QMGreensite_merged

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)