河南焦作：“三个三”扎实推进村（社区）“两委”换届选举工作

百度 目前A股实行注册制的条件并不成熟，在这种情况下实行注册制，只能是让更多的垃圾公司来到股市里圈钱，进而损害投资者利益。

????? ???? ????(??: Relativistic quantum mechanics)? ?????? ????? ????? ????. ?????? ???? ?? ?? ???? ??? ??. ???, ???? ?? ??? ????? ???? ?? ???? ?? ?????. ? ??? ?? ??? ??? ???? ???? ? ??? ??? ? ??. ?? ??? ?? ??? ??? ? ??. ? ??? ???? ???, ?? ??? ? ??? ????? ??? ?? ???, ?? ? ?? ?? ???? ????. ????? ????? ???? ???? ???? ???? ????? ??? ???? ???, ?? ?????? ???? ??? ???? ???? ?? ??? ???? ?????. ???? ????? ???????? ??? ??????. ???? ?? ? ?????? ??? ?? ?? ???? ?? ????? ???? ??????? ? ? ??(?: ?? ?? ?? ?? ????)? ?? ??? ??? ???? ????.

?? ???? ????? ???? ?? ??? ??? ??: ??? ??, ?? ?? 1/2 ????? ?? ?? ???, ?? ?? ? ?????? ??? ??? ?? ??. ? ??? ?? ??? ??? ??? ???? ???? ?? ?????. ?????, ????? ?? ????? ??? ???? ??? ???? ?? ??? ???? ?? ????? ???? ??.

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

? ???? ???? ??? 3?? ???? ???? ?? ???? ????? ?????? ???? ?? ??? ????(??? ??? ?? ?? ??) ?? ? ?? ?? ??? ?? ?? ? ?? ?? ?? ??? ???? ????(???? ?? ???)., ?? ????? ???? ????. ???? SI ??? ????. ??? ?? ? ?? ??? SI??? ???? ????. ?? ???? ?? ??? ???? ??. ??? ??? ????? ???? ??? ?? ??? ??. ?? ? ??? ??? ??.

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

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

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

$i\hbar {\frac {\partial }{\partial t}}\psi ={\hat {H}}\psi$

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

? ???? ?? ?? ???? ???? ?? t?? ??? 3?? ?? ?? r? ??? ??? ? ?? ?? ψ(r, t)??.

?? ??? ?? ?? ?? ??? $s$? ???. ?? $2s$? ????, ????? ??? ?? ????. ? $s$? $2s+1$?? $z$-?? ??? $\sigma =s,s-1,\dots ,-s+1,-s$? ???. ??? ?? ?? $\psi (\mathbf {r} ,t,\sigma )$? ??? ??? ?? ????.

????? 1920?? ??? ???, ???, ??? ? ???? ???? ??? ??? ????. ????? ??? ????? ??? ?? ??(1925)? ????? ??? ???? ??-?? ??(1939)? ????. ??? ??, ?(??? ????? ?? ?? ? ??)? ?? ???? ?? ?? ? ??? (??? ???? ???? ??)? ??? ??? ??? ??? ?? ? ??? ?? ????:

???? ???-??? ??? ?? ??? ??? ???? ????. ?? $m$? ??, ??? $E$ ? ????? ?? ??? ?? $p={\sqrt {\mathbf {p} \cdot \mathbf {p} }}$? ?? 3- ??? $p$? ?? ?? ????? ???:

$E^{2}=c^{2}\mathbf {p} \cdot \mathbf {p} +(mc^{2})^{2}\,.$

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

${\hat {E}}=i\hbar {\frac {\partial }{\partial t}}\,,\quad {\hat {\mathbf {p} }}=-i\hbar \nabla$

? ?? ???? ?? ???? ???? ????: ???-??? ??? ???? ??? ???? ???? $\psi$? ?? ?? ??? ?? ??? ????. ??? ????? ?? ??? ??? ??? ??? ???? ??? ?? ???? ??? ??? ?? ?????? ??? ?? ?? ??? ??? ??? ??? ? ??? ??. ?? ??? ??? ?? ??? ????. ?? ?? ???? ??? ?? ?? ??? 1???(0? ??? ?? ???? ???? ??).

?????? ??? ????? ? ?? ?????. ? ?? ???? $\psi$? ?? ????? ??? $A(t)$? ?? ???? ?? ???? ?? ???? ????.

${\frac {d}{dt}}A={\frac {1}{i\hbar }}[A,{\hat {H}}]+{\frac {\partial }{\partial t}}A\,,$

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

????? 1926??? ????? ??????? ?? ??? ?? ??? ????? ?? ???? ??? ?? ??? ???? ??? ?? ?????.

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

??? ??

?? ?? ? ????? ?????? ??? ?? ???? ??? ?? ??? ? ?? ????? ??? ???? ????? ????. ??? ????? ?????? ??? ?? ???? ψ(r₁, r₂, r₃, ..., t, σ₁, σ₂, σ₃...)? ????.

