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Unformatted text preview: Lecture 18: 1D chains (14 Oct 09) O. Admin Mid-term exam, take-home, handed out 16 Oct, due Monday 19 Oct. at 5 PM. review 12 Oct homework A. 1D discrete chain – complete 1. 1D chain with nearest neighbor interactions – Hooke’s law springs or transverse vibrations of discretized string. L = T- V = m 2 X j ˙ η 2 j- K 2 X j ( η j +1- η j ) 2 m ¨ η j =- K [2 η j- η j +1- η j- 1 ] 2. normal mode theory with η j ∝ exp( ijq- iω ( q ) t )) 3. Have used two boundary condition cases to set allowed q-values: fixed ends η = η N +1 = 0 [and q n = nπ/ ( N + 1) , n = 1 ,...,N ]; periodic boundary conditions η N +1 = η 1 [and q n = n 2 π/N ; n = 1 ,...N ]. The free end condition is set to have no force from one side: η 1 = η ; η N +1 = η N and the can be done with η j ∝ cos( q [ j- 1 2 ]. That works for j = 1 , automatically and for N,N + 1 set qN = nπ,n = 1 ,...N . 4. orthogonality of normal mode solutions (1) by general theory of the eigenvalue problems and (2) by explicit evaluation for the fixed end...
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This note was uploaded on 12/14/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at University of Wisconsin.
- Fall '09