2 semestr / lect19n.tex

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\centerline{\bf ‹…Љ–€џ 19. 5.10.2001}

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\centerline {\bf Љ‚ЂЌ’Ћ‚ЂЌ€… ќ‹…Љ’ђЋЊЂѓЌ€’ЌЋѓЋ ЏЋ‹џ}

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‘ў®©бвў  Є« ббЁзҐбЄ®Ј® н«ҐЄв஬ Ј­Ёв­®Ј® Ї®«п ®ЇаҐ¤Ґ«повбп
га ў­Ґ­Ёп¬Ё

\noindent Њ ЄбўҐ«« 
$$ rot{\vec H} \quad - \quad
{1 \over c}{\partial {\vec E} \over {\partial t}}
\quad = \quad
{4{\pi} \over c}{\vec j},
\qquad \quad
div{\vec E} \quad = \quad 4{\pi}{\rho},
$$
$$ rot{\vec E} \quad + \quad
{1 \over c}{\partial {\vec H} \over {\partial t}}
\quad = \quad 0,
\qquad \quad
div{\vec H} \quad = \quad 0,
$$
ў Є®в®але Ї«®в­®бвЁ § ап¤  Ё в®Є  бўп§ ­л га ў­Ґ­ЁҐ¬ ­ҐЇаҐалў­®бвЁ
$$ div{\vec j} \quad + \quad
{\partial {\rho} \over {\partial t}}
\quad = \quad 0.
$$
Њл 㬥Ґ¬ Єў ­в®ў вм бЁб⥬л, га ў­Ґ­Ёп ¤ўЁ¦Ґ­Ёп Є®в®але ¬®¦­®
ЇаҐ¤бв ўЁвм ў д®а¬Ґ га ў­Ґ­Ё© ѓ ¬Ё«мв®­ . Њ®¦­® «Ё ᢥбвЁ Є ­Ё¬
га ў­Ґ­Ёп Њ ЄбўҐ«« ? •®а®и® Ё§ўҐбв­®, Є Є нв® ¤Ґ« Ґвбп.

Џ®бЄ®«мЄг ¤ЁўҐаЈҐ­жЁп ­ Їа殮­­®бвЁ ¬ Ј­Ёв­®Ј® Ї®«п а ў­  ­г«о, в®
ҐҐ ¬®¦­® ЇаҐ¤бв ўЁвм Є Є а®в®а ¤агЈ®Ј® ўҐЄв®а  --- {\bf ўҐЄв®а­®Ј®
Ї®вҐ­жЁ « } $\vec A$:
$$ {\vec H} \quad = \quad rot{\vec A}.
$$
Џ®б«Ґ нв®Ј® ўв®а®Ґ ®¤­®а®¤­®Ґ га ў­Ґ­ЁҐ Њ ЄбўҐ««  ¬®¦­® ЇаЁўҐбвЁ Є
д®а¬Ґ
$$ rot({\vec E} \quad + \quad
{1 \over c}{\partial {\vec A} \over {\partial t}})
\quad = \quad 0.
$$
ќв® Ї®§ў®«пҐв ўла §Ёвм н«ҐЄваЁзҐбЄго ­ Їа殮­­®бвм Є Є
$$ {\vec E} \quad = \quad
- {1 \over c}{\partial {\vec A} \over {\partial t}}
\quad - \quad
grad{\phi}.
$$
”г­ЄжЁо $\phi$ ­ §лў ов {\bf бЄ «па­л¬ Ї®вҐ­жЁ «®¬}.

‡­ зҐ­Ёп ­ Їа殮­­®б⥩ ­Ґ Ё§¬Ґ­повбп ЇаЁ {\bf Є «ЁЎа®ў®з­ле
ЇаҐ®Ўа §®ў ­Ёпе}
$$ {\vec A} \quad \Longrightarrow \quad
{\vec A} \quad + \quad grad{\chi},
$$
$$ {\phi} \quad \Longrightarrow \quad
{\phi} \quad - \quad
{1 \over c}{\partial {\chi} \over {\partial t}}.
$$
”г­ЄжЁо $\phi$ ¬®¦­® ўлЎа вм в Є, зв®Ўл ўлЇ®«­п«®бм а ўҐ­бвў®
$$ div{\vec A} \quad = \quad 0.
$$
‚ н⮬ б«гз Ґ Ј®ў®апв ® {\bf Єг«®­®ў®© Є «ЁЎа®ўЄҐ}.

