A. Consider a spin-half system of ξ noninteracting spins in a magnetic field. The
Hamiltonian is given by ξ ξξ ξξ§
ξ
ξξ, where the ξ-th spin ξξcan have value \pm ξ΄ and
ξ is a some constant with the dimension of energy.
(a) Use the microcanonical ensemble to getξξξξ
. (b) Use the canonical ensemble to get ξξξξ
and the magnetic susceptibility ξ²ξξξ
. B. Consider a system of ξ noninteracting spins in a magnetic field. The
Hamiltonian is given by ξ ξξ ξξ§
ξ
ξξ, where the ξ-th spin ξξ can have value ξ ξ΄ξξ½ξξ΄
and ξ is a some constant with the dimension of energy. Use the canonical
ensemble to get the magnetization ξ±ξξξ
ξ ξξ΄ξξξ
ξ ξ§ξξ ξ. C. Consider a two-level system (energy levels are ξ ξ‘ and ξ‘) composed of ξ
noninteracting distinguishable particles. Calculate the Helmholtz free energy, the
entropy, and the specific heat as functions of temperature ξ.