1. Harmonic oscillators (20 points): We look at a system of N unco...

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1. Harmonic oscillators (20 points): We look at a system of N uncoupled harmonic oscillators in a three dimensional volume V . Each oscillator is described by an Hamiltonian H = ~p 2 2m + k 2 ~x2 where ~p is the momentum and ~x the position of a particle. (k and m are constants.) • Calculate the entropy S(U, V, N) assuming the system is isolated. • Calculate the free energy F(T, V, N) assuming that the system is coupled to a heat bath. • Calculate the energy-temperature dependence U(T) and the pressure p of this system.

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