2 semestr / lect17n.tex

\magnification \magstep 1

\centerline{\bf ‹…Љ–€џ 17(5). 26.09.2001}

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\centerline {\bf Ќ…Џђ€‚Ћ„€Њ›… ’…Ќ‡Ћђ›}

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‚®§­ЁЄ ойЁҐ ҐбвҐб⢥­­л¬ ®Ўа §®¬ ЇаЁ  ­ «Ё§Ґ 業ва «м­®-бЁЁ¬ҐваЁз­®©
®¤­®з бвЁз­®© § ¤ зЁ бдҐаЁзҐбЄЁҐ Ј а¬®­ЁЄЁ $Y_{lm}$ ¬®¦­® ®ЇаҐ¤Ґ«Ёвм
Є Є б®Ўб⢥­­лҐ ўҐЄв®ал ®ЇҐа в®а  ${\hat l}_{3}$
$$ {\hat l}_{3}Y_{lm} \quad = \quad mY_{lm},
$$
㤮ў«Ґвў®апойЁҐ ᮮ⭮襭Ёп¬
$$ {\hat l}_{\pm}Y_{lm} \quad = \quad
\sqrt{(l \mp m)(l \pm m + 1)}Y_{l,m \pm 1}.
$$
…б«Ё ${\hat l}_{\alpha}$ ॠ«Ё§говбп Є Є ¤ЁддҐаҐ­жЁ «м­лҐ
®ЇҐа в®ал, в® $Y_{lm}$ бв ­®ўпвбп дг­ЄжЁп¬Ё бдҐаЁзҐбЄЁе гЈ«®ў,
Є®в®алҐ ¬®¦­® в®«Є®ў вм Є Є ®ЇҐа в®ал г¬­®¦Ґ­Ёп,   ЇаЁўҐ¤Ґ­­лҐ
ўлиҐ а ўҐ­бвў  ¬®¦­® ЇаҐ¤бв ўЁвм Є Є ЇҐаҐбв ­®ў®з­лҐ ᮮ⭮襭Ёп
$$ [{\hat l}_{3}, Y_{lm}] \quad = \quad mY_{lm},
$$
$$ [{\hat l}_{\pm}, Y_{lm}] \quad = \quad
\sqrt{(l \mp m)(l \pm m + 1)}Y_{l,m \pm 1}.
$$
ќв®¬г ®Ўа §жг б«Ґ¤гҐв ®ЇаҐ¤Ґ«Ґ­ЁҐ ­ҐЇаЁў®¤Ё¬ле ⥭§®а®ў.

