第二量子化
F ^ 2 = 1 2 ∫ ρ ^ ( r 1 ) f ^ 2 ( r 1 , r 2 ) ρ ^ ( r 2 ) d r 1 d r 2 = 1 2 ∫ ψ ^ † ( r 1 ) ψ ^ † ( r 2 ) f ^ 2 ( r 1 , r 2 ) ψ ^ ( r 2 ) ψ ^ ( r 1 ) d r 1 d r 2 = 1 2 ∑ i , j , k , m [ ∫ ψ i ∗ ( r 1 ) ψ j ∗ ( r 2 ) f ^ 2 ( r ) ψ m ( r 2 ) ψ k ( r 1 ) d r ] a ^ i † a ^ j † a ^ m a ^ k {\displaystyle {\begin{aligned}{\hat {F}}_{2}={\frac {1}{2}}\int {\hat {\rho }}({\boldsymbol {r}}_{1}){\hat {f}}_{2}({\boldsymbol {r}}_{1},{\boldsymbol {r}}_{2}){\hat {\rho }}({\boldsymbol {r}}_{2})d{\boldsymbol {r}}_{1}d{\boldsymbol {r}}_{2}&={\frac {1}{2}}\int {\hat {\psi }}^{\dagger }({\boldsymbol {r}}_{1}){\hat {\psi }}^{\dagger }({\boldsymbol {r}}_{2}){\hat {f}}_{2}({\boldsymbol {r}}_{1},{\boldsymbol {r}}_{2}){\hat {\psi }}({\boldsymbol {r}}_{2}){\hat {\psi }}({\boldsymbol {r}}_{1})d{\boldsymbol {r}}_{1}d{\boldsymbol {r}}_{2}\\&={\frac {1}{2}}\sum _{i,j,k,m}{\bigg [}\int \psi _{i}^{*}({\boldsymbol {r}}_{1})\psi _{j}^{*}({\boldsymbol {r}}_{2}){\hat {f}}_{2}({\boldsymbol {r}})\psi _{m}({\boldsymbol {r}}_{2})\psi _{k}({\boldsymbol {r}}_{1})d{\boldsymbol {r}}{\bigg ]}{\hat {a}}_{i}^{\dagger }{\hat {a}}_{j}^{\dagger }{\hat {a}}_{m}{\hat {a}}_{k}\end{aligned}}}
この2体の相互作用は粒子 k , m {\displaystyle k,m} が衝突して粒子 i , j {\displaystyle i,j} になることを表している。