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???????????? ???????? ???????? ??????????? ???????? : ???????? ???????? ????????? ????????
???????? ???????????????? ????????????   · ???????   · ???????????   · ?????????   · ???????   · ???????????   · ???   · ???????   · ???????? ??????   · ???????   · ????????? ??????   · ???????   · ??????? ???????   · ????????? ???????   · ????????   · ???????????
??????????? ??????? ???????   · ?????? ???????   · ?????? ??????   · ??????? ?????   · ??? ?'???????   · ????? ???????   · ???? ????? ??????   · ??????? ????????   · ?????? ???????
? ? ?

???????? ?????? (??????????? ??????, ??????????? ??????, ?????????? ??????, ??????? ?????? ??????, ??????? ?????? ^{[1] [2]} ), ????????? ??????????, ??????????? ? ?????? ??????? ????? ????? ??????? ??? ???????? , ??? ? ???????????? ^{[3] [4]} ?

???? ? ????, ?? ??????? ?????? ????????? ? ???? ???????? ?? ????? ????????? ?????? ??????? ??????? ??????, ???????? ?????? ???????? ??????? ??????? ??????? ?????????? ??? ?????? ????????? ?????????? ???? ???????? ??????? ???? ???????? ??????? ???????? ??????? ???????? ???? ???? ??????? ??????? ???????? ?????? ???? ?????? ? ??????? ???? ???? ????????? ?????, ???????? ??? ???? ?????? ?????????? ??? ??????????? ????????????? ????

???????? ??????? ???? ???????? ??????????? ?, ??? ??????? ????????? ?????????????

??? ????????? ???????? ??????? ????????? ?????????? ??

???????? ??????? ??????? ???????????? [ ???????? | ???????? ???? ]

????????? [ ???????? | ???????? ???? ]

????????? ???? ${\displaystyle \mathbf {L} }$ ???????? ??????? ??????? ?????????? ?????????? ???????? ?????????? ? ????? ??? ???????-??????? ? ???????? ?????????? ????????? .

${\displaystyle ~\mathbf {L} =\mathbf {r} \times \mathbf {p} ,}$

????? ${\displaystyle ~\mathbf {r} }$-? ??????? ???????-??????? ? ????? ?????????? ??????????? ??????? ?????? ?????????? ????????, ${\displaystyle ~\mathbf {p} }$-?? ??????? ?????????

?? ???? ?????????? ????? ???????? ??????? ???????? ? ????? ???? ????????? (??????????) ??????.

${\displaystyle ~\mathbf {L} =\sum _{i}\mathbf {r} _{i}\times \mathbf {p} _{i},}$

????? ${\displaystyle ~\mathbf {r} _{i},\mathbf {p} _{i}}$-? ????????? ????????????? ??????? ???????? ? ???????-??????? ???

?????????? ??????? ?????????? ???? ????? ? ?????? ?????, ?????? ???????? ???????? ????????? ???? ?????? ??????? ??? ????????? ??????? ????????? ????? ??? ?????? ? ???? ????? ${\displaystyle ~\mathbf {L} =\int \mathbf {r} \times \mathbf {dp} ,}$ ????? ${\displaystyle \mathbf {dp} }$-? ????????? ?????? ???? ??????? ????? ???????? ??

?????????? ?????????? ??????????? ???????? ??????? ??????? ? ???? ????????? ???????????

???????? ??????? ??????????? ????? ? ??? ?????????????? . ?????? ?????????? ?????????, ?????? ?? ?? ???? ???????????????? ???????? ??????????? ????? ???? ????

${\displaystyle \mathbf {L} _{\Sigma }=\sum \limits _{i}\mathbf {L} _{i}}$

????????????? ???????????? ???????? ??????? ?????? ? ?????? ???????? ?????????? ?????????? ????????, ?????? ?????? ????? ???????????? ?( ????????????? ????? ??? ??????? ?? ???? ?????? ??????????? ???????? ??? ?????? ???????? ???? ???????

