1.3. Theorem of equi-partition Consider a classical one-dimensional harmonic oscillator, with Hamiltonian H = p2 2m

+ mω2x2 2

,

a. Determine the surface E = constant in the phase space and derive the thermodynamics of the system by using the microcanonical ensemble.

b. Put the system in contact with a thermal bath at temperature T. Compute the partition function in the canonical ensemble and show that the mean value of the energy is independent both from the frequency and the mass of the particle, i.e.

 p2 2m

 = mω2x2 2

 = 1 2

H = 1 2

kT.

c. Show that

(E −E)2 = (kT)2.