$\underline {ЌҐЇаЁў®¤Ё¬л© \quad ⥭§®а \quad а ­Ј }$ $j$--
$\underline{нв® \quad б®ў®ЄгЇ­®бвм}$ $2j + 1$
$\underline{®ЇҐа в®а®ў}$
$$ {\hat T}_{jm}, \quad -j \leq m \leq j,
$$
\centerline{
$\underline {㤮ў«Ґвў®апойЁе \quad ЇҐаҐбв ­®ў®з­л¬ \quad
ᮮ⭮襭Ёп¬}$}
$$ [{\hat j}_{3}, {\hat T}_{jm}]
\quad = \quad
m{\hat T}_{jm},
$$
$$ [{\hat j}_{\pm}, {\hat T}_{jm}]
\quad = \quad
\sqrt{(j \mp m)(j \pm m + 1)}{\hat T}_{j,m \pm 1}.
$$
€§ нв®Ј® ®ЇаҐ¤Ґ«Ґ­Ёп ­Ґ¬Ґ¤«Ґ­­® Ї®«гз овбп ®ЎйЁҐ ўла ¦Ґ­Ёп
¬ ваЁз­ле н«Ґ¬Ґ­в®ў ­ҐЇаЁў®¤Ё¬ле ⥭§®а®ў ў Ў §ЁбҐ $|\gamma,j,m>$,
б®бв®п饬 Ё§ б®Ўб⢥­­ле ўҐЄв®а®ў ®ЇҐа в®а®ў ${\vec J}^{2}$ Ё
${\hat J}^{3}$ Ё ¤агЈЁе ®ЇҐа в®а®ў ${\hat \Gamma}$, ¤®Ї®«­пойЁе
ЇҐаўлҐ ¤® Ї®«­®© бЁбвҐ¬л ­ Ў«о¤ Ґ¬ле.
$$ 1) \qquad <{\gamma}_{1},j_{1},m_{1}|{\hat
T}_{jm}|{\gamma}_{2},j_{2},m_{2}> \quad \not= \quad 0, \quad Ґб«Ё
\quad m_{1} \not= m + m_{2},
$$
$$ 2) \qquad
<{\gamma}_{1},j_{1},m_{1}|{\hat T}_{jm}|{\gamma}_{2},j_{2},m_{2}>
\quad = \quad
<{\gamma}_{1},j_{1}|T_{j}|{\gamma}_{2},j_{2}>
<j_{2}m_{2}jm|j_{2}jj_{1}m_{1}>,
$$
Ј¤Ґ $<j_{2}m_{2}jm|j_{2}jj_{1}m_{1}>$ -- Є®нддЁжЁҐ­вл Љ«ҐЎи -ѓ®а¤®­  --
бЄ «па­лҐ Їа®Ё§ўҐ¤Ґ­Ёп, ॠ«Ё§гойЁҐ а §«®¦Ґ­ЁҐ б®Ўб⢥­­ле ўҐЄв®а®ў
®ЇҐа в®а®ў ${J_{1}}^{2}$, ${J_{2}}^{2}$, $J^{2}$, $J_{3}$ Ї®
б®Ўб⢥­­л¬ ўҐЄв®а ¬ ®ЇҐа в®а®ў ${J_{1}}^{2}$, $J_{13}$,
${J_{2}}^{2}$, $J_{23}$:
$$ |j_{1}j_{2}jm> \quad = \quad
\sum_{m_{1}m_{2}}|j_{1}m_{1}j_{2}m_{2}>
<j_{1}m_{1}j_{2}m_{2}|j_{1}j_{2}jm>.
$$
ќвЁ ᮮ⭮襭Ёп Ї®«­®бвмо ®ЇаҐ¤Ґ«пов § ўЁбЁ¬®бвм ¬ ваЁз­ле н«Ґ¬Ґ­в®ў
­ҐЇаЁў®¤Ё¬ле ⥭§®а®ў ${\hat T}_{jm}$ ў Ў §ЁбҐ $|\gamma,j,m>$ ®в
 §Ё¬гв «м­ле Єў ­в®ўле зЁбҐ« $m$ ЇаЁ «оЎле зЁб« е $j,j_{1},j_{2}$.
€е ¬®¦­® ЇҐаҐда §Ёа®ў вм в Є, зв®Ўл ­Ґ ЁбЇ®«м§®ў вм
Є®нддЁжЁҐ­вл Љ«ҐЎи --ѓ®а¤®­  пў­®. ЏаҐ¤Ї®«®¦Ё¬, зв® бгйҐбвўгҐв
в Є®© ­ҐЇаЁў®¤Ё¬л© ⥭§®а ${\hat Q}_{jm}$, ¬ ваЁз­лҐ н«Ґ¬Ґ­вл
Є®в®а®Ј®
$<{\gamma}_{1},j_{1},m_{1}|{\hat Q}_{jm}|{\gamma}_{2},j_{2},m_{2}>$
­Ґ а ў­л ⮦¤Ґб⢥­­® ­г«о. ‚ н⮬ б«гз Ґ бЇа ўҐ¤«Ёў  д®а¬г« 
$$
<{\gamma}_{1},j_{1},m_{1}|{\hat T}_{jm}|{\gamma}_{2},j_{2},m_{2}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j_{1},j_{2})
<{\gamma}_{1},j_{1},m_{1}|{\hat Q}_{jm}|{\gamma}_{2},j_{2},m_{2}>.
$$
‚뤥«Ґ­ЁҐ ­ҐЇаЁў®¤Ё¬ле ⥭§®а®ў ў ®б®Ўл© Є« бб ўҐ«ЁзЁ­ ¤®«¦­®
гЇа®бвЁвм  ­ «Ё§ бвагЄвгал Єў ­в®ўле д®а¬г«. ЏаЁўҐ¤Ґ¬ ­ҐбЄ®«мЄ®
ЇаЁ¬Ґа®ў ­ҐЇаЁў®¤Ё¬ле ⥭§®а®ў.