???????? [ ???????? | ???????? ???? ]

???? ?? ???????? ??????? ???????? ? ?????????? ???????????, ??? ${\displaystyle ~\mathbf {r} }$ ? ${\displaystyle ~\mathbf {p} }$ ??????????? ?????????? ??????????? ?? ?????? ??????? ??????? ?????? ??????? ??????? ????? ?????? ? ?? ?? ????????? ??????? ???????? ????? ???????????, ???? ??? ?????????? ?????? ??????? ??? ????? ??????, ??? ????? ?????? ? ?????? ?????????????? ??? ??????? ? ?????????? ??????????? ?????? ??????, ??? ??? ??????????? ??????? ??????????? ???????? ????? ????? ? ????? ??? ?? ???? ????????? ????? ?????????? ?????????? ?????????? ??????? ????????? ?????????? ????????????.

${\displaystyle L=|\mathbf {r} ||\mathbf {p} |\sin \theta _{r,\;p},}$

????? ${\displaystyle ~\theta _{r,\;p}}$-? ${\displaystyle ~\mathbf {r} }$-? ? ${\displaystyle ~\mathbf {p} }$-? ?????? ???????? ?, ??? ???????? ? ??????, ?? ??????? ${\displaystyle ~\mathbf {r} }$-?? ???? ${\displaystyle ~\mathbf {p} }$ ???? ?????? ?????? ??????? ?????? ?????? ?????? ???????? ??????? ?????? ????????? ??????? ???????????? ?????? ???????????? ?????? ? ???????? ???????, ???? ?? ?????? ? ???????? ??????????? ??????

${\displaystyle ~\mathbf {r} }$ ????? ${\displaystyle ~\mathbf {r} =\mathbf {r_{\parallel }} +\mathbf {r_{\perp }} }$ ??????, ????? ${\displaystyle ~\mathbf {r_{\parallel }} }$-? ???????? ???????-??????? ?, ??? ???????? ? ???????? ????????, ??? ${\displaystyle ~\mathbf {r_{\perp }} }$-?? ??????????? ?? ${\displaystyle ~\mathbf {r_{\perp }} }$-? ??? ??????? ?????? ??????? ??????????????? ? ${\displaystyle ~\mathbf {p} }$ ????????, ??? ????????? ??????? ? ≪??????≫? ???? ?????? ?????? ? ???????? ??????? ??????? ????? ????????????. ???????-???????? ???????? ${\displaystyle ~\mathbf {p_{\parallel }} }$ ? ?????????? ${\displaystyle ~\mathbf {p_{\perp }} }$? ????, ?????????? ?????????? ?????????????, ?????? ??? ??? ????????????, ??? ??? ???????? ?????????? ?????????? ??????? ? ?????, ?????? ? ?????? ?? ????? ?????????????????? ${\displaystyle ~L}$-? ?????.

${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =(\mathbf {r_{\perp }} +\mathbf {r_{\parallel }} )\times \mathbf {p} =\mathbf {r_{\perp }} \times \mathbf {p} +\mathbf {r_{\parallel }} \times \mathbf {p} =\mathbf {r_{\perp }} \times \mathbf {p} .}$

${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {r} \times (\mathbf {p_{\perp }} +\mathbf {p_{\parallel }} )=\mathbf {r} \times \mathbf {p_{\perp }} .}$

??????????? ??????? ?????????? [ ???????? | ???????? ???? ]

??????????? ??????? (???????? ???????) ????????? ??????. ??? ????????? ????? ???????? ?????????? ?????????? ??????? ???????? ?????? ???? ???????? (??? ?????????? ??????? ???????? ?????? ??????? ????????) ???????? ????????? ????? ? ??????????

???????? ??????? ????????? ??? ???????? ???? ??????? ?.

${\displaystyle \tau ={\frac {d\mathbf {L} }{dt}}={\frac {d\mathbf {r} }{dt}}\times \mathbf {p} +\mathbf {r} \times {\frac {d\mathbf {p} }{dt}}=\mathbf {r} \times \mathbf {F} }$?

????????, ????????? ??? ??????? ??????? ?????? ? ????????? ????? ??????? ?????? ??????? (??????????) ??????? ?????? ??????? ??????? ??????.

