リンドハード理論
ε ( 0 , ω ) ≃ 1 + V q ∑ k , i q i ∂ f k ∂ k i ħ ω 0 − ħ 2 k → ⋅ q → m ≃ 1 + V q ħ ω 0 ∑ k , i q i ∂ f k ∂ k i ( 1 + ħ k → ⋅ q → m ω 0 ) ≃ 1 + V q ħ ω 0 ∑ k , i q i ∂ f k ∂ k i ħ k → ⋅ q → m ω 0 = 1 + V q ħ ω 0 2 ∫ d 2 k ( L 2 π ) 2 ∑ i , j q i ∂ f k ∂ k i ħ k j q j m ω 0 = 1 + V q L 2 m ω 0 2 2 ∫ d 2 k ( 2 π ) 2 ∑ i , j q i q j k j ∂ f k ∂ k i = 1 + V q L 2 m ω 0 2 ∑ i , j q i q j 2 ∫ d 2 k ( 2 π ) 2 k j ∂ f k ∂ k i = 1 − V q L 2 m ω 0 2 ∑ i , j q i q j 2 ∫ d 2 k ( 2 π ) 2 k k ∂ f j ∂ k i = 1 − V q L 2 m ω 0 2 ∑ i , j q i q j n δ i j = 1 − 2 π e 2 ε q L 2 L 2 m ω 0 2 q 2 n = 1 − ω p l 2 ( q ) ω 0 2 , {\displaystyle {\begin{alignedat}{2}\epsilon (0,\omega )&\simeq 1+V_{q}\sum _{k,i}{\frac {q_{i}{\frac {\partial f_{k}}{\partial k_{i}}}}{\hbar \omega _{0}-{\frac {\hbar ^{2}{\vec {k}}\cdot {\vec {q}}}{m}}}}\\&\simeq 1+{\frac {V_{q}}{\hbar \omega _{0}}}\sum _{k,i}{q_{i}{\frac {\partial f_{k}}{\partial k_{i}}}}(1+{\frac {\hbar {\vec {k}}\cdot {\vec {q}}}{m\omega _{0}}})\\&\simeq 1+{\frac {V_{q}}{\hbar \omega _{0}}}\sum _{k,i}{q_{i}{\frac {\partial f_{k}}{\partial k_{i}}}}{\frac {\hbar {\vec {k}}\cdot {\vec {q}}}{m\omega _{0}}}\\&=1+{\frac {V_{q}}{\hbar \omega _{0}}}2\int d^{2}k({\frac {L}{2\pi }})^{2}\sum _{i,j}{q_{i}{\frac {\partial f_{k}}{\partial k_{i}}}}{\frac {\hbar k_{j}q_{j}}{m\omega _{0}}}\\&=1+{\frac {V_{q}L^{2}}{m\omega _{0}^{2}}}2\int {\frac {d^{2}k}{(2\pi )^{2}}}\sum _{i,j}{q_{i}q_{j}k_{j}{\frac {\partial f_{k}}{\partial k_{i}}}}\\&=1+{\frac {V_{q}L^{2}}{m\omega _{0}^{2}}}\sum _{i,j}{q_{i}q_{j}2\int {\frac {d^{2}k}{(2\pi )^{2}}}k_{j}{\frac {\partial f_{k}}{\partial k_{i}}}}\\&=1-{\frac {V_{q}L^{2}}{m\omega _{0}^{2}}}\sum _{i,j}{q_{i}q_{j}2\int {\frac {d^{2}k}{(2\pi )^{2}}}k_{k}{\frac {\partial f_{j}}{\partial k_{i}}}}\\&=1-{\frac {V_{q}L^{2}}{m\omega _{0}^{2}}}\sum _{i,j}{q_{i}q_{j}n\delta _{ij}}\\&=1-{\frac {2\pi e^{2}}{\epsilon qL^{2}}}{\frac {L^{2}}{m\omega _{0}^{2}}}q^{2}n\\&=1-{\frac {\omega _{pl}^{2}(q)}{\omega _{0}^{2}}},\end{alignedat}}}
ここで E k = ħ ω k {\displaystyle E_{k}=\hbar \omega _{k}} 、 V q = 4 π e 2 ε q 2 L 3 {\displaystyle V_{q}={\frac {4\pi e^{2}}{\epsilon q^{2}L^{3}}}} 、 ω p l 2 = 4 π e 2 N ε L 3 m {\displaystyle \omega _{pl}^{2}={\frac {4\pi e^{2}N}{\epsilon L^{3}m}}} を用いた。
(SI単位では因子 4 π {\displaystyle 4\pi } を 1 / ε 0 {\displaystyle 1/\epsilon _{0}} に置き換わる)
この結果は古典的な誘電関数と同様である。