???? ???? ?? ??? ?? ??? ????? ??. ?? ????? ???? ? ???? ?? ?? ??? ?? ??? ??? ? ??. ??? ?? ??? ??? ???? 4?? ??? ?? $\mathbf {X} =(ct,\mathbf {r} )$? ????? ????, ???? 3-???? $a$? 4-??? $\mathbf {P} =(E/c,\mathbf {p} )$? ????? ????. ?? ????? ??? ?? ??? ?? ?? ?? ???? ?? ??? ? ??? ?? ????? ??? ? ??? ??? ?? ????. ??? ???, ??? ??? ? 3-??? ???? ??? ??? ?? ???.

????? ???? ??? ??? ??? ?? $(\mathbf {r} ,t)\rightarrow \Lambda (\mathbf {r} ,t)$? ?? ?? ?? ?? ?? ?? $\psi _{\sigma }$? ??? ?? ?? ?? $D$? ?? ????? ????.

$\psi _{\sigma }(\mathbf {r} ,t)\rightarrow D(\Lambda )\psi _{\sigma }(\Lambda ^{-1}(\mathbf {r} ,t))$

??? $D(\Lambda )$? ?? ?? ??, ? (2s + 1)×(2s + 1) ?? ????. ?? ????, $\psi$? $\sigma$? (2s + 1)?? ?? ?? ?? ??? ???? ? ??? ??. ??? s? $\sigma$?? ??? ?? ???? ???? ??? ?? ??? ???? ???? ???. ??? $\sigma$ ?? ??? ?? ? ? ?? ??? ? ??.

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

??? ??? ?? ??? ?? ???? ?????? ?? ??? $\mathbf {p} \cdot \mathbf {p} /2m$ ??? ?? ??? $V(\mathbf {r} ,t)$?? ???? ???? ?? ?? ???? ????.

${\hat {H}}={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} ,t)$

??? ?? ???? ???? ???? ????? ?? ????? ???? ???? ??. ? ??? ??? ??? ??? ???? ???. ???, ???? ?????? ??? ?? ??. ???-??? ???? ???? ???? 2? ????? ???? ????. ??? ??:

${\hat {H}}={\hat {E}}={\sqrt {c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}+(mc^{2})^{2}}}\quad \Rightarrow \quad i\hbar {\frac {\partial }{\partial t}}\psi ={\sqrt {c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}+(mc^{2})^{2}}}\,\psi$

? ?? ?? ??? ??? ?? ???. ???? ???? ?? ??? ??? ? ??. ? ??? ????? ??? ???? $\psi$? ??? ? ?? ?? ???? ????? ??. ???? ??? ?? ? ?? ??? ?? ??? ??? ?????. ?? ????, ?? ???? ?? ???? ????? ??? ???? ???? ??? ??. ? ???? ? ??? ? ?? ?? ????? ?? ??? ?? ? ?? ??? ???? ???? ??? ?? ? ??: ??? ??? $\mathbf {r} _{0}$? ???? ????, $\psi (\mathbf {r} _{0},t=0)$? ???? ??? ????? 0??. ??? , ?? ???? ???? ?? ??? $\psi (\mathbf {r} ,t)\neq 0$? ????? ????. ??? ?? ??? ?? ??? ???? ?? ? ??? ??? ? ??? ???? |r| > ct ? ?? 0? ???. ?? ?? ?? ?? $\psi (|\mathbf {r} |>ct,t)=0$?? ???? ??.

????? ???? ??? ??? ?? ?????? ??? ???? ??? ??. ??? ?? ??? ?? ????? $\mu _{B}$??? ???? ?? ?? ?? ???? ???.

${\hat {\boldsymbol {\mu }}}_{S}=-{\frac {g\mu _{B}}{\hbar }}{\hat {\mathbf {S} }}\,,\quad \left|{\boldsymbol {\mu }}_{S}\right|=-g\mu _{B}\sigma \,,$

??? $g$? ??? (??) g-???? $\mathbf {S}$? ?? ?????? ????? ?? ????. ???? ??? ??? $\mathbf {B}$? ?? ??? ?? ?? ?? ?

${\hat {H}}_{B}=-\mathbf {B} \cdot {\hat {\boldsymbol {\mu }}}_{S}$

? ?? ????? ?????? ????? ??. ???; ???? ?????? ???? ???-??? ??? ???? ?? ?? ???? ??? ???? ????.

???? ???? ??? ?? ??? ????? ????? ????. ???? ?? ??? ?? ??? ?? ? ?? ?? ??? ?? ?? ?? ???? ?? ???? ?? ??? ?? ?? ??? ?? ??. ?? ???? ???? ?????? ?? ??? ?? ??? $\sigma$? ?? ???? ?? ??? ?? ???? ????? ????? ???? ?????? ??? ??.

