Solution:

If only motion in the x-direction is allowed, the Hamiltonian is the sum of the kinetic energy T and the potential energy V, where V = (1/2)k(x₂-x₁-b)² + (1/2)k(x₃-x₂-b)², with k the spring constant and b the equilibrium separation.

(a)  Write Hamilton’s equations of motion for the molecule.
(b)  Assume that displacements from equilibrium are proportional to exp(iwt).  Express Hamilton’s equations in matrix form (i.e., a 3´3 matrix multiplying a 3-component column vector), and solve for the normal mode frequencies w₁<w₂<w₃.
(c)  Discuss the nature of the three normal modes, showing the motions of the three atoms for each mode.

A linear system of N-1 spheres of mass M and two fixed end spheres (of infinite mass) are connected by springs of spring constant k as shown.

The equilibrium positions are given by x_n⁰ = na. Let x_n = x_n⁰+u_n.  Note that u₀ = u_N = 0.
(a)  Write down the Hamiltonian of the system.
(b)  Determine the equations of motion for the nth sphere (1 £ n £ N-1).
(c)  Assume a solution of the form x_n(t) = exp(iwt)[Ae^ikan + Be^-ikan] and determine the values of k allowed by the boundary conditions.
(d)  How many independent values of k are there?
(e)  From the equation of motion, determine the frequency w_k associated with the allowed k-values.
(f)  Introduce normal coordinates P_k and Q_k and write the Hamiltonian in terms of them (without proof or derivation).