Chapter 25
A Glimpse of Quantum Field
Theory
25.1 The Quantization of Sound
Strikeasolidatanypoint,andyougenerateasoundwavetravelingthroughthesolid.
Onamicroscopiclevel,asolidisaregulararrangementofatoms,andasoundwave
issimplyaparticularvibrationofthoseatoms. Butthemotionofatomsshouldbe
treatedquantummechanically. What,then,isthequantummechanicsofsound?
Asusual,simplify.Ourmodelwillbeaone-dimensionalsolidwithperiodicbound-
aryconditions. Thisissimilar totheKronig-Penneymodel, exceptthe atomsare
allowedtovibrate. ThereareN atoms;thecoordinateofthen-thatomisxn,and
its classical equilibriumposition isxn 0. Periodic boundary conditions meanthat
xn+N=xn. Denotethedisplacementofthen-thatomfromitsequilibriumposition
by
qn=xn−xn 0 (25.1)
Assumingonlynearest-neighboratomsinteract,wecanwritethepotentialasasum
oftwo-bodypotentials,whichweexpandinTaylorseries
V =
∑N
n=1
f(qn+1−qn)
=
∑N
n=1
[f(0)+f′(0)(qn+1−qn)+
1
2
f′′(0)(qn+1−qn)^2 +...] (25.2)
Theconstanttermisphysicallyirrelevant,sosetf(0)= 0 forconvenience. Dueto
theperiodicboundaryconditionsqN+1=q 1 ,
∑N
n=1
(qn+1−qn)= 0 (25.3)