${\hat {H}}={\hat {H}}(\mathbf {r} ,t,{\hat {\mathbf {p} }},{\hat {\mathbf {S} }})$

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

?? ??? ?? ???-?? ? ?? ???

??? ? ??? ???? ???-??? ??? ?? ???? ?? ???-?? ???? ?? ?? ?? ??? ????? ?? ? ??.

${\hat {E}}^{2}\psi =c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}\psi +(mc^{2})^{2}\psi \,,$

?? 1925? ????? ??? ??? ? ????? ???? ???? ?, ??? ???? ??? ????? ???? ???? ??? 1927?? ????. ??? ?????? ????? ? ??????? ??? ? ?? ??? ???? ???? ?? ??? ??? ?? ???. ? ???? ?? ?? ???? ??? ? ??. ??? ? ???? ??? ?? ???? ?? ??? ? ??.

$\left({\hat {E}}-c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}-\beta mc^{2}\right)\left({\hat {E}}+c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}+\beta mc^{2}\right)\psi =0\,,$

??? ${\boldsymbol {\alpha }}=(\alpha _{1},\alpha _{2},\alpha _{3})$ ? $\beta$? ??? ??? ??? ??? i ≠ j ? ?? ????? ??? 4 × 4 ???? ????.

$\alpha _{i}\beta =-\beta \alpha _{i},\quad \alpha _{i}\alpha _{j}=-\alpha _{j}\alpha _{i}\,$

???.

$\alpha _{i}^{2}=\beta ^{2}=I$

?? 2? ???? ??? ?? ???? 2? ???? ??? ??? ?? ??? ?? ???. ? ?? ??:

$\left({\hat {E}}-c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}-\beta mc^{2}\right)\psi =0\quad \Leftrightarrow \quad {\hat {H}}=c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}+\beta mc^{2}$

? ?? ?????. ?? ??? ?? ?????? ?? ??? ?? ??? ?? ???. ? ??? ?????? ????. ?? ??? ?? ???? ??? ? ??. ??? 1928?? ?? ??? ?? ???? ?????? ??? ??, ???? ?? ?? $E-c{\boldsymbol {\alpha }}\cdot \mathbf {p} -\beta mc^{2}$? ???? ?? ??? ???-?? ???? ${\boldsymbol {\alpha }}$? $\beta$? ?? ?? ??? ???? ?? ????. ?? ?? ???? ???? ?? ?? ?? ??? ? ??. $\psi$? ??? ??? ??? ???-?? ????? ???? ??? ?? ??? ??? ?? 4? ???? ????? ?? ????. ?? ???? ??? ?? ??? ?? ????? ??? ?? ??? ??? ?? ???? ??? ????. ???? ??? ??? ??? ??? ??? ?? ??? ????? ?? ??? ???? ????. ??? ??? ?? ?? ??.

?? ? ??

????? ?????? ???? $\psi$? ??? ?? $\rho =|\psi |^{2}$? ?? ?? ????. ??? 1927??? ???? ????. ???? ?????? $\psi (\mathbf {r} ,t)$? ?? ????? ?? ??? ????? ??????? ???? ??. ?? ???? ?? ???? ?? ?? $\rho$ ?? ?? ?? j ( ?? ?? ??? ??)? ???? ???. ???? ?? ?? $\rho$ ?? ?? ?? ??? ??? ??? ?? ?? ??? ??? ??? ????. ?? ???? ??? ????:

$\rho =\psi ^{\dagger }\psi ,\quad \mathbf {j} =\psi ^{\dagger }\gamma ^{0}{\boldsymbol {\gamma }}\psi \quad \rightleftharpoons \quad J^{\mu }=\psi ^{\dagger }\gamma ^{0}\gamma ^{\mu }\psi$

??? ??? ???? ?? (??? ????? ?? ??? ?? ${\overline {\psi }}=\psi ^{\dagger }\gamma ^{0}$?? ???)? ???? J^μ ? ?? 4-?? ? ?? ???-?? ???? ??? ???? ???.

$\rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}{\frac {\partial \psi }{\partial t}}-\psi {\frac {\partial \psi ^{*}}{\partial t}}\right)\,,\quad \mathbf {j} =-{\frac {i\hbar }{2m}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)\quad \rightleftharpoons \quad J^{\mu }={\frac {i\hbar }{2m}}(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*})$

??? $\partial ^{\mu }$? 4-?????. $\psi$ ? $\partial \psi /\partial t$? ?? ?? ???? ??? ? ???? ??? ??? ? ? ??.

?? ?? ??? "?? ??"? "?? ??"?? ??? ?? ??? ??? ? ?? ??? ?? ??? ?????? ??. ??? ???? $\psi$? ????? ?? ??? ??? ?????. ??? ??? ??? ?? ?? ???? ????.

