QMGreensite_merged

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)

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