Use the equations for the canonical ensemble to discuss a very simple system consisting of two Ising spins S₁ and S₂ with energy given by H = −JS₁S₂, where J > 0. Use the notation x = βJ and show:

(a) The partition function is given by Z = 4 cosh(x).
(b) The average energy is given by 〈E〉 = −J tanh(x).
(c) Determine 〈E〉 for T → 0 and for T → ∞ .