${\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0\quad \rightleftharpoons \quad \partial _{\mu }J^{\mu }=0\,,$

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

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

???? ?? ???? ?? ??? ????? ?? ????? ???. ?? ??? ??? ?? ??? ???? ??? ????. ??? $\mathbf {B} =\nabla \times \mathbf {A}$ ? ?? ??? ?? $\phi (\mathbf {r} ,t)$? ???? ?? ?? ?? $\mathbf {A} (\mathbf {r} ,t)$ ? ???? ???? ? ?? q? ?? ?? ??? ?? ??? ??? ??.

${\hat {E}}\rightarrow {\hat {E}}-q\phi \,,\quad {\hat {\mathbf {p} }}\rightarrow {\hat {\mathbf {p} }}-q\mathbf {A} \quad \rightleftharpoons \quad {\hat {P}}_{\mu }\rightarrow {\hat {P}}_{\mu }-qA_{\mu }$

??? $P_{\mu }$? ???? 4-??? ???? ?? 4-????? $A_{\mu }$? 4-?????. ????? ??? ?? ??? ??? ????.

$E-e\phi \approx mc^{2}\,,\quad \mathbf {p} \approx m\mathbf {v}$

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

?? 0

???? ?????? ???-?? ???? ?? ?? ??? ????.

${({\hat {E}}-q\phi )}^{2}\psi =c^{2}{({\hat {\mathbf {p} }}-q\mathbf {A} )}^{2}\psi +(mc^{2})^{2}\psi \quad \rightleftharpoons \quad \left[{({\hat {P}}_{\mu }-qA_{\mu })}{({\hat {P}}^{\mu }-qA^{\mu })}-{(mc)}^{2}\right]\psi =0.$

??? 0? ?? ???? ?? ???-?? ????? ???? ????? ????? 0? ?? ??? ????. ??? ??? ?? ?? 1?? ??? (0,0) ?? ??? ??? ?? ??? ?????. ?? ?? ?? (0,0) ??? ??? ???? ?? ????. ?? (0,0) ??? ??? ?? ?? ? ? ??? ?? ??? ???. ??? ?? ????? ?? ??? ????? ?? ??? ??? 0? ?? ??? ??? ? ??. ?? ?? ?? ?1/2?? ?? ?? ???? ?? ?? ?? ??? ???? ??. ??? ?? ???-?? ???? ???? ?? ??? 0? ??? ??? ? ??.

????? ??? ???? ?? ????? ???? ??? ???-?? ???? ?? ????? ????. ? ???? π ???? ?? ?? ?? ???? ???? ?? ?? ??? ?? ? ?? ?? ?? ??? ???? ??? ?? ??? ?? ???. ??? ?? ????? ?? ??? ?? ?? ??? ???? ????.

???-?? ???? ?? ??? ????? ?? ?? ?? ??? ??? ? ??. ??? ??? ?? ?1/2??? ??? ???? ?? ??? ??? ? ??. ????? ???? ???? ?????? ?? ?? ?? ??? ?? ???? ????? ????.

$\left(i\hbar {\frac {\partial }{\partial t}}-q\phi \right)\psi ={\frac {1}{2m}}{({\hat {\mathbf {p} }}-q\mathbf {A} )}^{2}\psi \quad \Leftrightarrow \quad {\hat {H}}={\frac {1}{2m}}{({\hat {\mathbf {p} }}-q\mathbf {A} )}^{2}+q\phi .$

?? ?1/2?

??????? ??? 1927? ???? ????? ?? ??? ?? ?????? ??? ???? 2×?2 ??? ??? ?? ????.

$\left(i\hbar {\frac {\partial }{\partial t}}-q\phi \right)\psi =\left[{\frac {1}{2m}}{({\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} ))}^{2}\right]\psi \quad \Leftrightarrow \quad {\hat {H}}={\frac {1}{2m}}{({\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} ))}^{2}+q\phi$

$\psi$? ????? ???? ?????? ?? ??? ??? ?? ??? ??? 2?? ??? ?

$\psi ={\begin{pmatrix}\psi _{\uparrow }\\\psi _{\downarrow }\end{pmatrix}}$

??. ??? ?? ↑? ↓? "? ??" $\sigma =+1/2$? "?? ??" $\sigma =-1/2$??.

