QMGreensite_merged

382 CHAPTER25. AGLIMPSEOFQUANTUMFIELDTHEORY

Asintheordinaryharmonic-oscillatorproblem,wesolveHΨ=EΨbyintroducing
raising-loweringoperators

ak =

√

mωk

2 ̄h

[

Qk+

i

mωk

Pk

]

a†k =

√

mωk

2 ̄h

[

Q−k−

i

mωk

P−k

]

Qk =

√

̄h

mωk

(ak+a†−k)

Pk =

1

2 i

√

2 ̄hmωk(ak−a†−k) (25.18)

where

ωk=

√

σk

m

(25.19)

Intermsoftheseoperators,onefindsthat

H=

∑

k

̄hωk(a†kak+

1

2

) (25.20)

where
[ak,a†k′]=δk,k′ (25.21)

Theproblemhasbeenreducedtosolvingthedynamicsofasetofuncoupledharmonic
oscillators; oneoscillatorfor eachwavenumberk intherange −^12 (N−1) ≤ k ≤
1
2 (N−1).
Allloweringoperatorsannihilatetheharmonic-oscillatorgroundstate,sowere-
quirethat,forallwavenumbersk

akΨ 0 [Q]= 0 (25.22)

or

̄h

∂

∂Q−k

Ψ 0 =−mωkQkψ 0 (25.23)

Thisequationcanbesolvedbyinspection,andwehave,forthegroundstateofthe
solid,

Ψ 0 = Nexp

[

−

m

̄h

∑

k

ωkQkQ−k

]

= Nexp

[

−

√

m

̄h

∑

k

{ 2 K(1−cos(

2 πk

N

)}^1 /^2 QkQ∗k

]

(25.24)

withground-state(or”zero-point”)energy

E 0 =

∑

k

1

2

̄hωk (25.25)