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.
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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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