‚ Єг«®­®ў®© Є «ЁЎа®ўЄҐ ўв®а®Ґ ­Ґ®¤­®а®¤­®Ґ га ў­Ґ­ЁҐ Њ ЄбўҐ«« 
ЇаЁ­Ё¬ Ґв ўЁ¤
$$ div{\vec E} \quad = \quad
- {\nabla}^{2}{\phi}
\quad = \quad
4{\pi}{\rho}.
$$
ђҐиҐ­ЁҐ нв®Ј® га ў­Ґ­Ёп
$$ {\phi}({\vec r},t) \quad = \quad
{\int}
{{\rho}({\vec r}^{'},t) \over |{\vec r} - {\vec r}^{'}|}
d{\vec r}^{'}.
$$
Ї®Є §лў Ґв, зв® §­ зҐ­Ёп бЄ «па­®Ј® Ї®вҐ­жЁ «  ®ЇаҐ¤Ґ«пҐвбп
¬Ј­®ўҐ­­л¬ а бЇаҐ¤Ґ«Ґ­ЁҐ¬ § а冷ў, в.Ґ. {\bf ў Єг«®­®ў®© Є «ЁЎа®ўЄҐ
бЄ «па­л© Ї®вҐ­жЁ « ${\phi}({\vec r},t)$ ­Ґ пў«пҐвбп ­Ґ§ ўЁбЁ¬®©
¤Ё­ ¬ЁзҐбЄ®© ЇҐаҐ¬Ґ­­®©}.

Џ®б«Ґ¤­ҐҐ Ё§ га ў­Ґ­Ё© Њ ЄбўҐ««  ЇаЁ­Ё¬ Ґв ўЁ¤
$$ {1 \over c^{2}}{{\partial}^{2} {\vec A} \over
{\partial}t^{2}}
\quad - \quad
{\nabla}^{2}{\vec A}
\quad = \quad
{4{\pi} \over c}{\vec j}_{\perp},
$$
Ј¤Ґ ўҐЄв®а ${\vec j}_{\perp}$, а ў­л©
$$ {\vec j}_{\perp} \quad = \quad
{\vec j} \quad - \quad {1 \over c}
{\nabla}{{\partial}{\phi} \over {\partial}t},
$$
㤮ў«Ґвў®апҐв га ў­Ґ­Ёо
$$ div{\vec j}_{\perp} \quad = \quad 0.
$$

{\leftskip 1.5 true cm \noindent ’ ЄЁ¬ ®Ўа §®¬ ў Єг«®­®ў®©
Є «ЁЎа®ўЄҐ бў®©бвў  н«ҐЄв஬ Ј­Ёв­®Ј® Ї®«п

\noindent ®ЇаҐ¤Ґ«повбп {\bf ўҐЄв®а­л¬ Ї®вҐ­жЁ «®¬} ${\vec A}$,
㤮ў«Ґвў®апйЁ¬ ¤®Ї®«­ЁвҐ«м­®¬г гб«®ўЁо

