(A Density Matrix Problem). (1) Show that the density matrix for an ensemble of spin-1/2 particles may be written as
ρ=21(I+a⋅σ)
where a is a c-number vector.
(2) Show that a is the mean polarization, ⟨σˉ⟩.
(3) An ensemble of electrons in a magnetic field B=Bk, is in thermal equilibrium at temperature T. Construct the density matrix for this ensemble. Calculate ⟨μˉ⟩.