ブロッホの定理
T a ^ { H ^ ( x ) ψ ( x ) } = H ^ ( x + a ) ψ ( x + a ) = H ^ ( x ) ψ ( x + a ) = H ^ ( x ) { T a ^ ψ ( x ) } {\displaystyle {\begin{aligned}{\hat {T_{a}}}\left\{{\hat {H}}(x)\psi (x)\right\}&={\hat {H}}(x+a)\psi (x+a)\\&={\hat {H}}(x)\psi (x+a)\\&={\hat {H}}(x)\left\{{\hat {T_{a}}}\psi (x)\right\}\end{aligned}}}
∴ T a ^ H ^ = H ^ T a ^ {\displaystyle \therefore {\hat {T_{a}}}{\hat {H}}={\hat {H}}{\hat {T_{a}}}}
すなわち T a ^ {\displaystyle {\hat {T_{a}}}} と H ^ {\displaystyle {\hat {H}}} は互いに交換し、同時固有関数を持つ。
H ^ ψ ( x ) = E ψ ( x ) {\displaystyle {\hat {H}}\psi (x)=E\psi (x)}
T a ^ ψ ( x ) = C a ψ ( x ) {\displaystyle {\hat {T_{a}}}\psi (x)=C_{a}\psi (x)} ― (1)