}

$$ div{\vec A} \quad = \quad 0. $$

—в®Ўл ЇҐаҐ©вЁ Є ®ЎлЄ­®ўҐ­­л¬ ¤ЁддҐаҐ­жЁ «м­л¬ га ў­Ґ­Ёп¬, ᮤҐа¦ йЁ¬
в®«мЄ® дг­ЄжЁЁ ўаҐ¬Ґ­Ё, 㤮Ў­® ўл¤Ґ«Ёвм пў­® в®, зв® ®ЇаҐ¤Ґ«пҐв
ўҐЄв®а­лҐ бў®©бвў  Ї®«п. ’Ґе­ЁзҐбЄЁ нв® Їа®йҐ ўбҐЈ® ¤Ґ« вмбп
б«Ґ¤гойЁ¬ бЇ®б®Ў®¬. ЏаҐ¤Ї®« Ј Ґвбп, зв® Ї®«Ґ § Є«о祭® ў ЄгЎҐ ®ЎкҐ¬ 
$V = L^{3}$. ќв® Ї®§ў®«Ёв а §«®¦Ёвм Ї®вҐ­жЁ « ў ап¤ ”гамҐ.
$$ {\vec A}(\vec r) \quad = \quad
{\sum}_{\vec f}(e^{i{\vec f}{\vec r}}{\vec A}_{\vec f}
\quad + \quad
e^{-i{\vec f}{\vec r}}{{\vec A}^{*}}_{\vec f}).
$$
‚室пйЁҐ ў ап¤ ”гамҐ б« Ј Ґ¬лҐ бЈагЇЇЁа®ў ­л в Є, зв®Ўл нв®
а §«®¦Ґ­ЁҐ  ўв®¬ вЁзҐбЄЁ ЇаЁў®¤Ё«® Є ¤Ґ©б⢨⥫쭮© б㬬Ґ
$$ {\vec A} \quad = \quad {\vec A}^{*}.
$$
’ॡ®ў ­ЁҐ б ¬®б®Їа殮­­®бвЁ ®ЇҐа в®а  Ё¬Їг«мб  ЇаЁў®¤Ёв
Є гб«®ўЁо ЇҐаЁ®¤Ёз­®cвЁ
$$ {\vec A}(x_{\alpha} + L) \quad = \quad
{\vec A}(x_{\alpha}).
$$
ќв® гб«®ўЁҐ ®ЇаҐ¤Ґ«пҐв ў®§¬®¦­лҐ §­ зҐ­Ёп ${\vec f}$:
$$ {\vec f} \quad = \quad
{2{\pi} \over L}{\vec n},
$$
Ј¤Ґ ${\vec n}$ --- ўҐЄв®а б 楫®зЁб«Ґ­­л¬Ё б®бв ў«пойЁ¬Ё
$$ {\vec n} \quad = \quad (n_{1}, n_{2}, n_{3})
\qquad n_{\alpha} \quad = \quad 0, \pm1, \pm2,.....
$$
“б«®ўЁҐ $div{\vec A} = 0$ ЇаЁў®¤Ёв Є а ўҐ­бвўг
$$ div{\vec A}({\vec r})
\quad = \quad
i{\sum}_{\vec f}
(e^{i{\vec f}{\vec r}}({\vec f}{\vec A}_{\vec f})
\quad - \quad
e^{-i{\vec f}{\vec r}}({\vec f}{{\vec A}^{*}}_{\vec f}))
\quad = \quad 0.
$$
Ћ­® Ўг¤Ґв 㤮ў«Ґвў®аҐ­®  ўв®¬ вЁзҐбЄЁ, Ґб«Ё Ї®¤зЁ­Ёвм Є®нддЁжЁҐ­вл
${\vec A}_{\vec f}$ гб«®ўЁп¬
$$ {\vec f}{\vec A_{\vec f}} \quad = \quad 0,
\qquad
{\vec f}{{\vec A}^{*}}_{\vec f} \quad = \quad 0.
$$
ђҐиҐ­Ёп Ї®«г祭­ле га ў­Ґ­Ё© Ё¬Ґов ўЁ¤
$$ {\vec A}_{\vec f} \quad = \quad
{\sum}_{{\lambda} = 1}^{2}
{\vec e}_{\lambda}(\vec f)A_{\lambda}(\vec f),
\quad = \quad
{{\vec A}_{\vec f}}^{*} \quad = \quad
{\sum}_{{\lambda} = 1}^{2}
{\vec e}_{\lambda}(\vec f){A_{\lambda}(\vec f)}^{*}
$$
Ј¤Ґ ўҐЄв®ал ${\vec e}$ 㤮ў«Ґвў®апов ᮮ⭮襭Ёп¬ ®ав®Ј®­ «м­®бвЁ
$$ {\vec e}_{{\lambda}_{1}}(\vec f){\vec e}_{{\lambda}_{2}}(\vec f)
\quad = \quad