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

$\left(i\hbar {\frac {\partial }{\partial t}}-q\phi \right)\psi =\gamma ^{0}\left[c{\boldsymbol {\gamma }}\cdot {({\hat {\mathbf {p} }}-q\mathbf {A} )}-mc^{2}\right]\psi \quad \rightleftharpoons \quad \left[\gamma ^{\mu }({\hat {P}}_{\mu }-qA_{\mu })-mc^{2}\right]\psi =0$

??? ?? ???? 4 × 4 ?? ?? $\gamma ^{0}=\beta ,{\boldsymbol {\gamma }}=(\gamma _{1},\gamma _{2},\gamma _{3})=\beta {\boldsymbol {\alpha }}=(\beta \alpha _{1},\beta \alpha _{2},\beta \alpha _{3})$ ? ??? ??? ???? ???? ? ?? ??????. ??? ??? ???(??? ??? ? ??)? ?? ??? 4 × 4 ?? ??? ???, ????? ???? ???? ?? ???? ???(?, ?? 1?? ???). ??? $\psi$? 4?? ??? ???, ????? ??? ?? ???? 2?? 2?? ???? ????.

$\psi ={\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}\psi _{+\uparrow }\\\psi _{+\downarrow }\\\psi _{-\uparrow }\\\psi _{-\downarrow }\end{pmatrix}}$

2-??? $\psi _{+}$? 4-??? $(E,\mathbf {p} )$, ?? $q$, 2?? ?? ?? $\sigma =\pm 1/2$? ?? ??? ????. ?? 2-??? $\psi _{-}$? ???? ??? ?? ?? ?? $\sigma =\pm 1/2$? ???? ?? 4-??? $-(E,\mathbf {p} )$, ?? ?? $-q$, ?, ?? ??? ??, ??? ???? ???,??? ??? ?? ??? ????. ?? ??? ???? ????. ? ??? ??? ?? ???? bispinor? ????. ????? ????, ?? ???? ??? ????? ????. 1?? ??? ??? ??? ?? $\mathbf {A} =\mathbf {0}$?, $\phi$? ??? ???? ?? ???? ???? ??? ??-?? ????, ?? ??????? ??, ?? ?? ????. ??? ?????? ? ??? ??? ?? ???? ?? ???? ?????. ?? ?? ????? ????? ???? ????.

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

$\left({\frac {\hat {E}}{c}}+{\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }}\right)\psi _{+}=0\,,\quad \left({\frac {\hat {E}}{c}}-{\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }}\right)\psi _{-}=0\quad \rightleftharpoons \quad \sigma ^{\mu }{\hat {P}}_{\mu }\psi _{+}=0\,,\quad \sigma _{\mu }{\hat {P}}^{\mu }\psi _{-}=0\,,$

? ??? ??? ?? ????? ??? ? ?? ??? ???? ?? ?????. ???? ??? ???? ?? ??? ???? ?? ??? 2 × 2 ?? ??? ??. ???? ?????? ?? ? ??? ??? ???(?? ???)? ???? ??? ??? ???(?? ???)? ???? 0?? ??? ?? σ₀ ?? ??? ??? ?? ????.

??? ??? ?? ??? ?? ???? ??. ?? ?????, ???? ??? ???? ????? ??? ???? SO(2) ? SO(3) ? ?? ????. , ] ? ???? [ , ]₊ ??? ??:

$\left[\sigma _{a},\sigma _{b}\right]=2i\varepsilon _{abc}\sigma _{c}\,,\quad \left[\sigma _{a},\sigma _{b}\right]_{+}=2\delta _{ab}\sigma _{0}$

??? ε_abc? 3?? ??-??? ????. ?? ??? ???? ???? ??? ???? ??? ???? ??? ??? ????? ?? η^αβ? ??? ????.

$\left[\gamma ^{\alpha },\gamma ^{\beta }\right]_{+}=\gamma ^{\alpha }\gamma ^{\beta }+\gamma ^{\beta }\gamma ^{\alpha }=2\eta ^{\alpha \beta }\,,$

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

1929?? ???? ???? ?????? ?? ???? ? ? ??? ??? ?? ?1/2? ????? 1? ???? ???? ???? ??? ????. ?? ???? ?? ??? ?? ????? ? ?? ?? ? ????. ??? ??? ??? ??? ??? ??????? ??? ?? ??? ?? ????.

???? ????

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

${\hat {h}}={\hat {\mathbf {S} }}\cdot {\frac {\hat {\mathbf {p} }}{|\mathbf {p} |}}={\hat {\mathbf {S} }}\cdot {\frac {c{\hat {\mathbf {p} }}}{\sqrt {E^{2}-(m_{0}c^{2})^{2}}}}$

??? p? ??? ???, S? ?? s ??? ?? ?? ???, E? ??? ? ???, m0? ????. ???? ?? ? ?? ??? ??? ??? ????. ???? ??? 3-????? ?? ???? ?? ??? ?? ???? ?? ?????. ?? ???? ?? ??? ?? ?? ?? ?? ?? ??? ??? ?? ?? ?? ???.

?? ???(? ?? ???)? ?? ??? ??? ????. ?1/2?3-???(?? c )? , ???? σ · c p (??? ?? ?1/2? ? ${\sqrt {E^{2}-(m_{0}c^{2})^{2}}}$ .

