380 CHAPTER25. AGLIMPSEOFQUANTUMFIELDTHEORY
sothesecondtermisalsozero. Thenthepotentialmusthavetheform
V =
1
2
K
∑N
n=1
(qn+1−qn)^2 (25.4)
forsmalldisplacements ofthe atomsaroundequilibrium, whereK ≡f′′(0). The
Hamiltonianis
H=
∑N
n=1
[
1
2 m
p^2 n+
1
2
K(qn+1−qn)^2
]
(25.5)
wherepnisthemomentumofthen-thatom. Uponquantization,
pn→−i ̄h
∂
∂xn
=−i ̄h
∂
∂qn
(25.6)
andtheN-bodySchrodingerequationforthesolidis
∑N
n=1
[
−
̄h^2
2 m
∂^2
∂qn^2
+
1
2
K(qn+1−qn)^2
]
Ψ[{qi}]=EΨ[{qi}] (25.7)
Thesystemhasagroundstate,denotedΨ 0. Quantizedsoundwavescanonlycorre-
spondtoexcitedstatesofthesystem.
TheSchrodinger equationaboveis apartialdifferential equationinN ∼ 1023
variables. Theonlychanceofsolvingitisbythemethodofseparationofvariables.
IntroducethefiniteFouriertransform
qn =
1
√
N
(N∑−1)/ 2
k=−(N−1)/ 2
Qkexp[i
2 πn
N
k]
pn =
1
√
N
(N∑−1)/ 2
k=−(N−1)/ 2
Pkexp[i
2 πn
N
k] (25.8)
whichautomaticallyincorporatesperiodicboundaryconditions. Usingtheidentity
∑N
n=1
exp[i
2 π(k−k′)
N
n]=Nδkk′ (25.9)
wecanwritetheinversetransform
Qk =
1
√
N
∑N
n=1
qnexp[−i
2 πk
N
n]
Pk =
1
√
N
(N∑−1)/ 2
k=−(N−1)/ 2
pnexp[−i
2 πk
N
n] (25.10)