 0 0 1 2 0 2 2 2 2 2 = = − = = = m m d dx V cons t L e E k m k ikx k ψ 2. H ( ) 1 = =−∞ n iKn n e V Matrix elements: H dx L e e V L e k k L ik x iKnx n n ikx = [ ] [ ] ( ) / / 1 0 1 2 1 2
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573lec39 - 5.73 Lecture 39 39 1 One Dimensional Lattice...

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