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

${\hat {h}}={\hat {\mathbf {S} }}\cdot {\frac {c{\hat {\mathbf {p} }}}{E}}$

? ? ??

?? ???? ?? ?1/2? ??? ??? ? ??. ?? ???? ??, ???? ?? ???? ??? ?? ?? ???? ?????. 1936?? ??? ?? ???? ?? ?????? ????, 3? ?? ???? ? ???? ??? ???? ?? ????. ???–??? ???? 1948?? ??? ?? ???? ??????, ??? ?? ?? ?? ??? ??? ? ??. ?? ???-?? ???? ????? ??? ?? ?? ?? ???? ???? ??? ??? ??? ???? ?? ?????.

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

$\psi (\mathbf {r} ,t)={\begin{bmatrix}\psi _{\sigma =s}(\mathbf {r} ,t)\\\psi _{\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{\sigma =-s}(\mathbf {r} ,t)\end{bmatrix}}\quad \rightleftharpoons \quad {\psi (\mathbf {r} ,t)}^{\dagger }={\begin{bmatrix}{\psi _{\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots &{\psi _{\sigma =-s+1}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =-s}(\mathbf {r} ,t)}^{\star }\end{bmatrix}}$

??? ???? ??? ???? ????. ?? s ? ??? ??? ?? ??? ?? $2s+1$ ??? ?? ?? ???? ?? ? ?? $2s+1$ ??? ??(? ??? $2s+1$?? ??? σ?? ??). ?, ?? $2(2s+1)$ -??? ??? ???:

$\psi (\mathbf {r} ,t)={\begin{bmatrix}\psi _{+,\,\sigma =s}(\mathbf {r} ,t)\\\psi _{+,\,\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{+,\,\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{+,\,\sigma =-s}(\mathbf {r} ,t)\\\psi _{-,\,\sigma =s}(\mathbf {r} ,t)\\\psi _{-,\,\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{-,\,\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{-,\,\sigma =-s}(\mathbf {r} ,t)\end{bmatrix}}\quad \rightleftharpoons \quad {\psi (\mathbf {r} ,t)}^{\dagger }{\begin{bmatrix}{\psi _{+,\,\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi _{+,\,\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots &{\psi _{-,\,\sigma =-s}(\mathbf {r} ,t)}^{\star }\end{bmatrix}}$

?? ?? +? ??? ???? ?? ?? - ? ???? ????. ??? ??? ?? ?? $s$ ??? ?? 2?? ??? ?? ????. ??? $+s$? ???? ??? ??? ??? ?? ??? ?? ??? ?? ??? $-s$? ???? ?? ??? ??? ?? ???? ?? ???.

$\psi (\mathbf {r} ,t)={\begin{pmatrix}\psi _{+}(\mathbf {r} ,t)\\\psi _{-}(\mathbf {r} ,t)\end{pmatrix}}$

???? ???-??? ??? ??? ??? ?? ?? ??? ?? ???? ????? ?? ???? ???? ??? 2?? ???? ????. ????? ??? ?? ???? 1927? ?? ???? ?? ????? ???? ???? ?? 1913?? ?? ???? ??? ???? ????.

??? ? ?? ??? ???? ???? ?? ?? ??? ???? ?? ??? ?? ??? ??? ??? ??? ??? ?? ???? ????. ??/2? ?? ? ??, ???? ?? ???? ??? ??, ?? ? ??? ?? ???? ???. ?? ???? ???? ??? ???( ?? ??? ??? ? ?? ??? ??? )? ?????. (????? ?? ??? ??? ???). ?? ??, ?? ?1/2?? ?? ?? ???? ????? ?? 1 ??? ?? ????? ?? ???? ????. ? ??? ?? ??? ??? ?? ?? ? (?? ??) Cédric Lorcé(2009)? ??.

?? ???

????/??? ?? ???? ??? ?? $\mathbf {p} =m\mathbf {v}$ ? ???? ???? ???? ?? ???? ???? ??? ??? ?? ??? ? ??.

${\hat {\mathbf {v} }}={\frac {1}{m}}{\hat {\mathbf {p} }}$

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

${\hat {\mathbf {v} }}={\frac {i}{\hbar }}\left[{\hat {H}},{\hat {\mathbf {r} }}\right]$

?? ± c ??? ???? ??? ??. ? ?? ??? ??? ???–???? ??(Foldy–Wouthuysen transformation) ??.

???? ?? ?????

???? ??? ??? ???? $\psi$? ?? ?? ???? ???? ? ?? ?? ????. ????? ?? ???? ?????(???? ???? ??? ???)? ??? ?? ??? ???-???? ???

$\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)-{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0\,$

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

${\mathcal {L}}={\overline {\psi }}(\gamma ^{\mu }P_{\mu }-mc)\psi$

???-?? ?????? ??? ??.

