Consider a paramagnetic solid comprising N 12-spins subject to a magnetic field of magnitude B. The eigenvalues of the Hamiltonian read N H == -ΣHBBSi, i=1 where μB is the Bohr magneton and s; € {−1, +1} is the spin projection quantum number at site i E {1,.., N}. (a) State the definition of a separable Hamiltonian. Is the above Hamiltonian separable? Justify your answer. (b) Show that the quantum canonical partition function of the system reads Q(N, B, B) = [2 cosh(ßµÂB)]ª, where ß is the parameter setting the canonical ensemble family of distributions. (c) Using the correspondence between statistical mechanics and thermodynamics show that the entropy of the system reads S(T, N, B) = NkB In 2 + NkB Incosh (K₂F)) :)). MBB T - N tanh (HBD), KBT. where T is the absolute thermodynamic temperature and k the Boltzmann constant.

The three matrix operators for spin one satisfy sz Sy – Sy 8z = isz and cyclic permutations. Show that s = sz, (sz tisy )³ = 0. For the sane in, m, can have the values im, m – 1, .., -m, while A12 has eigenvalue m(m + 1). Thus M? = m(m + 1) ±2 5 x 1 = 5 times each once 6 15 4 x 8 = 32 times +3/2 ±1/2 3/2 each 8 times 4 ±1 3 x 27 = 81 times 1 each 27 times 3 2 x 18 = 96 times 1/2 ±1/2 each 48 times 1 × 12 = 42 times 0, each 42 times Total 256 eigenvalues A certain state | 4) is an eigenstate of L? and L,: L'|v) = 1(l+ 1) h² |»), mh|v) . For this state calculate (La) and (L²).

(AA) ²( ▲ B) ²≥ ½ (i[ÂÂ])² If [ÂÂ]=iñ, and  and represent Hermitian operators corresponding to observable properties, what is the minimum value that AA AB can have? Report your answer as a decimal number with three significant figures.

A spin-2 system has 5 spin states in the Sz basis. Express the following as matrices or vectors in this basis: Ŝ2 operator, Ŝx operator, Ŝy operator, Ŝz operator, and State |2, 1⟩x (x component of spin = ħ).