Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 207

c) Given the form of the spin operator inpart (c),one immediatelyderives

the equation by neglectingterms oforder

The equations of motion have aneigenvalue which represents the fre-

quencies ofthe spin waves.

d) The internal energy perunitvolume of the spinwaves isgiven by

where theoccupationnumber issuitable forbosons. At lowtemperature

we can evaluatethisexpression by defining thedimensionlessvariable

whichgives for theintegral

At low temperature the upperlimit of the integral becomeslarge, and the
internalenergy is proportional to The heat capacity is the derivative
of withrespect totemperature, so itgoes as

Fluctuations

4.89 Magnetization Fluctuation (Stony Brook)

The energy of a dipole in a magnetic field may be written

The partitionfunctionZis simply