AMATHProblemSet1-20010 - Advanced Quantum Theory Amath 473,...

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Advanced Quantum Theory Amath 473, 673 / Phys 454 Problem Set 1 This assignment is due at the beginning of class on Friday, Oct 1, 2010. 1. Consider a Hamiltonian describing two coupled oscillators: H = p 2 A 2 m A + k 1 q 2 A + c q A q B + p 2 B 2 m B + k 2 q 4 B where ( q A ,p A ) and ( q B ,p B ) are canonical coordinate pairs for systems A and B respectively that satisfy the fundamental Poisson bracket identities given in class. Assume that the dynamical equation for any function of the canonical coordinates is given by, df dt = { f,H } + ∂f ∂t . a) Using only the formal properties of the Poisson bracket derive the dynamical equation for each of the canonical coordinates. b) Assume that all of the canonical coordinates are commutative under ordinary multiplication and hence that we are describing a classical system. Use the associated representation of the Poisson bracket in terms of partial derivatives to derive the classical dynamical equations for each of the canonical coordinates. c) Assume that the canonical coordinates are
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