Thermodynamics, Statistical Physics, and Quantum Mechanics

40 PROBLEMS

Determine the thermodynamic mean value of the magnetic moment

and the magnetization of thesystemM, and calculate it for

and

Find the magnetization of the system in the limits and

and discuss the physical meaning of the results.

4.82 Paramagnetism at High Temperature (Boston)

a) Show that for a system with a discrete,finite energyspectrum the

specific heat per particle at high temperatures for all is

where is the spectrum variance

b)

c)

Use the result of (a) to derive the high-temperature specific heat for

a paramagnetic solid treated both classically and quantum mechani-

cally.

Compare your quantum mechanical result for with the exact

formula for

4.83 One-Dimensional Ising Model (Tennessee)

Consider N spins in a chain which can be modeled using the one-
dimensionalIsingmodel

where the spin has the values

4.84 Three Ising Spins (Tennessee)

Assume three spins are arranged in an equilateral triangle with each spin in-
teractingwith its twoneighbors (seeFigure P.4.84). The energy expression
for the Ising model in a magnetic field is

b)

c)

a)

b)

Find thepartitionfunction.

Find the heat capacity per spin.