QMGreensite_merged

25.1. THEQUANTIZATIONOFSOUND 381

Bytakingthecomplexconjugateoftheseinversetransforms,andusingthefactthat
qnandpnarerealnumbers,weseethat

Q∗k=Q−k Pk∗=P−k (25.11)

ThewonderfulthingaboutthistransformationisthattheHamiltonian,writtenin
termsofQk, Pk,isseparable,i.e.

∑

n

q^2 n =

1

N

∑

n

∑

k 1

∑

k 2

Qk 1 Qk 2 exp[i

2 π(k 1 +k 2 )

N

n]

=

∑

k

QkQ−k

∑

n

p^2 n =

1

N

∑

n

∑

k 1

∑

k 2

Pk 1 Pk 2 exp[i

2 π(k 1 +k 2 )

N

n]

=

∑

k

PkP−k

∑

n

qn+1qn =

1

N

∑

n

∑

k 1

∑

k 2

Qk 1 Qk 2 exp[i

2 π(k 1 +k 2 )

N

n]ei^2 πk^1 /N

=

∑

k

QkQ−kei^2 πk/N

=

∑

k

QkQ−kcos[2πk/N] (25.12)

Putitalltogether,

H=

∑

k

{ 1

2 m

PkP−k+K[1−cos(

2 π

N

)]QkQ−k

}

(25.13)

WestillneedtoknowthecommutationrelationsbetweenPandQ:

[Qk,Pk′] =

1

N

∑

n

∑

n′

[qn,pn′]exp[−i

2 π

N

(kn+k′n′)]

= i ̄hδk,−k′ (25.14)

Thus,

Pk=−i ̄h

∂

∂Q−k

(25.15)

InthiswaytheHamiltonianofthe1-dimensionalsolidhasbeenrewrittenasasum
ofharmonicoscillatorHamiltonians

H=

∑

k

[ 1

2 m

PkP−k+

1

2

σkQkQ−k

]

(25.16)

where
1
2

σk≡K[1−cos(

2 πk

N

)] (25.17)