25.1. THEQUANTIZATIONOFSOUND 383
Inclassicalphysics,therearenosoundvibrationsatallinasolidatzerotemperature;
eachatomisat restinits equilibriumpositionxn =xn 0. Quantum-mechanically,
of course,the atomscannot beinsuchapositioneigenstate,because, duetothe
Uncertainty Principle, this would implyan infinite kinetic energy foreachatom.
Thus,evenatzerotemperature,theatomshaveafinitekinetic/potentialenergyof
vibration,whichiswhyE 0 ,thevibationalenergyatzerotemperature,issometimes
calledthe”zero-point”energy
Togettheexcitedstates,itssufficienttoobservethat
[H,a†k]= ̄hωka†k =⇒ Ha†k=a†k(H+h ̄ωk) (25.26)Repeateduseofthiscommutatorshowsusthat
HΨnanb...np=Enanb...npΨnanb...np (25.27)where
Ψnanb...np=(a†ka)na(a†kb)nb...(a†kp)npΨ 0 (25.28)
arethewavefunctionsoftheexcitedenergyeigenstates,and
Enanb...np=na ̄hωa+nb ̄hωb+...+np ̄hωp+E 0 (25.29)Interpretation:Aglanceat(25.8)showsthatQkisessentiallytheamplitudeofa
vibrational(sound)waveofwavenumberk,propagatingthroughthesolid. Suppose
we try to excite such waves inthe solid, by strikingit at some givenfrequency.
Inclassical physics, the correspondingsound waves couldhave anyenergy at all,
depending on how vigorouslywe strike thesolid. But quantum-mechanically, we
havejustdiscoveredthattheenergyofsuchsoundwavesisquantizedinunitsof ̄hωk;
i.e.theenergythatcanbeaddedtothesolid,byexcitingvibrationsofwavenumber
k,canonlybeoneofthepossiblevalues
∆E=n ̄hωk (25.30)Wherehaveweseenthisequationbefore? Itis,ofcourse,Planck’scondition
for the energy in aradiation field, where ωk = 2 πf isthe angular frequencyof
theradiation. Einstein’sinterpretationofPlanck’sformula,backin1905,wasthat
theenergy oftheradiation fieldof agivenfrequencyissubdividedinthe formof
photons, eachofwhichpropagatesthrough spacelike aparticle ofenergy hf. By
quantizingtheoscillations of asolid,we havefound the samephenomenon. The
energyofsoundvibrationsofasolid,ofagivenwavenumber,iscarriedintheform
of particle-likeexcitations knownas phonons, whichpropagatethrough thesolid
withmomentumproportionaltowavenumberk,andenergyequalto ̄hωk. Forthis
reason, wedrop the ”raising/lowering” operatorterminology, and referto a†k and
ak as creation/destruction operators, respectively, because they create/destroy
individualphononsinthesolid.