$\displaystyle\int\frac{1}{2}ε_0\hat{E}(r)\hat{E}(r)\ d^3r$

　　　　$=\displaystyle\int\frac{1}{2}ε_0\ i\biggl(\frac{1}{2π}\biggr)^{\frac{3}{2}}\biggl[\displaystyle\int\displaystyle\sum^2_{σ=1}\sqrt{\frac{ℏω_k}{2ε_0}}ｅ_{kσ}(\hat{a}_{kσ}e^{ik･r}-\hat{a}^\dagger_{kσ}e^{-ik･r})\ d^3k\biggr] \ i\biggl(\frac{1}{2π}\biggr)^{\frac{3}{2}}\biggl[\displaystyle\int\displaystyle\sum^2_{σ^´=1}\sqrt{\frac{ℏω_{k^´}}{2ε_0}}ｅ_{k^´σ^´}(\hat{a}_{k^´σ^´}e^{ik^´･r}-\hat{a}^\dagger_{k^´σ^´}e^{-ik^´･r})\ d^3k^´\biggr]\ d^3r$

　　　　$=\displaystyle\frac{1}{2}\biggl(\frac{1}{2π}\biggr)^3\displaystyle\int\int\int\displaystyle\sum^2_{σ=1}\displaystyle\sum^2_{σ^´=1}\frac{ℏ}{2}\sqrt{ω_kω_k^´}ｅ_{kσ}･ｅ_{k^´σ^´} \ \biggl[-\hat{a}_{kσ}\hat{a}_{k^´σ^´}e^{i(k+k^´)･r}+\hat{a}_{kσ}\hat{a}^\dagger_{k^´σ^´}e^{i(k-k^´)･r} +\hat{a}^\dagger_{kσ}\hat{a}_{k^´σ^´}e^{-i(k-k^´)･r}-\hat{a}^\dagger_{kσ}\hat{a}^\dagger_{k^´σ^´}e^{-i(k+k^´)･r}\biggr] d^3k\ d^3k^´\ d^3r$

　　　　$=\displaystyle\frac{1}{2}\displaystyle\int\int\displaystyle\sum^2_{σ=1}\displaystyle\sum^2_{σ^´=1}\frac{ℏ}{2}\sqrt{ω_kω_k^´}ｅ_{kσ}･ｅ_{k^´σ^´}$
　　　　　　$\displaystyle \biggl[-\hat{a}_{kσ}\hat{a}_{k^´σ^´}\biggl(\frac{1}{2π}\biggr)^3\int e^{i(k+k^´)･r}d^3r+\hat{a}_{kσ}\hat{a}^\dagger_{k^´σ^´}\biggl(\frac{1}{2π}\biggr)^3\int e^{i(k-k^´)･r}d^3r +\hat{a}^\dagger_{kσ}\hat{a}_{k^´σ^´}\biggl(\frac{1}{2π}\biggr)^3\int e^{-i(k-k^´)･r}d^3r-\hat{a}^\dagger_{kσ}\hat{a}^\dagger_{k^´σ^´}\biggl(\frac{1}{2π}\biggr)^3\int e^{-i(k+k^´)･r}d^3r\biggr] d^3k\ d^3k^´$

　　　　$=\displaystyle\frac{1}{2}\displaystyle\int\int\displaystyle\sum^2_{σ=1}\displaystyle\sum^2_{σ^´=1}\frac{ℏ}{2}\sqrt{ω_kω_k^´}ｅ_{kσ}･ｅ_{k^´σ^´} \displaystyle \biggl[-\hat{a}_{kσ}\hat{a}_{k^´σ^´}δ(k+k^´)+\hat{a}_{kσ}\hat{a}^\dagger_{k^´σ^´}δ(k-k^´) +\hat{a}^\dagger_{kσ}\hat{a}_{k^´σ^´}δ(k-k^´)-\hat{a}^\dagger_{kσ}\hat{a}^\dagger_{k^´σ^´}δ(k+k^´)\biggr] d^3k\ d^3k^´$

　　　　$=\displaystyle\frac{1}{2}\int\sum^2_{σ=1}\frac{ℏω_k}{2} \displaystyle \biggl[\hat{a}_{kσ}\hat{a}_{-kσ}+\hat{a}_{kσ}\hat{a}^\dagger_{kσ} +\hat{a}^\dagger_{kσ}\hat{a}_{kσ}+\hat{a}^\dagger_{kσ}\hat{a}^\dagger_{-kσ}\biggr] d^3k$