{\bf ЌҐЇаЁў®¤Ё¬л© ⥭§®а ­г«Ґў®Ј® а ­Ј } --- нв® ®ЇҐа в®а $\hat T$,
Є®¬¬гвЁагойЁ© б® ўбҐ¬Ё ®ЇҐа в®а ¬Ё ¬®¬Ґ­в  Є®«ЁзҐбвў  ¤ўЁ¦Ґ­Ёп
${\hat J}_{\alpha}$:
$$ [{\hat J}_{\alpha}, \hat T] \quad = \quad 0,
$$
ќв® --- ®ЇҐа в®а бЄ «па­®© ўҐ«ЁзЁ­л ў ­ иҐ¬ бв а®¬ ®ЇаҐ¤Ґ«Ґ­ЁЁ.
Њ ваЁз­лҐ н«Ґ¬Ґ­вл бЄ «па  Ё¬Ґов ®б®ЎҐ­­® Їа®бвго бвагЄвгаг:
$$ <{\gamma}_{1},j_{1},m_{1}|{\hat T}|{\gamma}_{2},j_{2},m_{2}>
\quad = \quad
{\delta}_{j_{1}j_{2}}{\delta m_{1}m_{2}}
<{\gamma}_{1},j_{1}|T|{\gamma}_{2},j_{1}>
$$
„ҐЄ ав®ўл б®бв ў«пойЁҐ ўҐЄв®а  $A_{\alpha}$ 㤮ў«Ґвў®апов
ЇҐаҐбв ­®ў®з­л¬ ᮮ⭮襭Ёп¬
$$ [J_{\alpha}, A_{\beta}]
\quad = \quad
i{\epsilon}_{\alpha \beta \gamma}A_{\gamma}.
$$
€§ ­Ёе ­Ґваг¤­® Ї®«гзЁвм б®бв ў«пойЁҐ {\bf ­ҐЇаЁў®¤Ё¬®Ј® ⥭§®а 
ЇҐаў®Ј® а ­Ј }.
„Ґ©б⢨⥫쭮, ­Ґваг¤­® Їа®ўҐаЁвм, зв® ваЁ ўҐ«ЁзЁ­л
$$ {\hat T}_{1,-1} \quad = \quad b({\hat A}_{1} - i{\hat A}_{2}),
\qquad
{\hat T}_{1,0} \quad = \quad d{\hat A}_{3},
\qquad
{\hat T}_{1,1} \quad = \quad c({\hat A}_{1} + i{\hat A}_{2})
$$
㤮ў«Ґвў®апов ЇҐаҐбв ­®ў®з­л¬ ᮮ⭮襭Ёп¬
$$ [J_{3}, {\hat T}_{1m}]
\quad = \quad
m{\hat T}_{1m}.
$$
€е ¬®¦­® бзЁв вм б®бв ў«пойЁ¬Ё ­ҐЇаЁў®¤Ё¬®Ј® ⥭§®а  ЇҐаў®Ј® а ­Ј  ў
⮬ б«гз Ґ, Ґб«Ё ўлЇ®«­Ґ­л ᮮ⭮襭Ёп
$$ [{\hat J}_{+}, {\hat T}_{1,0}] \quad = \quad
-d({\hat A}_{1} + i{\hat A}_{2})
\quad = \quad
-{d \over c}{\hat T}_{1,1}
\quad = \quad
\sqrt{(1-0)(1+0+1)}{\hat T}_{1,1},
$$
$$ [{\hat J}_{+}, {\hat T}_{1,-1}] \quad = \quad
-2ib{\hat A}_{3}
\quad = \quad
-{d \over c}{\hat T}_{1,0}
\quad = \quad
\sqrt{(1+1)(1-1+1)}{\hat T}_{1,0}.
$$
ќвЁ а ўҐ­бвў  Ї®§ў®«пов ᢥбвЁ Є®нддЁжЁҐ­вл $b$ Ё $c$ Є $d$:
$$ {\hat T}_{1,-1} \quad = \quad
{d \over \sqrt{2}}({\hat A}_{1} - i{\hat A}_{2}),
\qquad
{\hat T}_{1,0} \quad = \quad d{\hat A}_{3},
\qquad
{\hat T}_{1,1} \quad = \quad
-{d \over \sqrt{2c}}({\hat A}_{1} + i{\hat A}_{2}).
$$
Љ®нддЁжЁҐ­в $d$ 㤮Ў­® ўлЎа вм в ЄЁ¬, зв®Ўл д §л ⥭§®а®ў
${\hat T}_{1m}$ б®ўЇ «Ё б д § ¬Ё бдҐаЁзҐбЄЁе Ј а¬®­ЁЄ $Y_{1m}$.
ђ ­ҐҐ бдҐаЁзҐбЄЁҐ Ј а¬®­ЁЄЁ Ўл«Ё ®ЇаҐ¤Ґ«Ґ­л д®а¬г«®©
$$ Y_{lm}(\vec n)
\quad = \quad
(-1)^{m+|m| \over 2}i^{l}
\sqrt{{2l+1 \over 4\pi}{(l-|m|)! \over (l+|m|)!}}
P^{\hat |m|}_{l}(cos \theta)e^{im\phi},
$$
Ї®н⮬㠥бвҐб⢥­­® ўлЎа вм ®ЇаҐ¤Ґ«Ґ­ЁҐ
$$ {\hat T}_{1,-1} \quad = \quad
{i \over \sqrt{2}}({\hat A}_{1} - i{\hat A}_{2}),
\qquad
{\hat T}_{1,0} \quad = \quad i{\hat A}_{3},
\qquad
{\hat T}_{1,1} \quad = \quad
-{i \over \sqrt{2c}}({\hat A}_{1} + i{\hat A}_{2}).
$$
„ҐЄ ав®ў ⥭§®а ўв®а®Ј® а ­Ј  $T_{\alpha \beta}$ 㤮ў«Ґвў®апҐв
ЇҐаҐбв ­®ў®з­л¬ ᮮ⭮襭Ёп¬
$$ [J_{\alpha}, {\hat T}_{\beta \gamma}]
\quad = \quad
i{\epsilon}_{\alpha \beta \nu}T_{\nu \gamma}
\quad + \quad
i{\epsilon}_{\alpha \gamma \nu}T_{\beta \gamma}.
$$
€§ўҐбв­®, зв® ¤ҐЄ ав®ў ⥭§®а ўв®а®Ј® а ­Ј  ¬®¦­® Ё­ў аЁ ­в­®
®в­®бЁвҐ«м­® Ї®ў®а®в®ў а §ЎЁвм ­  бЁ¬¬ҐваЁз­го Ё  ­вЁбЁ¬¬ҐваЁз­го
з бвЁ:
$$ A_{\alpha \beta} \quad = \quad
{1 \over 2}(A_{\alpha \beta} + A_{\beta \alpha})
\quad + \quad
{1 \over 2}(A_{\alpha \beta} - A_{\beta \alpha}).
$$
Ђ­вЁбЁ¬¬ҐваЁз­ п з бвм бў®¤Ёвбп Є ўҐЄв®аг, в.Ґ. ЇаҐ¤бв ў«пҐв
б®Ў®© ­ҐЇаЁў®¤Ё¬л© ⥭§®а ЇҐаў®Ј® а ­Ј .

