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hw01sol - 1 Introduction to Quantum and Statistical...

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Unformatted text preview: 1 Introduction to Quantum and Statistical Mechanics Homework 1 Solution Prob. 1.1. (a) (10 pts) n j z z z k dt z d m j j j j ,..., 1 ), 2 ( 1 1 2 2 = + − = − + From Newton’s second law we have: , 2 2 dt z d m ma F j = = The total force experienced by particle j according to Hooke’s law: ) 2 ( ) ( ) ( 1 1 1 1 − + − + + − = − − − − = j j j j j j j z z z k z z k z z k F Then, equating the force we obtain the equation that governs the motion of the particle. (b) (10 pts) . As Δ x → , we get: 2 1 1 1 1 2 2 2 x z z z x x z z x z z x z j j j j j j j j Δ + − = Δ Δ − − Δ − = ∂ ∂ − + − + Using L= Δ x(n+ 1)~n Δ x, total mass M=n m , total stiffness K=k/n , we obtain: 2 2 2 2 2 x z M KL dt z d j j ∂ ∂ = Then, plugging in plane wave z j = e i(kx-wt) into the wave equation , the dispersion relation is: 2 2 2 k M KL w = , or 2 k M K L w ± = The group and phase velocities are defined as, M K L k w v g ± = ∂ ∂ = ; M K L k w v p ± = = (c) (10 pts)...
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hw01sol - 1 Introduction to Quantum and Statistical...

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