${\displaystyle \mathbf {L} _{\mathrm {system} }=\mathrm {constant} \leftrightarrow \sum \tau _{\mathrm {ext} }=0,}$

????? ${\displaystyle ~\tau _{\rm {ext}}}$-? ?????????? ?????????? ???????? ??????? ???? ?? ??? ??????? ???? ???? ??? ??? ???????, ??? ??????? ?????? ?????????? ???

???????? ??????? ????????? ?????? ????????????? ????? ? ??????????? ???????????????? , ???????? ??????????? ?????????????????? ????????? ????????? ?????? ??????? ????????? ?????? ???? ${\displaystyle ~\delta \varphi }$ ????????? ??????? ${\displaystyle ~i}$ ???????-??????? ??????? ? ${\displaystyle ~\delta \mathbf {r} _{i}=\delta \varphi \times \mathbf {r} _{i}}$-??, ??? ????????????? ${\displaystyle ~\delta \mathbf {v} _{i}=\delta \varphi \times \mathbf {v} _{i}}$? ????????? ${\displaystyle ~{\mathcal {L}}}$ ???????? ????????? ??????????? ????????????? ????????? ??????? ??????? ??????? ?? ???????? ??? ????????

${\displaystyle \delta {\mathcal {L}}={\mathcal {L}}(\mathbf {r} _{i}+\delta \mathbf {r} _{i},\;\mathbf {v} _{i}+\delta \mathbf {v} _{i})-{\mathcal {L}}(\mathbf {r} _{i},\;\mathbf {v} _{i})=\sum \limits _{i}\left({\frac {\partial {\mathcal {L}}}{\partial \mathbf {r} _{i}}}\delta \varphi \times \mathbf {r} _{i}+{\frac {\partial {\mathcal {L}}}{\partial \mathbf {v} _{i}}}\delta \varphi \times \mathbf {v} _{i}\right)=0}$?

????? ??????? ${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\mathbf {v} _{i}}}}=\mathbf {p_{i}} ,\;{\frac {\partial {\mathcal {L}}}{\partial \mathbf {r_{i}} }}=\mathbf {{\dot {p}}_{i}} }$, ????? ${\displaystyle ~\mathbf {p} _{i}}$-? ${\displaystyle ~i}$-?? ??????? ???????????? ???????? ?, ?????? ?????????????? ??????? ????????????? ???????? ?????? ? ????????

${\displaystyle {\dot {\mathbf {p} _{i}}}\,\delta \varphi \times \mathbf {r} _{i}+\mathbf {p} _{i}\,\delta \varphi \times \mathbf {{\dot {r}}_{i}} }$

??????? ????, ???????? ???? ?????????? ?????????????, ???????? ?????????? ?????? ???????????????, ???? ???????????, ????? ??????? ????????? ????????????, ????????.

${\displaystyle \delta {\mathcal {L}}=\delta \varphi \sum \limits _{i}\left(\mathbf {r} _{i}\times {\dot {\mathbf {p} _{i}}}+{\dot {\mathbf {r} _{i}}}\times \mathbf {p} _{i}\right)=\delta \varphi {\frac {d}{dt}}\sum \limits _{i}(\mathbf {r} _{i}\times \mathbf {p} _{i})=\delta \varphi {\frac {d\mathbf {L} }{dt}}=0,}$

????? ${\displaystyle \mathbf {L} =\sum \mathbf {L} _{i}=\sum \mathbf {r} _{i}\times \mathbf {p} _{i}}$-? ????????? ???????? ??????? ?? ???? ?? ${\displaystyle \delta \varphi }$-? ????????? ?, ${\displaystyle \delta {\mathcal {L}}=0}$ ????????????????? ??????? ? ${\displaystyle ~{\frac {d\mathbf {L} }{dt}}=0}$?

??????? ??? ???????? ??????? ???????? ? ???????? ??????? ??????? ???????? ??????? ? ?????????? ??????? ???????? ??????? ????.

${\displaystyle \mathbf {L} _{\mathrm {total} }=\mathbf {L} _{\mathrm {spin} }+\mathbf {L} _{\mathrm {orbit} }}$?