‘Ё¬¬ҐваЁз­л© ⥭§®а ўв®а®Ј® а ­Ј  ¬®¦­® ЇаҐ¤бв ўЁвм ў д®а¬Ґ
$$ A_{\alpha \beta} \quad = \quad
{1 \over 3}{\delta}_{\alpha \beta}TrA
\quad + \quad
\Big(A_{\alpha \beta} - {1 \over 3}{\delta}_{\alpha \beta}TrA \Big).
$$
Џ®бЄ®«мЄг б«Ґ¤ ⥭§®а  ­Ґ Ё§¬Ґ­пҐвбп ЇаЁ Ї®ў®а®в е, в® ЇҐаў®Ґ
б« Ј Ґ¬®Ґ ЇаҐ¤бв ў«пҐв б®Ў®© бЄ «па,   ўв®а®Ґ
-- бЁ¬¬ҐваЁз­л© ⥭§®а ўв®а®Ј® Ї®ап¤Є  б ­г«Ґўл¬ б«Ґ¤®¬ --
®ЇаҐ¤Ґ«пҐвбп Їпвмо Ї а ¬Ґва ¬Ё. €§ ҐЈ® б®бв ў«пойЁе ¬®¦­® Ї®бва®Ёвм
{\bf ­ҐЇаЁў®¤Ё¬л© ⥭§®а ўв®а®Ј® Ї®ап¤Є }. ЌҐваг¤­® ­ ©вЁ пў­лҐ
д®а¬г«л, ЇҐаҐў®¤пйЁҐ ¤ҐЄ ав®ў бЁ¬¬ҐваЁз­л© ⥭§®а ўв®а®Ј® Ї®ап¤Є 
$T_{\alpha \beta}$ б ­г«Ґўл¬ б«Ґ¤®¬ ў ⥭§®а ${\hat T}_{2,m}$:
$$ T_{2,0} = -\sqrt{3 \over 2}T_{33},
\quad T_{2, \pm 1} = \pm(T_{13} \pm iT_{23}),
\quad T_{2, \pm 2} = -{1 \over 2}(T_{11} - T_{22} \pm 2iT_{12}).
$$
$$ T_{\alpha \beta} =
T_{\beta \alpha},
\quad \sum_{\alpha}T_{\alpha \alpha} = 0.
$$