　　　　$\displaystyle\int\frac{1}{2}ε_0\hat{B}(r)\hat{B}(r)\ d^3r$

　　　　$=\displaystyle\int\frac{1}{2μ_0}\ i\biggl(\frac{1}{2π}\biggr)^{\frac{3}{2}}\biggl[\int\sum^2_{σ=1}\sqrt{\frac{ℏ}{2ε_0ω_k}}k×ｅ_{kσ}(\hat{a}_{kσ}e^{ik･r}-\hat{a}^\dagger_{kσ}e^{-ik･r})\ d^3k\biggr] \ i\biggl(\frac{1}{2π}\biggr)^{\frac{3}{2}}\biggl[\int\sum^2_{σ^´=1}\sqrt{\frac{ℏ}{2ε_0ω_{k^´}}}k^´×ｅ_{k^´σ^´}(\hat{a}_{k^´σ^´}e^{ik^´･r}-\hat{a}^\dagger_{k^´σ^´}e^{-ik^´･r})\ d^3k^´\biggr]\ d^3r$

　　　　$=\displaystyle\frac{1}{2μ_0}\biggl(\frac{1}{2π}\biggr)^3\int\int\int\sum^2_{σ=1}\sum^2_{σ^´=1}\frac{ℏ}{2ε_0}\sqrt{\frac{1}{ω_kω_k^´}}(k×ｅ_{kσ})(k^´×ｅ_{k^´σ^´})$
　　　　　$\biggl[-\hat{a}_{kσ}\hat{a}_{k^´σ^´}e^{i(k+k^´)･r}+\hat{a}_{kσ}\hat{a}^\dagger_{k^´σ^´}e^{i(k-k^´)･r} +\hat{a}^\dagger_{kσ}\hat{a}_{k^´σ^´}e^{-i(k-k^´)･r}-\hat{a}^\dagger_{kσ}\hat{a}^\dagger_{k^´σ^´}e^{-i(k+k^´)･r}\biggr] d^3k\ d^3k^´\ d^3r$

　　　　$=\displaystyle\frac{1}{2μ_0}\int\int\sum^2_{σ=1}\sum^2_{σ^´=1}\frac{ℏ}{2ε_0}\sqrt{\frac{1}{ω_kω_k^´}}(k×ｅ_{kσ})(k^´×ｅ_{k^´σ^´})$
　　　　　$\displaystyle \biggl[-\hat{a}_{kσ}\hat{a}_{k^´σ^´}\biggl(\frac{1}{2π}\biggr)^3\int e^{i(k+k^´)･r}d^3r+\hat{a}_{kσ}\hat{a}^\dagger_{k^´σ^´}\biggl(\frac{1}{2π}\biggr)^3\int e^{i(k-k^´)･r}d^3r +\hat{a}^\dagger_{kσ}\hat{a}_{k^´σ^´}\biggl(\frac{1}{2π}\biggr)^3\int e^{-i(k-k^´)･r}d^3r-\hat{a}^\dagger_{kσ}\hat{a}^\dagger_{k^´σ^´}\biggl(\frac{1}{2π}\biggr)^3\int e^{-i(k+k^´)･r}d^3r\biggr] d^3k\ d^3k^´$

　　　　$=\displaystyle\frac{1}{2}\int\int\sum^2_{σ=1}\sum^2_{σ^´=1}\frac{ℏ}{2ε_0μ_0}\sqrt{\frac{1}{ω_kω_k^´}}[(k･k^´)(ｅ_{kσ}･ｅ_{k^´σ^´})-(k･ｅ_{k^´σ^´})(k^´･ｅ_{kσ})]$
　　　　　$\displaystyle \biggl[-\hat{a}_{kσ}\hat{a}_{k^´σ^´}δ(k+k^´)+\hat{a}_{kσ}\hat{a}^\dagger_{k^´σ^´}δ(k-k^´) +\hat{a}^\dagger_{kσ}\hat{a}_{k^´σ^´}δ(k-k^´)-\hat{a}^\dagger_{kσ}\hat{a}^\dagger_{k^´σ^´}δ(k+k^´)\biggr] d^3k\ d^3k^´$

　　　　$=\displaystyle\frac{1}{2}\int\sum^2_{σ=1}\frac{ℏ}{2ε_0μ_0}\frac{1}{ω_k}k^2 \displaystyle \biggl[-\hat{a}_{kσ}\hat{a}_{-kσ}+\hat{a}_{kσ}\hat{a}^\dagger_{kσ} +\hat{a}^\dagger_{kσ}\hat{a}_{kσ}-\hat{a}^\dagger_{kσ}\hat{a}^\dagger_{-kσ}\biggr] d^3k$

　　　　$=\displaystyle\frac{1}{2}\int\sum^2_{σ=1}\frac{ℏω_k}{2} \displaystyle \biggl[-\hat{a}_{kσ}\hat{a}_{-kσ}+\hat{a}_{kσ}\hat{a}^\dagger_{kσ} +\hat{a}^\dagger_{kσ}\hat{a}_{kσ}-\hat{a}^\dagger_{kσ}\hat{a}^\dagger_{-kσ}\biggr] d^3k$


　　　　※$(A×B)(C×D)=(A･C)(B･D)-A･D)(C･B)、c^2=\frac{1}{ε_0μ_0}、ω^2=c^2k^2$