573lec39

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

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39 - 1 5.73 Lecture 39 updated September 19, 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 l . Consider a finite lattice (N atoms) but impose periodic (head-to-tail) boundary condition. L = N l Each unit cell, eq: A + A B × V i (x) i-th unit cell This is an infinitely repeated finite interval: Fourier Series Vx e V K n iKnx n () = = π =−∞ 2 l “reciprocal lattice vector”
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39 - 2 5.73 Lecture 39 updated September 19, 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 h Kn). Note that Kn is larger than the largest k (shortest λ ) free particle state that can be supported by a lattice of spacing l . Kn n k = π π ≤≤ π 2 l ll , first Brillouin Zone for k We will see that the lattice is able to exchange momentum in quanta of h 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 22 2 2 0 01 2 0 2 == = = = mm d dx V cons t Le E k m k ikx k h h ψ 2.
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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

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