ќвЁ ᮮ⭮襭Ёп ®б®ЎҐ­­® 㤮Ў­л, Ґб«Ё ⥭§®а ${\hat Q}_{jm}$
Ї®бв஥­ Ё§ б®бв ў«пойЁе ®ЇҐа в®а  ¬®¬Ґ­в  Є®«ЁзҐбвў  ¤ўЁ¦Ґ­Ёп.
ЏаҐ¦¤Ґ 祬 ЇаЁўҐбвЁ пў­лҐ д®а¬г«л, § ¬ҐвЁ¬, зв® ®­Ё бЇа ўҐ¤«Ёўл «Ёим
ЇаЁ $j_{1} = j_{2}$, в Є Є Є ў Їа®вЁў­®¬ б«гз Ґ Їа ў п з бвм
а ўҐ­бвў  ⮦¤Ґб⢥­­® а ў­  ­г«о (ў бЁ«г ЇҐаҐбв ­®ў®з­ле
ᮮ⭮襭Ё© $[{\vec J}^{2}, J_{\alpha}] = 0$).

1) ЌҐЇаЁў®¤Ё¬л© ⥭§®а ­г«Ґў®Ј® а ­Ј  $\hat T$ 㤮ў«Ґвў®апҐв
ЇҐаҐбв ­®ў®з­л¬ ᮮ⭮襭Ёп¬
$$ [{\hat J}_{\alpha}, {\hat T}]
\quad = \quad 0,
$$
зв® б®ўЇ ¤ Ґв б ЇаЁ­пвл¬ а ­ҐҐ ®ЇаҐ¤Ґ«Ґ­ЁҐ¬ бЄ «па .

‚ Є зҐб⢥
⥭§®а  ${\hat Q}_{jm}$ ¬®¦­® ў§пвм Ґ¤Ё­Ёз­л© ®ЇҐа в®а $\hat E$,
Ї®н⮬г
$$ <{\gamma}_{1},j,m_{1}|\hat T|{\gamma}_{2},j,m_{2}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j) <j,m_{1}|{\hat E}|j,m_{2}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j){\delta}_{m_{1}m_{2}}.
$$

‚ла ¦ п ­ҐЇаЁў®¤Ё¬лҐ ⥭§®ал ў вҐа¬Ё­ е ¤ҐЄ ав®ўле б®бв ў«пойЁе
⥭§®а®ў, ¬л Ї®Є § «Ё, зв® ўҐ«ЁзЁ­л ${\hat T}_{jm}$ ¤Ґ©б⢨⥫쭮
бгйҐбвўгов.

Ћ¤­ Є®, ЇаЁ д®а¬г«Ёа®ўЄҐ Їа ЄвЁзҐбЄЁ «оЎ®© дЁ§ЁзҐбЄ®© § ¤ зЁ
®Ўлз­® ЁбЇ®«м§говбп {\bf ¤ҐЄ ав®ўл ⥭§®ал} --- ўҐ«ЁзЁ­л
${\hat T}_{\alpha \beta...}$, ЇаҐ®Ўа §гойЁҐбп ЇаЁ Ї®ў®а®в е Є Є
Їап¬лҐ Їа®Ё§ўҐ¤Ґ­Ёп ўҐЄв®а®ў $u_{\alpha}{\otimes}v_{\beta}{\dots}$.
—в®Ўл  ­ «®ЈЁз­лҐ д®а¬г«л ¤«п ¤ҐЄ ав®ўле ⥭§®а®ў, ­Ґ®Ўе®¤Ё¬®
ўла §Ёвм нвЁ ўҐ«ЁзЁ­л ў вҐа¬Ё­ е ­ҐЇаЁў®¤Ё¬ле ⥭§®а®ў
${\hat T}_{jm}$. ЏаЁўҐ¤Ґ¬ ­ҐЄ®в®алҐ пў­лҐ д®а¬г«л.

