Thermodynamics, Statistical Physics, and Quantum Mechanics

206 SOLUTIONS

ForBandMsmall we may rearrange (S.4.87.4),yielding

where Thedivergence of at indicates theonset of
ferromagnetism. Thespins will alignspontaneously in theabsence of an
appliedmagnetic field atthistemperature.

4.88 Spin Waves in Ferromagnets (Princeton,

Colorado)

Quantumspins havethe commutationrelations

a) The time dependences of thespins are given by theequations of motion:

b) The classicalspin field at point is In thesimplecubic lattice

the six neighboring latticesites are at the points where is

or We expand the sum in a Taylor series, assuming that is a small

number,and find