Сферичні спінори

Повернутися до розділу "Спін 1/2".

Розділ доповнюється.

Оператори квадрату моменту імпульсу, просторової інверсії та проекції моменту на вісь $$\ Oz$$ комутують із оператором Гамільтону. Дійсно, перше та третє було показано у розділі про повний момент імпульсу, а щодо другого, то

$$\ [\hat {P}, \hat {H}]\Psi (\mathbf r, t) = \hat {P}((\alpha \cdot \hat {\mathbf p}) + \gamma_{0} m)\Psi(\mathbf r , t) - i\hat {H}\hat {\gamma}_{0}\Psi (-\mathbf r , t) = i\gamma_{0}(-(\alpha \cdot \hat {\mathbf p}) + \gamma_{0} m)\Psi(-\mathbf r , t) - i((\alpha \hat {\mathbf p}) + \gamma_{0}m)\hat {\gamma}_{0}\Psi (-\mathbf r , t) = |[\hat {\alpha}, \gamma_{0}]_{+} = 0| = $$

$$\ = i\gamma_{0}(-(\alpha \cdot \hat {\mathbf p}) + \gamma_{0} m)\Psi(-\mathbf r, t) - i\gamma_{0}(-(\alpha \hat {\mathbf p}) + \gamma_{0}m)\hat {\gamma}_{0}\Psi (-\mathbf r , t) = 0$$.

$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$$$\ $$