{\bf ‘Є «па} $\hat S$ --- нв® ­ҐЇаЁў®¤Ё¬л© ⥭§®а ­г«Ґў®Ј® а ­Ј :
$$ \hat S \quad = \quad {\hat T}_{00}.
$$
„ҐЄ ав®ўл б®бв ў«пойЁҐ ўҐЄв®а  ¬®¦­® ЇаҐ¤бв ўЁвм ў д®а¬Ґ
$$ {\hat V}_{1} \quad = \quad
-{i \over \sqrt{2}}{\hat T}_{1,-1}
\quad + \quad {i \over \sqrt{2}}{\hat T}_{1,1},
$$
$$ {\hat V}_{2} \quad = \quad
{1 \over \sqrt{2}}{\hat T}_{1,-1}
\quad + \quad {1 \over \sqrt{2}}{\hat T}_{1,1},
$$
$$ {\hat V}_{3} \quad = \quad -i{\hat T}_{1,0}.
$$
Ќ Є®­Ґж, ¤«п ¤ҐЄ а®ў®Ј® ­ҐЇаЁў®¤Ё¬®Ј® ⥭§®а  ўв®а®Ј® Ї®ап¤Є ,
$$ D_{\alpha \beta} \quad = \quad D_{\beta \alpha}, \qquad
Tr{\hat D} = 0,
$$
бЇа ўҐ¤«Ёўл ᮮ⭮襭Ёп
$$ {\hat D}_{11} \quad = \quad {1 \over \sqrt{2}}
({\hat T}_{2,-1} - {\hat T}_{2,1} \quad + \quad
+{\sqrt{2 \over 3}}{\hat T}_{2,0}),
$$
$$ {\hat D}_{12} \quad = \quad
{i \over 2}({\hat T}_{2,2} - {\hat T}_{2,-2}),
\qquad
{\hat D}_{13} \quad = \quad
{1 \over 2}({\hat T}_{2,2} - {\hat T}_{2,-2}),
$$
$$ {\hat D}_{22} \quad = \quad
{1 \over 2}({\hat T}_{2,1} - {\hat T}_{2,1}),
\qquad
{\hat D}_{23} \quad = \quad
{i \over 2}({\hat T}_{2,1} + {\hat T}_{2,-1}),
\qquad
{\hat D}_{33} \quad = \quad - \sqrt{2 \over 3}{\hat T}_{2,0}.
$$
‚ ®ЎйҐ¬ б«гз Ґ ­ҐЇаЁў®¤Ё¬л© ¤ҐЄ ав®ў ⥭§®а а ­Ј  $j$ ЇаҐ¤бв ўЁвм

Є Є
$$ {\hat T}^{(j)}_{\alpha \beta \gamma ...}
\quad = \quad
\sum_{m}N^{(j)}_{\alpha \beta \gamma ...}{\hat T}_{jm}.
$$


Џ®н⮬г бЇа ўҐ¤«Ёўл д®а¬г«л ўЁ¤ 
$$
<{\gamma}_{1},j_{1},m_{1}|{\hat T}^{(j)}_{\alpha \beta \gamma...}
|{\gamma}_{2},j_{1},m_{2}>
\quad = \quad
$$
$$ \sum_{m}N^{(j)}_{\alpha \beta \gamma...}
<{\gamma}_{1},j_{1},m_{1}|{\hat T}_{jm}|{\gamma}_{2},j_{1},m_{2}>
\quad = \quad
$$
$$
C({\gamma}_{1},{\gamma}_{2},j_{1},j)
\sum_{m}N^{(j)}_{\alpha \beta \gamma...}
<{\gamma}_{1},j_{1},m_{1}|{\hat Q}_{jm}|{\gamma}_{2},j_{1},m_{2}>
\quad = \quad
$$
$$ C({\gamma}_{1},{\gamma}_{2},j_{1},j)
<{\gamma}_{1},j_{1},m_{1}|{\hat Q}^{j}_{\alpha \beta \gamma...}
|{\gamma}_{2},j_{1},m_{2}>
$$
Ј¤Ґ ${\hat Q}^{(j)}_{\alpha \beta \gamma ...}$ Їа®Ё§ў®«м­л©
­ҐЇаЁў®¤Ё¬л© ¤ҐЄ ав®ў ⥭§®а. Ќг¦­® в®«мЄ®, зв®Ўл ҐЈ® ¬ ваЁз­лҐ
н«Ґ¬Ґ­вл ­Ґ Ўл«Ё ⮦¤Ґб⢥­­® а ў­л ­г«о.

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\centerline {\bf ‘ђ…„Ќ€… ‡ЌЂ—…Ќ€џ Ќ…Џђ€‚Ћ„€Њ›• ’…Ќ‡ЋђЋ‚}

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‘®®в­®иҐ­Ёп ЇаҐ¤л¤г饣® а §¤Ґ«  ®б®ЎҐ­­® 㤮Ў­л, Ґб«Ё ⥭§®а
${\hat Q}^{(j)}_{\alpha \beta \gamma ...}$
Ї®бв஥­ Ё§ б®бв ў«пойЁе ®ЇҐа в®а  ¬®¬Ґ­в  Є®«ЁзҐбвў  ¤ўЁ¦Ґ­Ёп.
ЏаҐ¦¤Ґ 祬 ЇаЁўҐбвЁ пў­лҐ д®а¬г«л, б«Ґ¤гҐв Ї®¤зҐаЄ­гвм, зв® ®­Ё
бЇа ўҐ¤«Ёўл «Ёим ЇаЁ $j_{1} = j_{2}$, в Є Є Є ў Їа®вЁў­®¬ б«гз Ґ
Їа ў п з бвм а ўҐ­бвў  ⮦¤Ґб⢥­­® а ў­  ­г«о (ў бЁ«г
ЇҐаҐбв ­®ў®з­ле ᮮ⭮襭Ё© $[{\vec J}^{2}, J_{\alpha}] = 0$).