???????? ??????? ??????????????????? [ ???????? | ???????? ???? ]

????????????????? ??????? ??????????? ??????? ??????? ??????? ${\displaystyle ~p}$ ??????? ???????? ????????? ??? ????? ???????, ${\displaystyle ~\mathbf {L} =\mathbf {r} \times \mathbf {p} }$ ???????? ??????? ??????? ???????? ????????? ??? ???????? ?????? ????????, ??? ??? ??????? ? ≪??????? ???????≫.

${\displaystyle ~\mathbf {p} -{\frac {e\mathbf {A} }{c}},}$

????? ${\displaystyle ~e}$-? ?????????? ????? ?, ${\displaystyle ~c}$-?? ?????? ???????????? , ${\displaystyle ~A}$-?? ????????????????? ????? ?????????? ?????????? ? ????????, ${\displaystyle m}$ ?????????, ??????????? ??????? ???????????? ????????????????? ????????

${\displaystyle H={\frac {1}{2m}}\left(\mathbf {p} -{\frac {e\mathbf {A} }{c}}\right)^{2}+e\varphi ,}$

????? ${\displaystyle ~\varphi }$-? ????????????????? ????? ??????? ?????????? ?? ????????? ???????? ??????? ??? ≪??????? ???????? ???????≫ ???????? ?

${\displaystyle K=\mathbf {r} \times \left(\mathbf {p} -{\frac {e\mathbf {A} }{c}}\right)}$?

???????? ??????? ????????? ???????????? [ ???????? | ???????? ???? ]

??????? ???????? [ ???????? | ???????? ???? ]

???????? ??????? ????????? ???????????? ??????????? ?, ???????? ??????? ? ????? ?????? ??????? ????????? ????????? ?????????????? ?????????? ???????? ??????? ?????????? ???????? ??????? ??? ???? ? ???? ?????? ???? ??????????? ${\displaystyle \hbar }$ ( ${\displaystyle h}$ ??????) ???????????, ??? ?????? ?????????? ?? ??????? ${\displaystyle 2\pi }$-?? ??????? ????? ?? ?????, ?? ?????????? ??? ???? ???? ????????? ?????? ???????? ??????, ??? ?????? ?? ??????????? ??? ????? ?????????? ??? ???????? ???????? ??????? ???? ?????????? ? ${\displaystyle \hbar /2}$-??? ??????? ?????????? ???????? ??????? ?????? ? ??? ??????????????? ${\displaystyle ~r_{x}}$, ${\displaystyle ~r_{y}}$, ${\displaystyle ~r_{z}}$, ${\displaystyle ~p_{x}}$, ${\displaystyle ~p_{y}}$, и ${\displaystyle ~p_{z}}$? ??????????? ????????????????? ????????? ?????????? ??? ??????????? ????????????????? ??????????? ??????? ??, ?? ???????? ?? ?????????? ???????? ?????????? ????? ????? ??? ?????????????? ??? ???????? ????????????? ??, ?? ??? ????? ??? ???????? ??????? ??? ?????????? ????

????????????? ???? ???????? ??????? ????????? ???????????? ???????? ? ????? ????? ??????? ?????????? ? ???????? ?????????? ??????? ???????? ? ?????? ????????? ?????? ? ???????? ?????????? ????????? ???????????? ????

${\displaystyle {\hat {\mathbf {L} }}={\hat {\mathbf {r} }}\times {\hat {\mathbf {p} }},}$

????? ${\displaystyle {\hat {\mathbf {r} }}}$ и ${\displaystyle {\hat {\mathbf {p} }}}$-? ???????????????? ?????????? ? ???????? ???????????? ??, ??? ????????? ????????? ???? ????????????, ???????????? ? ???? ???????? ??? ??????? ????? ??????????? ??????? ????????? ?????? ? ???? ?????

${\displaystyle {\hat {\mathbf {L} }}=-i\hbar (\mathbf {r} \times \nabla ),}$

????? ${\displaystyle \nabla }$-? ????? ????????? ?? ?? ???????? ??????? ????? ???????? ????????? ?, ??? ???? ??????? ?????????????

