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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.

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.

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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 μg 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 (BugB)], where ß is the parameter setting the canonical ensemble family of distributions. and (c) Using the correspondence between statistical mechanics thermodynamics show that the entropy of the system reads S(T, N, B) = Nkâ ln 2 + Nkg ln (cosh (KP)) - H MBB T N tanh (MBBY kgT B where T is the absolute thermodynamic temperature and k the Boltzmann constant. (d) Using the fact that ln(1 + x) ≈ x, cosh x ≈ 1 + x²/2 and tanh x≈ x for x < 1 show that at high temperatures the entropy found in (c) approximately reads S(T, N, B) ≈ So - K B² T² and give the expressions of So and K. ==
Transcribed Image Text: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 μg 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 (BugB)], where ß is the parameter setting the canonical ensemble family of distributions. and (c) Using the correspondence between statistical mechanics thermodynamics show that the entropy of the system reads S(T, N, B) = Nkâ ln 2 + Nkg ln (cosh (KP)) - H MBB T N tanh (MBBY kgT B where T is the absolute thermodynamic temperature and k the Boltzmann constant. (d) Using the fact that ln(1 + x) ≈ x, cosh x ≈ 1 + x²/2 and tanh x≈ x for x < 1 show that at high temperatures the entropy found in (c) approximately reads S(T, N, B) ≈ So - K B² T² and give the expressions of So and K. ==
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