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Midterm 1 Solutions

# Midterm 1 Solutions - Physics 112 Spring 2010 Professor...

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Professor William Holzapfel Midterm 1 Solutions Problem 1 a) For a spin 1 particle, S z can have values - ~ , 0 , ~ , which means that the z - component of the magnetic moment can have values - μ, 0 . Since the energy of a magnetic moment in a magnetic ﬁeld is U = - · ~ H , this gives possible energies μH, 0 , - μH , which when graphed as functions of H just give three straight lines. b) The partition function for a single particle is Z 1 = e - μH/τ + 1 + e μH/τ = 1 + 2 cosh μH/τ and so for the N distinguishable particles we have Z N = Z N 1 = (1 + 2 cosh μH/τ ) N . c) Skipping a step or two of algebra, we have U = τ 2 Z ∂Z ∂τ = - μHN 2 sinh μH/τ 1 + 2 cosh μH/τ . (1) d) From (1) we have lim τ 0 U ( τ ) = - μHN lim x →∞ e x - e - x 1 + e x + e - x = - μHN which just says that at τ = 0, all spins are aligned with the ﬁeld and in their minimum energy state. Equation (1) also tells us that lim τ →∞ U ( τ ) = - μHN lim x 0 e x - e - x 1 + e x + e - x = 0 which just says that as τ → ∞ (and the entropy becomes maximized) the spins are evenly distributed between the three possible energy states, and since the average energy of these states is zero then so is the average energy of the system. 1

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Midterm 1 Solutions - Physics 112 Spring 2010 Professor...

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