${\mathcal {L}}=-{\frac {\hbar ^{2}}{m}}\eta ^{\mu \nu }\partial _{\mu }\psi ^{*}\partial _{\nu }\psi -mc^{2}\psi ^{*}\psi \,.$

??? ?? ???? ?? ????? ???? ??. ??? ?? ?? ?? ??? ???? ???? ?? ? ??? ??? ??? ???? ???? ??? ???? ??? ? ??? ???? ???? ?? ???? ??? ? ??? ???. $\psi$? ? ??? ??? ????? ???? ???? ????? ?? ????? ????. ???? ?? ?? ???? ??? ???? ?? ?? ?????? ????. ??? ?? ???? ? ?? ????(?: Weinberg(1995) ??).

???? ?? ????

????? ?????? ? ??? ???? ???? ?? ?? ?? $\mathbf {L} =\mathbf {r} \times \mathbf {p}$?? ????. ???? ?????? ?? ? ??? ???? ??? 4?? ?? ? ????? ??? ?? ???? ???? ??( ??? ??? ????? ??)? ???? ??? ?? ????.

$M^{\alpha \beta }=X^{\alpha }P^{\beta }-X^{\beta }P^{\alpha }=2X^{[\alpha }P^{\beta ]}\quad \rightleftharpoons \quad \mathbf {M} =\mathbf {X} \wedge \mathbf {P} \,,$

?? 6?? ????. 3?? ????? 3-?? ? ?????. $M^{12}=L^{3}$, $M^{23}=L^{1}$, $M^{31}=L^{2}$, ??? ? ?? $M^{01},M^{02},M^{03}$? ?? ??? ?? ?? ?????. ??? ?? ??? ?? ?? ???? ?? ?? ???? ??. ??? m? ??? ?? ? ???? ??? ??? ??.

$J^{\alpha \beta }=2X^{[\alpha }P^{\beta ]}+{\frac {1}{m^{2}}}\varepsilon ^{\alpha \beta \gamma \delta }W_{\gamma }p_{\delta }\quad \rightleftharpoons \quad \mathbf {J} =\mathbf {X} \wedge \mathbf {P} +{\frac {1}{m^{2}}}\star (\mathbf {W} \wedge \mathbf {P} )$

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

$W_{\alpha }={\frac {1}{2}}\varepsilon _{\alpha \beta \gamma \delta }M^{\beta \gamma }p^{\delta }\quad \rightleftharpoons \quad \mathbf {W} =\star (\mathbf {M} \wedge \mathbf {P} )$

???–???? ?? ????. ???? ??? ?? ??? ??? Troshin & Tyurin(1994)? ??.

??? ??? ??-?? ????

1926?? ??? ??? ?????. ??? ??-?? ????? ??? ??? ??? ???? ?? ??? ??? ?? ???? ????. 1939? ???? ??? ??? ????.

?? ???? ? ?? ??? ???? ??? $\mathbf {B}$? ?? ??? $\mathbf {E}$? ?? ?? $\mathbf {v}$? ???? ??? ?? ?? ????? ??? ??? ??? $\mathbf {B} '$? ??? ???.

$\mathbf {B} '={\frac {\mathbf {E} \times \mathbf {v} }{c^{2}{\sqrt {1-\left(v/c\right)^{2}}}}}\,.$

????? ?? v << c ??:

$\mathbf {B} '={\frac {\mathbf {E} \times \mathbf {v} }{c^{2}}}.$

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

${\hat {H}}=-\mathbf {B} '\cdot {\hat {\boldsymbol {\mu }}}_{S}=-\left(\mathbf {B} +{\frac {\mathbf {E} \times \mathbf {v} }{c^{2}}}\right)\cdot {\hat {\boldsymbol {\mu }}}_{S}\,,$

??? ? ?? ?? ?? ????? ?? ??? ?? ???? ? ?? ?? ???? ?? (v/c)2 ??? ?? ??? ?? ????? ???? ??. ???? ? ?? ???? ??? ??? ???. ?? ??? ??? ??? ?? ??? ??? ?? ??? ??? ?? ??? ???? ??? ?? ??? ????. ??? ???? ??? ?? ???? ??? ?? ??? ??? ???? ??. ? ??? ??? ?? ??? ?? ???? ?? ??? ??? ??-?? ?? ??? ???? ???? ?????? ???? ???:

${\hat {H}}=-\mathbf {B} '\cdot {\hat {\boldsymbol {\mu }}}_{S}=-\left(\mathbf {B} +{\frac {\mathbf {E} \times \mathbf {v} }{2c^{2}}}\right)\cdot {\hat {\boldsymbol {\mu }}}_{S}\,.$

???? ????? ??, 1?2 ? ?? ????? ????.

??