‚ б«гз Ґ ⥭§®а  ­г«Ґў®Ј® а ­Ј  ў Є зҐб⢥ ⥭§®а  $\hat Q$ ¬®¦­®
ў§пвм Ґ¤Ё­Ёз­л© ®ЇҐа в®а $\hat E$. ‚ १г«мв вҐ Ї®«гз овбп в Є
­ §лў Ґ¬лҐ Їа ўЁ«  ®вЎ®а  ¤«п ¬ ваЁз­ле н«Ґ¬Ґ­в®ў бЄ «па .
$$ <{\gamma}_{1},j,m_{1}|{\hat S}|{\gamma}_{2},j,m_{2}>
\quad = \quad
{\delta m_{1}m_{2}}<{\gamma}_{1},j|S|{\gamma}_{2},j>.
$$
{\bf Њ ваЁз­л© н«Ґ¬Ґ­в бЄ «па  ¤Ё Ј®­ «Ґ­ Ї® Ё­¤ҐЄб ¬ $m$ Ё ­Ґ
§ ўЁбЁв ®в ­Ёе}.

‚ б«гз Ґ ўҐЄв®а  ®ЇҐа в®ал ${\hat Q}_{\alpha}$ ¬®¦­® § ¬Ґ­Ёвм
®ЇҐа в®а ¬Ё ¬®¬Ґ­в  Є®«ЁзҐбвў  ¤ўЁ¦Ґ­Ёп ${\hat J}_{\alpha}$:
$$ <{\gamma}_{1},j,m_{1}|{\hat V}_{\alpha}|{\gamma}_{2},j,m_{2}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j)
<j,m_{1}|{\hat J}_{\alpha} |j,m_{2}>.
$$
”Ё§ЁзҐбЄЁ© б¬лб« нв®© д®а¬г«л ¬®¦­® (­Ґ ᮢᥬ в®з­®) ўла §Ёвм
г⢥তҐ­ЁҐ¬: {\bf б।­ҐҐ §­ зҐ­ЁҐ ўҐЄв®а  ­ Їа ў«Ґ­® ў¤®«м ¬®¬Ґ­в 
Є®«ЁзҐбвў  ¤ўЁ¦Ґ­Ёп}. ЌҐваг¤­® гв®з­Ёвм бвагЄвгаг Є®нддЁжЁҐ­в 
$C$. ‚®бЇ®«м§гҐ¬бп ⥬, зв® бЄ «па­®Ґ Їа®Ё§ўҐ¤Ґ­ЁҐ ўҐЄв®а®ў --- нв®
бЄ «па. Џ®н⮬г
$$ <{\gamma}_{1},j,m_{1}|{\vec V}{\vec
J}|{\gamma}_{2},j,m_{2}> \quad = \quad {\delta m_{1} m_{2}}
<{\gamma}_{1},j,m|{\vec J}{\vec V}|{\gamma}_{2},j,m>.
$$
‘ ¤агЈ®© бв®а®­л, нвг ¦Ґ ўҐ«ЁзЁ­г ¬®¦­® ЇаҐ¤бв ўЁвм б㬬®©
$$
<{\gamma}_{1},j,m_{1}|{\vec V}{\vec J}|{\gamma}_{2},j,m_{2}>
\quad = \quad
\sum_{{\gamma},j^{'},m^{'}}
<{\gamma}_{1},j,m_{1}|{\vec V}|{\gamma},j^{'},m^{'}>
<m^{'},j^{'},{\gamma}|{\vec J}|{\gamma}_{2},j,m_{2}>.
$$
Џ®бЄ®«мЄг
$$ <m^{'},j^{'},{\gamma}|{\vec J}|{\gamma}_{2},j,m_{2}>.
\quad = \quad
{\delta}_{\gamma {\gamma}_{2}}{\delta}_{j^{'}j},
<m^{'},j|{\vec J}|j,m_{2}>,
$$
в® б㬬Ёа®ў ­ЁҐ Ї® Ё­¤ҐЄб ¬ $\gamma$ Ё $j^{'}$ б­Ё¬ Ґвбп Ё
$$
<{\gamma}_{1},j,m_{1}|{\vec V}{\vec J}|{\gamma}_{2},j,m_{2}>
\quad = \quad
\sum_{m^{'}}
<{\gamma}_{1},j,m_{1}|{\vec V}|{\gamma}_{2},j,m^{'}>
<m^{'},j|{\vec J}|j,_{2}>.
$$
‚ Ї®б«Ґ¤­Ґ© б㬬Ґ ¬ ваЁз­лҐ н«Ґ¬Ґ­вл ўҐЄв®а  ¬®¦­® ЇаҐ¤бв ўЁвм
д®а¬г«®©