{\delta}_{{\lambda}_{1}{\lambda}_{2}},
\qquad
{\vec e}_{{\lambda}_{1}}(\vec f)\times
{\vec e}_{{\lambda}_{2}}(\vec f)
\quad = \quad
{{\vec f} \over |{\vec f}|}.
$$
Ё Ї®«­®вл
$$ {\sum}_{\lambda}
{\vec e}_{\lambda \alpha}{\vec e}_{\lambda \beta}
\quad = \quad
{\delta}_{\alpha \beta} -
{f_{\alpha}f_{\beta} \over {\vec f}^{2}}.
$$
„ «ҐҐ гв®з­пҐвбп ўЁ¤ Є®нддЁжЁҐ­в®ў $A_{\lambda}(\vec f)$:
$$ A_{\lambda}(\vec f) \quad = \quad
B(|\vec f|)a_{\lambda}(\vec f), \qquad
{A_{\lambda}}^{*}(\vec f) \quad = \quad
B(|\vec f|){a_{\lambda}}^{*}(\vec f),
$$
Ј¤Ґ
$$ B(|\vec f|) \quad = \quad
c\sqrt{2{\pi}{\hbar} \over V{\omega}(|\vec f|)},
\qquad {\omega}(|\vec f|) \quad = \quad c|\vec f|.
$$
Љ®нддЁжЁҐ­вл $B(|\vec f|)$ ўлЎЁа овбп в Є, зв®Ўл н­ҐаЈЁп
н«ҐЄв஬ Ј­Ёв­®Ј® Ї®«п
$$ E \quad = \quad
{1 \over 8{\pi}}{\int}({\vec E}^{2} + {\vec H}^{2})d{\vec r},
$$
Є®в®а п ў Єг«®­®ў®© Є «ЁЎа®ўЄҐ ®ЇаҐ¤Ґ«пҐвбп д®а¬г«®©
$$ E \quad = \quad {1 \over 8{\pi}}{\int}d{\vec r}
((rot {\vec A})^{2} +
{1 \over c^{2}}({\partial {\vec A} \over {\partial t}})^{2})
\quad + \quad
{1 \over 2}{\int}
d{\vec r}d{\vec r}^{'}
{{\rho}(\vec r){\rho}({\vec r}^{'}) \over
|{\vec r} - {\vec r}^{'}|},
$$
ў б«гз Ґ бў®Ў®¤­®Ј® Ї®«п ($\rho = 0$) ЇаЁ­Ё¬ «  ўЁ¤
$$ E \quad \equiv \quad
{\cal H}(a_{\lambda}(\vec f), {a^{*}}_{\lambda}(\vec f))
\quad = \quad
\sum_{{\vec f},{\lambda}}
{\hbar}{\omega}(\vec f)
{{a}^{*}}_{\lambda}(\vec f)a_{\lambda}(\vec f).
$$
Џ®б«Ґ Ї®¤бв ­®ўЄЁ ў га ў­Ґ­ЁҐ M ЄбўҐ««  а §«®¦Ґ­Ёп Ї®вҐ­жЁ «  ў ап¤
”гамҐ Ї®«гз Ґвбп ¤®«Ј®¦¤ ­­ п бЁб⥬  га ў­Ґ­Ё© ¤«п Є®нддЁжЁҐ­в®ў
$a_{\lambda}(\vec f)$:
$$ {1 \over c^{2}}{d^{2}a_{\lambda}(\vec f) \over dt^{2}}
\quad + \quad {\vec f}^{2}a_{\lambda}(\vec f)
\quad = \quad
j_{\lambda}(\vec f),
$$
$$ {1 \over c^{2}}{d^{2}{a_{\lambda}}^{*}(\vec f) \over dt^{2}}
\quad + \quad {\vec f}^{2}{a_{\lambda}}^{*}(\vec f)
\quad = \quad
{j_{\lambda}}^{*}(\vec f),
$$
Ј¤Ґ $j_{\lambda}(\vec f)$ --- Є®нддЁжЁҐ­вл ”гамҐ Ї«®в­®бвЁ в®Є .