???? ?????? ???? ??? ??? ?? ????? ?? ???? ??? ??? ???? ??[?? ??, R. Resnick ? R. Eisberg(1985), ? ??? (1974??]. 1890???? 1950???? ??? ??? ?? ??? ?? ??? ??? ??? ??? ??? ?? ??? ???????? ??? ? ??? ?? ???. 20?? ?? ??? ?? ??? ??? ???? ??????? ???? ???? ?? ?? ??? ????. ?? ??? ?????, ????, ?????? ???? ???? ??? ??; ???, ??? ?? ? ??, ?? ? ?? ?? ?? ? ?? ????. ?? ??? ??? ??? ????.

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

1905? ??? ?????? ?? ??? ????. 1916?? ????? ?? ??? ????. ?? 1? ???? ???? ?? ??? ???? ? ?????. 1923?? ??? ??? ?? ?????? ????? ? ?? ??? ????. ? ?? ??-?? ??? ?? ?? ????. ? ???? ?? ??? ??? ?? ??? ???? ? ??? ??? ??-?? ???? ??? ????. 1927??? ????? ?? ??? ????? ??? ??? ????? ???? ??-?? ???? ?? ??? ??? ????.

??

• 1897? ??? ??? ???? ??? ?? ? ?? ??? ????. ??? ??? ??: ?? ???? ??? ? ???? ?? ?? ?? ??? ????.
• 1908? ???? ??? ??? ???? ?? ?? ???? ??? ???? ?? ??? ??? ???.
• 1911? ????? ??? Geiger-Marsden ???? ?? ?? ??? ??? ?? ??? ???? ??? ??? ?????.
• 1913 ??? ?? ??: ?? ????? ?? ????? ??(?? ??? ??).
• 1922 Stern–Gerlach ?? : ??? ???? ??? ??.
• 1924? ???? ????? ??? ??? ???? ????.
• 1932 ??? ? ???? ????? ???? ???? ???? ???? ???? ??? ??? ????.
• 1958 ????? ??? ??: ??? ??? ???? ?? ???? ?? ? ??? ?? ? ??, ?? ???? ? ?? ??? ??? ?? ? ? ?? ?? ?? ?? ? ??? ??? ??? ????.

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

1935?? ?????, ??, ????? ??? ?? ??? ?? ??? ???? ?? ????? ??????? ???? ?? ??? ??? ??? ??? ????. ??? ??? ???? ????? ?? ???? ??? ?? ? ??. ?? ????? ??? ???? ??? ??? ?? ?? ??? ??? ????. ??? ?? ??? ? ?? ???? ?? ?? ??? ??(? ???? $c$? ??? ? ?? ??? ?? ????? ??? ??). ???? ?? ??? ??? ???? ???. 1959?? ?? ????? ????-? ??? ?? ??? ???? ?????? ??? ???? ??? ?? ??? ????. ???? ?? ? ??? 4-??? ??? ?? ?? ???? ?? ????? ??????? ??? ?????(? ??)? ??? ?? 0? ????? ?? ??? ???? ??? ???. 1964? ? ??? EPR ??? ?? ???? ??????, ???? ????? ????? ?? ?? ???? ??? ? ??? ????.

??????? ??

• 1943 ????? ??????? ??? ??? ???? ??? ????.
• 1947 ???? ??? ????? ?? ???? ????. ??? ??????? ??? ?? ? ??? ???? ????? ?? ?? ???? ????.

?? ??

?? ???? ??

• ???? ?? ??
• ???? ???
• Electron spin resonance
• ??????

?????

• ?? ???
• ?? ??
• ??? ??
• ???? ??
• ???? ???
• ??? D-??

?? ???? ????

• ?????
• ?? ?? ???
• ???? ??? ???
• ???
• CPT ???
• ????(???)
• ?? ??
• ??? ??(???)
• ???

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????? ?

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

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??

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? ????

???? ????? ??

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???? ?? ?? ? ??

• Pfeifer, W. (2008) [2004]. 《Relativistic Quantum Mechanics, an Introduction》. 
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• “Relativistic Quantum Mechanics” (PDF). ????? ???. 
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?? ??

• Pfeifer, W. (2008) [2004]. 《Relativistic Quantum Mechanics, an Introduction》. 
• Luka?evi?, Igor (2013). “Relativistic Quantum Mechanics (Lecture Notes)” (PDF). 2014? 8? 26?? ?? ?? (PDF)?? ??? ??. 
• “Relativistic Quantum Mechanics” (PDF). ????? ???. 
• Miller, David J. (2008). “Relativistic Quantum Mechanics” (PDF). University of Glasgow. 2020? 12? 19?? ?? ?? (PDF)?? ??? ??. 2023? 6? 2?? ???. 
• Swanson, D.G. (2007). “Quantum Mechanics Foundations and Applications”. Alabama, USA: Taylor & Francis. 160?. 
• Calvert, J.B. (2003). “The Particle Electron and Thomas Precession”. 
• Arteha, S.N. “Spin and the Thomas precession”.