ラムシフト
すべての k → {\displaystyle {\vec {k}}} について和をとると
⟨ ( δ r → ) 2 ⟩ v a c = ∑ k → ( e m c 2 k 2 ) 2 ⟨ 0 | ( E k → ) 2 | 0 ⟩ = ∑ k → ( e m c 2 k 2 ) 2 ( ħ c k 2 ε 0 Ω ) = 2 Ω ( 2 π ) 3 4 π ∫ d k k 2 ( e m c 2 k 2 ) 2 ( ħ c k 2 ε 0 Ω ) since continuity of k → implies ∑ k → → 2 Ω ( 2 π ) 3 ∫ d 3 k = 1 2 ε 0 π 2 ( e 2 ħ c ) ( ħ m c ) 2 ∫ d k k {\displaystyle {\begin{aligned}\langle (\delta {\vec {r}})^{2}\rangle _{\mathrm {vac} }&=\sum _{\vec {k}}\left({\frac {e}{mc^{2}k^{2}}}\right)^{2}\left\langle 0\left|(E_{\vec {k}})^{2}\right|0\right\rangle \\&=\sum _{\vec {k}}\left({\frac {e}{mc^{2}k^{2}}}\right)^{2}\left({\frac {\hbar ck}{2\epsilon _{0}\Omega }}\right)\\&=2{\frac {\Omega }{(2\pi )^{3}}}4\pi \int dkk^{2}\left({\frac {e}{mc^{2}k^{2}}}\right)^{2}\left({\frac {\hbar ck}{2\epsilon _{0}\Omega }}\right)&&{\text{since continuity of }}{\vec {k}}{\text{ implies }}\sum _{\vec {k}}\to 2{\frac {\Omega }{(2\pi )^{3}}}\int d^{3}k\\&={\frac {1}{2\epsilon _{0}\pi ^{2}}}\left({\frac {e^{2}}{\hbar c}}\right)\left({\frac {\hbar }{mc}}\right)^{2}\int {\frac {dk}{k}}\end{aligned}}}