‚ б«гз Ґ бў®Ў®¤­®Ј® Ї®«п «оЎ®Ґ аҐиҐ­ЁҐ нвЁе га ў­Ґ­Ё© бў®¤Ёвбп Є
бгЇҐаЇ®§ЁжЁЁ дг­ЄжЁ©
$$ a_{\lambda}({\vec f},t)
\quad = \quad
a_{\lambda}({\vec f})e^{-i{\omega}(\vec f)t},
$$
$$ {a_{\lambda}}^{*}({\vec f},t)
\quad = \quad
{a_{\lambda}}^{*}({\vec f})e^{i{\omega}(\vec f)t}
$$
б Їа®Ё§ў®«м­л¬Ё Є®¬Ї«ҐЄб­л¬Ё зЁб« ¬Ё
$a_{\lambda}({\vec f})$ Ё ${a_{\lambda}}^{*}({\vec f})$.

ќвЁ дг­ЄжЁЁ --- аҐиҐ­Ёп га ў­Ґ­Ё© ЇҐаў®Ј® Ї®ап¤Є 
$$ {d \over dt}a_{\lambda}({\vec f},t)
\quad = \quad
-i{\omega}(\vec f)a_{\lambda}({\vec f},t),
$$
$$ {d \over dt}{a_{\lambda}}^{*}({\vec f},t)
\quad = \quad
-i{\omega}(\vec f){a_{\lambda}}^{*}({\vec f},t),
$$

…б«Ё а бб¬ ваЁў вм
$a_{\lambda}({\vec f})$ Ё ${a_{\lambda}}^{*}({\vec f})$ Є Є
­Ґ§ ўЁбЁ¬лҐ ЇҐаҐ¬Ґ­­лҐ, в® ЇаЁўҐ¤Ґ­­лҐ ўлиҐ га ў­Ґ­Ёп ¬®¦­® § ЇЁб вм
Є Є га ў­Ґ­Ёп ѓ ¬Ё«мв®­ 
$$ {d \over dt}a_{\lambda}({\vec f},t)
\quad = \quad
-{i \over {\hbar}}
{\partial {\cal H} \over {\partial {a_{\lambda}}^{*}({\vec f})}},
$$
$$ {d \over dt}{a_{\lambda}}^{*}({\vec f},t)
\quad = \quad
{i \over {\hbar}}
{\partial {\cal H} \over {\partial a_{\lambda}({\vec f})}}.
$$

…б«Ё $F$ --- Їа®Ё§ў®«м­ п дг­ЄжЁп
$a_{\lambda}({\vec f})$ Ё ${a_{\lambda}}^{*}({\vec f})$, в® ҐҐ
Їа®Ё§ў®¤­ п Ї® ўаҐ¬Ґ­Ё а ў­ 
$$ {dF \over dt} \quad = \quad
\sum_{{\vec f},{\lambda}}
{\partial F \over {\partial {a_{\lambda}}^{*}({\vec f})}}
{d \over dt}{a_{\lambda}}^{*}(\vec f)
\quad + \quad
{\partial F \over {\partial a_{\lambda}({\vec f})}}
{d \over dt}a_{\lambda}(\vec f)
\quad =
$$
$$ {i \over {\hbar}}\sum_{{\vec f},{\lambda}}
{\partial {\cal H} \over {\partial a_{\lambda}(\vec f)}}
{\partial F \over {\partial {a_{\lambda}}^{*}(\vec f)}}
\quad - \quad
{\partial {\cal H} \over {\partial {a_{\lambda}}^{*}(\vec f)}}
{\partial F \over {\partial a_{\lambda}(\vec f)}}
$$
…б«Ё ®ЇаҐ¤Ґ«Ёвм {\bf бЄ®ЎЄг Џг бб®­ } дг­ЄжЁ© $A$ Ё $B$
$$ \lbrace A,B \rbrace
\quad = \quad
\sum_{{\vec f},{\lambda}}
{\partial {\cal A} \over {\partial a_{\lambda}(\vec f)}}
{\partial B \over {\partial {a_{\lambda}}^{*}(\vec f)}}
\quad - \quad
{\partial A \over {\partial {a_{\lambda}}^{*}(\vec f)}}
{\partial B \over {\partial a_{\lambda}(\vec f)}},
$$
в® Їа®Ё§ў®¤­го $F$ ¬®¦­® § ЇЁб вм ў д®а¬Ґ
$$ {dF \over dt} \quad = \quad
{i \over \hbar}\lbrace H,F \rbrace.
$$
„«п ¤ «м­Ґ©иҐЈ® бгйҐб⢥­­®, зв® бЄ®ЎЄ  Џг бб®­  дг­ЄжЁ©
$a_{\lambda}({\vec f})$ Ё ${a_{\lambda}}^{*}({\vec f})$ а ў­ 
ᮮ⢥бвўгойЁ¬ бЁ¬ў®« ¬ Ља®­ҐЄҐа 
$$ \lbrace a_{\lambda}({\vec f}),
{a_{{\lambda}^{'}}}^{*}({\vec f_{1}}) \rbrace
\quad = \quad
{\delta}_{{\lambda},{\lambda}^{'}}
{\delta}_{{\vec f},{\vec f_{1}}}.
$$
ќв® ᮮ⭮襭ЁҐ ў¬ҐбвҐ б  ­ «®ЈЁз­л¬Ё а ўҐ­бвў ¬Ё
$$ \lbrace a_{\lambda}({\vec f}),
a_{{\lambda}^{'}}({\vec f_{1}}) \rbrace
\quad = \quad 0,
$$
$$ \lbrace {a_{{\lambda}}}^{*}({\vec f}),
{a_{{\lambda}^{'}}}^{*}({\vec f_{1}}) \rbrace
\quad = \quad 0.
$$
¬®¦­® ЇаЁ­пвм §  ®ЇаҐ¤Ґ«Ґ­ЁҐ ўҐ«ЁзЁ­
$a_{\lambda}({\vec f})$ Ё ${a_{\lambda}}^{*}({\vec f})$.

