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Unformatted text preview: Homework 6 PHZ 5156 Due Thursday, October 15 1. Consider the timeindependent Schrodinger equation, ~ 2 2 m d 2 dx 2 + V ( x ) φ ( x ) = Eφ ( x ) where V ( x ) is a periodic potential with periodicity L such that V ( x + NL ) = V ( x ) where N is any integer. Bloch’s theorem shows that the eigenstates of the above equation can be written as, φ λk ( x ) = 1 √ L X G c λk,G exp[ i ( k + G ) x ] where G are the (onedimensional in this case) reciprocal lattice vectors given by G n = 2 πn L , where n is an integer. a) Show that the eigenvalue problem can then be written as, X G H G,G c λk,G = E λk c λk,G and determined an expression for the elements of the matrix H G,G . Convince yourself that this suggests the need to compute the Fourier transform of V ( x ) as, V G,G = 1 L Z L V ( x )exp[ i ( G G ) x ] dx Also, you should find that the diagonal elements of the matrix depend on k , whereas the offdiagonal elements do not....
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This note was uploaded on 08/08/2011 for the course PHZ 5156 taught by Professor Johnson,m during the Fall '08 term at University of Central Florida.
 Fall '08
 Johnson,M

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