Problem Set 12

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Problems -- Due Wednesday, April 27, by 5pm.

Upload your solutions to Blackboard at this link: PS12

Problem 1 [20 points]
N fermions A of spin 1/2 (i.e. g_s=2) are introduced into a large volume V at temperature T. Two fermions may combine to create one boson with spin 0 (i.e. g_s=1) via the interaction,

A + A ↔ A₂
Creation of the molecule A₂ costs energy ε_o > 0. At equilibrium, the system will contain N_F fermions and N_B bosons. Provide expressions from which the ratio N_B/N_F can be calculated, and perform the calculation explicitly for T=0. What would this (T=0) ratio be, if the particles were classical (i.e. quantum statistics could be neglected). Explain the difference.
Problem 2 [10 points]
Consider the Ising model. The magnitization density is m, and the magnetic field is h. The magnetic susceptibility is then defined by

χ = dm
dh

where the derivative is evaluated at h = 0.
Derive an expression for χ in terms of the fluctuations in magnetization m. (Hint: recall, we had a similar expression for specific heat in terms of fluctuations in energy, and for compressibility in terms of fluctuations in volume.)
Show from this result that fluctuations in magnetization become negligible in the thermodynamic limit, except at a critical point.