ЌҐваг¤­® Їа®Єў ­в®ў вм н«ҐЄв஬ Ј­Ёв­®Ґ Ї®«Ґ --- ®ЇаҐ¤Ґ«Ёвм
{\bf ®ЇҐа в®а ўҐЄв®а­®Ј® Ї®вҐ­жЁ « } ў Єг«®­®ў®© Є «ЁЎа®ўЄҐ.
„«п нв®Ј® ®ЇаҐ¤Ґ«пов Ї ал на¬Ёв®ў® б®Їа殮­­ле ®ЇҐа в®а®ў
${\hat a}_{\lambda}({\vec f})$ Ё
${{\hat a}_{\lambda}}^{*}({\vec f})$, 㤮ў«Ґвў®апойЁе
ЇҐаҐбв ­®ў®з­л¬ ᮮ⭮襭Ёп¬
$$ [{\hat a}_{\lambda}({\vec f}),
{\hat a}_{{\lambda}^{'}}({\vec f_{1}}) \rbrace
\quad = \quad 0,
$$
$$ [{{\hat a}_{{\lambda}}}^{*}({\vec f}),
{{\hat a}_{{\lambda}^{'}}}^{*}({\vec f_{1}}) \rbrace
\quad = \quad 0,
$$
$$ {\hat a}_{\lambda}({\vec f}),
{{\hat a}_{{\lambda}^{'}}}^{*}({\vec f_{1}}) \rbrace
\quad = \quad
{\delta}_{{\lambda},{\lambda}^{'}}
{\delta}_{{\vec f},{\vec f_{1}}}.
$$
Ћб­®ў­ п ¤Ё­ ¬ЁзҐбЄ п ЇҐаҐ¬Ґ­­ п Ї®«п бв ­®ўЁвбп ®ЇҐа в®а®¬
$$ {\hat A(\vec r)} \quad = \quad
\sum_{{\vec f},{\lambda}}B(\vec f){\vec e}_{\lambda}(\vec f)
({\hat a}_{\lambda}({\vec f})e^{i{\vec f}{\vec r}}
\quad + \quad
{{\hat a}_{\lambda}}^{*}({\vec f})e^{-i{\vec f}{\vec r}},
$$
  ®ЇҐа в®ал н­ҐаЈЁЁ Ё Ё¬Їг«мб  н«ҐЄв஬ Ј­Ёв­®Ј® Ї®«п ®ЇаҐ¤Ґ«повбп
ўла ¦Ґ­Ёп¬Ё
$$ {\hat {\cal H}} \quad = \quad
\sum_{{\vec f},{\lambda}}{\hbar}{\omega}(\vec f)
{\hat n}_{{\vec f},{\lambda}},
$$
$$ {\hat {\vec P}} \quad = \quad
\sum_{{\vec f},{\lambda}}{\hbar}{\vec f}
{\hat n}_{{\vec f},{\lambda}},
$$
Ј¤Ґ б ¬®б®Їа殮­­лҐ ®ЇҐа в®ал ${\hat n}_{{\vec f},{\lambda}}$,
а ў­л
$$ {\hat n}_{{\vec f},{\lambda}} \quad = \quad
{{\hat a}_{\lambda}}^{*}({\vec f}){\hat a}_{\lambda}({\vec f}).
$$






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