(Solved): 4P 1. Consider N harmonic oscillators with coordinates and momenta (qP), and subject to the Hami ...

4P 1. Consider N harmonic oscillators with coordinates and momenta (qP?), and subject to the Hamiltonian q H(q?P?) = -22-21 ? 2m i=1 (a) Calculate the entropy S as function of the total energy E. (Hint. By appropriate change of scale the surface of constant energy can be deformed into a sphere. You may then ignore the difference between the surface area and volume for N >> 1. A more elegant method is to implement the deformation by a canonical transformation.) (b) Calculate the energy E, and heat capacity C, as functions of temperature 7, and N. (c) Find the joint probability density P(q.p) for a single oscillator. Hence calculate the mean kinetic energy, and mean potential energy, for each oscillator.