$$ <{\gamma}_{1},j,m_{1}|{\vec V}|{\gamma}_{2},j,m^{'}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j)<j,m_{1}|{\vec J}|j,m^{'}>.
$$
Џ®б«Ґ нв®Ј® б㬬  «ҐЈЄ® ўлзЁб«пҐвбп, зв® ЇаЁў®¤Ёв Є १г«мв вг
$$ <{\gamma}_{1},j,m_{1}|{\vec V}{\vec J}|{\gamma}_{2},j,m_{2}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j)
<j,m_{1}|{\vec J}^{2}|j,m_{2}>
\quad = \quad
$$
$$ C({\gamma}_{1},{\gamma}_{2},j)j(j+1){\delta}_{m_{1} m_{2}}.
$$
…б«Ё $j$ ­Ґ а ў­® ­г«о, в® ¬®¦­® ­ ©вЁ зЁб«® $C$. ’ ЄЁ¬ ®Ўа §®¬
$$ <{\gamma}_{1},j,m_{1}|{\hat V}_{\alpha}|{\gamma}_{2},j,m_{2}>
\quad = \quad
\left \{
{<{\gamma}_{1},j,m|{\vec V}{\vec J}|{\gamma}_{2},j,m>
\over j(j+1)}<j,m_{1}|{\hat J}_{\alpha}|j,m_{2}>, \quad
j \not= 0 \atop 0, \qquad \qquad j = 0 \right.
$$
‚ б«гз Ґ {\bf ­ҐЇаЁў®¤Ё¬®Ј® ⥭§®а  ўв®а®Ј® Ї®ап¤Є } ᮮ⢥вбвўгойЁ©
⥭§®а ${\hat Q}_{\alpha \beta}$ ¬®¦­® ў§пвм ў д®а¬Ґ
$$ {\hat Q}_{\alpha \beta} \quad = \quad
{\hat J}_{\alpha}{\hat J}_{\beta}
\quad + \quad
{\hat J}_{\beta}{\hat J}_{\alpha}
\quad - \quad
{2 \over 3}{\delta}_{\alpha \beta}{\vec J}^{2}
$$
Џ®бв®п­­го ў а ўҐ­б⢥
$$
<{\gamma}_{1},j,m_{1}|{\hat T}_{\alpha \beta}|{\gamma}_{2},j,m_{2}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j)
<j, m_{1}|{\hat Q}_{\alpha \beta}|j,m_{2}>
$$
¬®¦­® ®ЇаҐ¤Ґ«Ёвм, ўлзЁб«пп ¬ ваЁз­л© н«Ґ¬Ґ­в
$$
<{\gamma}_{1},j,m_{1}|
{\hat J}_{\alpha}{\hat T}_{\alpha \beta}{\hat J}_{\beta}
|{\gamma}_{2},j,m_{2}>,
$$
Їа®Ї®ажЁ®­ «м­л© ${\delta}_{m_{1} m_{2}}$. ђ §« Ј п, Є Є ў
ЇаҐ¤л¤г饬 б«гз Ґ, Їа®Ё§ўҐ¤Ґ­ЁҐ ®ЇҐа в®а®ў Ї® Ў §Ёбг
$|{\gamma},j,m>$ Ё б®Ї®бв ў«пп нв® ўла ¦Ґ­ЁҐ б ¬ ваЁз­л¬ н«Ґ¬Ґ­в®¬
$$ <j,m_{1}|J_{\alpha}{\hat Q}_{\alpha \beta}J_{\beta}|m_{2},j>
\quad = \quad
{\delta}_{\alpha \beta}{1 \over 3}j(j + 1)(2j + 3)(2j - 1),
$$
Ї®«гзЁ¬
$$
<{\gamma}_{1},j,m_{1}|{\hat T}_{\alpha \beta}|{\gamma}_{2},j,m_{2}>
\quad = \quad
{3 \over j(j+1)(2j+3)(2j-1)}<j, m_{1}|{\hat Q}_{\alpha \beta}|j,m_{2}>.
$$








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