1-4 - Phonon commutation relations: Consider N
akν =
∑ ℓ =1
the phonon creation and annihilation operators, − i ⋅k • R i⋅k • R N * e e D .C . 1/ 2 1/ 2 † −1/ 2 −1/ 2 Pℓ ) ← P ℓ ) = akν ; ε kν • ( ( M ωkν ) u ℓ + i ( M ωkν ) → ε kν • ( ( M ωkν ) uℓ − i( M ω kν ) 2 N ℏ 2 N ℏ ℓ =1 ℓ
∑
compon onen entt of atom atomic ic disp displa lace ceme ment nt of ℓth atom] atom];; uℓi = [ith comp i
Pℓ = [i
ℓ
R ℓ = [ℓ
th
latt lattic icee-sit sitee posit positio ion]; n]; ℓ = [atom-index];
(1.1)
*
th
compone onent of momentum of ℓth atom]; m]; ε ki ν = [ith component of polarization-vector] = ε −i k ,ν ;
Index notation:
th
th
Throughout this problem: we will refer to the i and/or j component of any vector quantity.
Invert these relations (1.1). To make obvious the steps one must undertake to invert (1.1), write this as a linear
system, using ε kν • uℓ =ɺ ε ki ν uℓi , and the characteristic length ℓ kν ≡
akν N a† = ∑ kν ℓ =1
i 1 ε 2 N kν i 1 ε 2 N kν
e
− i ⋅k •R ℓ
e
+ i ⋅k • R ℓ
([u ([u
i
/ ℓ kν ] + i[ pℓi ℓ kν / ℏ])
ℏ / ( M ω kν ) = ℓ − kν ;
e − i ⋅k • R = ∑ i⋅k • R i i / ℓ kν ] − i[ pℓ ℓ kν / ℏ]) ℓ =1 e ℓ ℓ
Front-multiply (1.2) by the operator
∑
j
kν
N ε k j′ν akν e+ ik ′•R e− i⋅(k −k ′)• R ∑ * j † + ik ′•R = ∑ ∑ i⋅(k +k ′)•R kν ε k ′ν akν e kν ℓ =1 e ℓ
ε k ′ν e
ie
ℓ
i⋅k ′•R ℓ
and its conjugate
i
−i
− i⋅ ( k − k ′ )• R ℓ
−ie
ℓ
Subsequently: invert the matrix [11
]−1 =
1 − i −i
i ⋅ (k + k ′ )• R ℓ
[−−1i
−i
1
∑
ℓ
ℓ
1 2 N
ℓ,ν
−i e
i
∑
* j
kν
i⋅k • R ℓ
ε k ′ν e
− i ⋅k ′•Rℓ
i 1 ε 2 N kν i 1 ε 2 N kν
i ⋅ pℓ ℓ kν / ℏ i
⋅ uℓ / ℓ kν
(1.2)
nd
row) in which
1 2 N
⋅ uℓ
(for the 2
j
i
1 = ⋅ N 1 i ⋅ pℓ ℓ kν / ℏ i
ε k′ν ε kν ⋅ uℓ / ℓ kν i
ε − k ′ν ε kν
1
− i ⋅k • Rℓ
( ),
*j
1 2 N
] = 12 [1− i
i k ′• R j uℓ j / ℓ k ′ν 2 1 1 1 ε k ′ν akν e = j ∑ − i i j † ik′•R → 2 N / p ℓ ℏ k ν ε k′ν a− kν e ℓ − k ′ν
i
ie
ℓ
ℓ
i i* we use time reversal ε kν = ε −k ,ν (see (1.1)); collapse only ℓ
N
we get,
−i i
j
/ ℓ k ′ν
(1.3) j 1 ⋅ pℓ ℓ − k ′ν / ℏ 2 N
] , and we get,
eik• R (ε −jkν a−†kν + ε kjν akν ) × ℓ kν uℓj 1 j = ik • R j † (1.4) ∑ j 2 N ( ) ( / ) i − × p e a a ℏ ℓ ε ε k ν ℓ − kν + kν kν kν kν ℓ
ℓ
i
i
i
find expressions expressions for u ℓ = ɺ uℓ and Pℓ = ɺ Pℓ in terms of ukν = ɺ ukν and Pkν = Pkν , defined as, ℏ
i
ukν = i Pkν = −i
2 M ω kν
i ε kν akν =
i † 1 ℏ ω ε M a = −i kν kν kν 2
1 2
2 1 ℏ i i† ε −kν akν = Pkν ; 2 ℓ kν
i i † ℓ kν ε kν akν ↔ akν ε − kν ℓ kν
1 ℏ i † ε kν akν ↔ +i 2 ℓ kν
1
i†
= ukν ;
ℏ
ℓ kν ≡
= ℓ − kν ;
(1.5)
eik •R (u−jkν † + ukjν ) ∑ ik•R ( P j − P j † ) ; N kν e kν − kν
(1.6)
M ω kν
We just rewrite (1.4) as,
uℓ j 1 j = N pℓ
j j † ik • R [ 12 ℓ −kν ε −kν a−kν + 12 ℓ kν ε kν akν ] e ik•R 1 = ∑ [ 2 (ℏ / ℓ − kν )ε k jν a−†kν − 12 (ℏ / ℓ + kν )ε kjν akν ] kν ie ℓ
ℓ
1
ℓ
ℓ
Show that [ akν , ak†′ν ′ ] = δ kk ′δ νν ′ . Using ε ki ν ε −jkν ′δ ij = δνν ′ = ε ki ν ε k*νj ′ (note the representation of the inner-product),
representing the dot-product using index-notation, e.g., ε kν • uℓ = ε kiν uℓi , and using [uℓi , Pℓ j ] = iℏδ ij (baby QM), we wind up getting,
†
[akν , ak ′ν ′ ] =
1
∑e
2 N ℓ ,ℓ′
i⋅( k•R ℓ − k ′• R ℓ′ )
[
ε kν • uℓ ℓ kν
+i
* ε kν • Pℓ ε kν ′ • u ℓ′ ℏ / ℓ kν
,
ℓ kν
−i
* ε kν ′ • P ℓ′ ℏ / ℓ kν
]
ε ki ν ε −jkν ′ [uℓi , uℓj ] 2 ε ki ν ε −jkν ′ [ Pℓi , Pℓ j ] −ε kiν ε −jkν ′ [uℓi , Pℓ j ] + εkiν ε −jkν ′ [ Pℓi , uℓj ] = −i +i ∑ δ kk′ ℓ 2 2 ℏ ℓ ℏ 2 N ℓ ( / ) kν kν i j i j i j −ε kν ε −kν ′ iℏδ ij + ε kν ε − kν ′ ( −iℏδ ji ) 2ε kν ε − kν ′δ ij N 1 † [akν , ak ′ν ′ ] = δ kk ′ ∑ 0 − 0 + i δ kk ′ ∑ 1 = δ kk′δνν ′ = 2 N ℏ 2 N ℓ ℓ =1 1
(1.7)
