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Unformatted text preview: Homework 6 PHZ 5156 Due Thursday, October 15 1. Consider the time-independent Schrodinger equation,- ~ 2 2 m d 2 dx 2 + V ( x ) ( x ) = E ( x ) where V ( x ) is a periodic potential with periodicity L such that V ( x + NL ) = V ( x ) where N is any integer. Blochs theorem shows that the eigenstates of the above equation can be written as, k ( x ) = 1 L X G c k,G exp[ i ( k + G ) x ] where G are the (one-dimensional in this case) reciprocal lattice vectors given by G n = 2 n L , where n is an integer. a) Show that the eigenvalue problem can then be written as, X G H G,G c k,G = E k c k,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 off-diagonal elements do not....
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- Fall '08