Two-level system the result of Problerrl4 is often seen in another form. (a) If the two energy levels are at and , show

Two-level system the result of Problerrl4 is often seen in another form. 

(a) If the two energy levels are at Δ and – Δ, show that the energy and heat capacity are U = - Δ tanh (Δ/k_BT); C = k_B (Δ/k_BT)² sech² (Δ/k_BT). 

(b) If the system has a random composition such that all values of Δ are equally likely up to some limit Δ₀, show that the heat capacity is linearly proportional to the temperature, provided k_BT << Δ₀. This result was applied to the heat capacity of dilute magnetic alloys by W. Marshall, Phys. Rev. 118, 1519 (1960). It is also used in the theory of glasses.