${\displaystyle [L_{i},\;L_{j}]=i\hbar \varepsilon _{ijk}L_{k},\quad \left[L_{i},\;\mathbf {L} ^{2}\right]=0}$, ????? ${\displaystyle \varepsilon _{ijk}}$-? ??????????? ??????? ??

????? ?????? ??????????? ? ?????????????????? ???????????? ? ???? ???????? ??????? ???????????? ????

${\displaystyle \left[L_{i},\;H\right]=0}$

?????? ?????????????? [ ???????? | ???????? ???? ]

???????? ??????? ???????????? ????????? ????????? ?? ?????? (??????) ?????????????? ????????? ??????? ???????? ??????? ???????? ????????? ???????????

${\displaystyle -{\frac {1}{\hbar ^{2}}}\mathbf {L} ^{2}={\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial }{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}}$

??? ????????? ??????? ????????? ???????? ???

${\displaystyle L^{2}\mid l,\;m\rangle ={\hbar }^{2}l(l+1)\mid l,\;m\rangle }$

${\displaystyle L_{z}\mid l,\;m\rangle =\hbar m\mid l,\;m\rangle ,}$

?????

${\displaystyle \langle \theta ,\;\varphi \mid l,\;m\rangle =Y_{l,\;m}(\theta ,\;\varphi )}$

?????? ??????????? ???

???????? ??????? ???????? ?? ??????????????? ???????????? [ ???????? | ???????? ???? ]

${\displaystyle ~\mathbf {v} }$ ??????????? ???????, ${\displaystyle ~\mathbf {r} }$ ???????-???????? ?????????? ?????? ?????? ${\displaystyle ~m}$ ????????? ??????? ????? ???????? ??????? ?????? ? ?????? ???

${\displaystyle ~\mathbf {L} =\mathbf {r} \times m\mathbf {v} }$

???????, ????? ${\displaystyle \times }$-? ?????????? ?????????? ????? ??

?????? ???????? ??????? ???????? ????? ??? ???? ? ??????? ?????? ???? ???????? ? ?????????? ???????? ????? ??????????? ????? ????????? ?????? ???????? ?????????? ??????, ???????? ????????

${\displaystyle \mathbf {L} =\int \limits _{V}{\mathbf {dL} }=\int \limits _{V}{\mathbf {r} \times \mathbf {v} \,dm}}$?

?? ?????? ? ???? ??? ${\displaystyle \rho }$ ?????????

${\displaystyle \mathbf {L} =\int \limits _{V}{\mathbf {r} \times \mathbf {v} \rho dV}}$?

??? ?????????, ?? ${\displaystyle \rho (x,y,z)}$ ???????????? ???????? ?, ??? ?????? ??????? ???????? ? ?????? ????????, ?????? ??? ??????? ????????? ??????????? ??????

????????????? ??????????? ???? ???? ????? (???, ????????? ???????, ????????? ??????? ?????????? ?????) ????? ????????????? (????? ???????? ???? ??????) ??????? ??????? ??????? ??????????? ????? ???? ?

${\displaystyle ~\mathbf {L} =I{\boldsymbol {\omega }},}$

????????????, ????? ${\displaystyle ~I}$-? ????????? ??????? ? ?????? ??????? ??????, ${\displaystyle ~{\boldsymbol {\omega }}}$-?? ??????????? ?????????? ????????

????????? ??????? ???????? ??????? ??????? ??????????? ?????????? ??????? ??? ?????? ? ????????? ??????? ?????? ?????????? (????????? ????????).

${\displaystyle \mathbf {L} ={\hat {I}}{\boldsymbol {\omega }}}$

????????? ??????? ??? ????????? ??????? ?????????? ??????? ????? ????????? ?????? ? ?????? ???????? ??? ??? ??????, ?????? ????????? ???????????? ????? ???????? ? ??????????? ?????????

??? ??? [ ???????? | ???????? ???? ]

• ????????? ??????
• ???? ??????

?????????????????? [ ???????? | ???????? ???? ]

????????????? [ ???????? | ???????? ???? ]

• Биденхарн Л., Лаук Дж., Угловой момент в квантовой физике. Теория и приложения, том 1, М., Мир, 1984.
• Блохинцев Д. И., Основы квантовой механики, М., Наука, 1976.