クラマース・ハイゼンベルグの分散式
H ^ ′ = H ^ 1 ′ + H ^ 2 ′ H ^ 1 ′ = − ∑ j e j 2 m j ∑ k , ν 2 π ħ ω k V ( a ^ k ν + a ^ − k ν † ) { p j ⋅ ε ν ( k ) e i k ⋅ r j + e i k ⋅ r j ε ν ( k ) ⋅ p j } H ^ 2 ′ = ∑ j e j m j ∑ k , ν ∑ k ′ , ν ′ π ħ ω k ω k ′ V 2 ε ν ( k ) ⋅ ε ν ′ ( k ′ ) ( a ^ k ν + a ^ − k ν † ) ( a ^ k ′ ν ′ + a ^ − k ′ ν ′ † ) e i ( k + k ′ ) ⋅ r j {\displaystyle {\begin{aligned}{\hat {H}}'&={\hat {H}}'_{1}+{\hat {H}}'_{2}\\{\hat {H}}'_{1}&=-\sum _{j}{\frac {e_{j}}{2m_{j}}}\sum _{k,\nu }{\sqrt {\frac {2\pi \hbar }{\omega _{k}V}}}({\hat {a}}_{k\nu }+{\hat {a}}_{-k\nu }^{\dagger })\{p_{j}\cdot \varepsilon _{\nu }(k)e^{ik\cdot r_{j}}+e^{ik\cdot r_{j}}\varepsilon _{\nu }(k)\cdot p_{j}\}\\{\hat {H}}'_{2}&=\sum _{j}{\frac {e_{j}}{m_{j}}}\sum _{k,\nu }\sum _{k',\nu '}{\frac {\pi \hbar }{\sqrt {\omega _{k}\omega _{k'}V^{2}}}}\varepsilon _{\nu }(k)\cdot \varepsilon _{\nu '}(k')({\hat {a}}_{k\nu }+{\hat {a}}_{-k\nu }^{\dagger })({\hat {a}}_{k'\nu '}+{\hat {a}}_{-k'\nu '}^{\dagger })e^{i(k+k')\cdot r_{j}}\end{aligned}}}