2.Quantization of elastic waves. Let N particles of mass M be connected by a spring offorce constant C and length a. To fix the boundary conditions, let the particles form acircular ring. Consider that the transverse displacements of the particles are out ofplane of the ring. The displacement of the particle sis q, and its momentum is ps. Giventhat the Hamiltonian of the system isnH = Σ{2MP² + 2 C(9s +1 - 95) ²}s=1Show that the energy eigenvalues are€n = (n + 2²2 ) hwwhere n = 0,1,2,3,...Hint: You may use the transformation and the corresponding inverse transformationof the particle's coordinates q, and momentum p, to the phonon coordinates Qk andmomentum Pk. That is,= N- Σ Q₁ exp(iksa), QK = N-²9, exp(-iksa)9s =9sPs = Nz ΣP₁ exp(-iksa),SP₁ = N-Ps exp(iksa)S

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Answered: Quantization of elastic waves. Let N…

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This is question 8.8 in John R. Taylor's "Classical Mechanics" textbook by the way! (ISBN: 9781891389221)
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