2 semestr / lect16n.tex

\magnification \magstep 1

\centerline{\bf ‹…Љ–€џ 16(4). 21.09.2001}

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\centerline {\bf ’…Ћђ€џ ‚Ћ‡Њ“™…Ќ€‰}

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‡ ¤ з  ® б®Ўб⢥­­ле §­ зҐ­Ёпе
$$ {\hat {\cal H}}{\Psi} \quad = \quad E{\Psi}
$$
б Їа®Ё§ў®«м­л¬ ®ЇҐа в®а®¬ ${\hat {\cal H}}$, Є Є Їа ўЁ«®,
­Ґа §аҐиЁ¬   ­ «ЁвЁзҐбЄЁ.

Ћ¤­ Є®, ЇаЁ аҐиҐ­ЁЁ ¬­®ЈЁе дЁ§ЁзҐбЄЁ ў ¦­ле § ¤ з ¤®бв в®з­®
¬Ґ­ҐҐ ®ЎйЁе а бб㦤Ґ­Ё©. Џгбвм Ј ¬Ё«мв®­Ё ­ бЁбвҐ¬л ¬®¦­®
ЇаҐ¤бв ўЁвм ў д®а¬Ґ
$$ {\hat {\cal H}} \quad = \quad
{\hat {\cal H}}_{0} \quad + \quad
{\lambda}{\hat {\cal H}}_{1},
\qquad {\lambda} << 1,
$$
Ј¤Ґ ${\hat {\cal H}}_{0}$ -- ®ЇҐа в®а б зЁбв® ¤ЁбЄаҐв­л¬ бЇҐЄв஬,
ЇаЁзҐ¬ ўбҐ ҐЈ® б®Ўб⢥­­лҐ §­ зҐ­Ёп Ё б®Ўб⢥­­лҐ ўҐЄв®ал Ё§ўҐбв­л.

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\centerline {\bf ‚Ћ‡Њ“™…Ќ€џ Ќ…‚›ђЋ†„…ЌЌЋѓЋ ‘Џ…Љ’ђЂ}

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‘­ з «  ¤«п Їа®бв®вл ЇаҐ¤Ї®«®¦Ё¬, ўбҐ б®Ўб⢥­­лҐ §­ зҐ­Ёп нв®
®ЇҐа в®а  ­Ґўл஦¤Ґ­л:
$$ {\hat {\cal H}}_{0}{\Phi}_{n}
\quad = \quad
E_{0n}{\Phi}_{n},
$$
$$ <{\Phi}_{m}|{\Phi}_{n}> \quad = \quad {\delta}_{mn},
\qquad
\forall {\Psi} \in H \quad
{\Psi} \quad = \quad
\sum_{n}{\Phi}_{n}<{\Phi}_{n}|{\Psi}>.
$$
ЏаЁ­Ё¬ п ў® ў­Ё¬ ­ЁҐ бўп§м ®ЇҐа в®а®ў ${\cal {\hat H}}$ Ё
${\hat {\cal H}}_{0}$, ¬®¦­® бЄ § вм, зв® $E_{0n}$ Ё ${\Phi}_{n}$ --
ЇаЁЎ«Ё¦Ґ­­лҐ аҐиҐ­Ёп в®з­®© § ¤ зЁ:
$$ {\Psi}_{n} \quad = \quad
{\Phi}_{n} \quad + \quad O(\lambda),
\qquad
E_{n} \quad = \quad
E_{0n} \quad + \quad O(\lambda).
$$
ЌҐваг¤­® гв®з­Ёвм бвагЄвгаг Ї®Їа ў®з­ле б« Ј Ґ¬ле ў нвЁе д®а¬г« е.
Њ®¦­® а ббг¤Ёвм в Є: Ї®бЄ®«мЄг ®ЇҐа в®а $H$  ­ «ЁвЁзҐбЄЁ § ўЁбЁв ®в
Ї а ¬Ґва  $\lambda$, ¬®¦­® ®¦Ё¤ вм, зв® нв® Ўг¤Ґв бЇа ўҐ¤«Ёў® Ё ¤«п
б®Ўб⢥­­ле §­ зҐ­Ё© Ё б®Ўб⢥­­ле ўҐЄв®а®ў. Џ®н⮬㠬®¦­® ЇаҐ¤бв ўЁвм
ўҐ«ЁзЁ­л $E_{n}$ Ё ${\Psi}_{n}$ ў д®а¬Ґ
$$ E_{n} \quad = \quad E_{0n}
\quad + \quad {\lambda}E_{1n}
\quad + \quad O({\lambda}^{2}),
$$
$$ {\Psi}_{n} \quad = \quad {\Phi}_{n} \quad + \quad
{\lambda}{{\Phi}^{(1)}}_{n} \quad + \quad
O({\lambda}^{2}).
$$
Џ®¤бв ў«пп нвЁ ўла ¦Ґ­Ёп ў га ў­Ґ­Ёп ¤«п б®Ўб⢥­­ле §­ зҐ­Ё©,
Ї®«гзЁ¬ ᮮ⭮襭ЁҐ
$$ {\hat {\cal H}}_{0}{\Phi}_{0} \quad + \quad
{\lambda}{\hat {\cal H}}_{0}{{\Phi}^{(1)}}_{n}
\quad + \quad
{\lambda}{\hat {\cal H}}_{1}{\Phi}_{n}
\quad = \quad
$$
$$ E_{0n}{\Phi}_{n} \quad + \quad
{\lambda}E_{n0}{{\Phi}^{(1)}}_{n}
\quad + \quad
{\lambda}E_{1n}{\Phi}_{n}
\quad + \quad O({\lambda}^{2}).
$$
Џ®бЄ®«мЄг ўҐЄв®ал ${\Phi}_{n}$ -- аҐиҐ­Ёп га ў­Ґ­Ё©
$$ ({\hat {\cal H}}_{0} - E_{0n}){\Phi}_{n}
\quad = \quad 0,
$$
в® Ї®«г祭­®Ґ а ўҐ­бвў® 㤮ў«Ґвў®аЁвбп б в®з­®бвмо ¤® б« Ј Ґ¬ле
ўв®а®Ј® Ї®ап¤Є  Ї® ${\lambda}$, Ґб«Ё ўҐЄв®а ${{\Phi}^{1}}_{n}$
Ўг¤Ґв аҐиҐ­ЁҐ¬ га ў­Ґ­Ёп
$$ ({\hat {\cal H}}_{0} - E_{0n}){{\Phi}^{(1)}}_{n}
\quad = \quad F_{1},
$$
Ј¤Ґ
$$ F_{1} \quad = \quad E_{1n}{\Phi}_{n}
\quad - \quad {\hat {\cal H}}_{1}{\Phi}_{n}.
$$
“¬­®¦Ёў ®ЎҐ з бвЁ га ў­Ґ­Ёп бЄ «па­® ­  ўҐЄв®а ${\Phi}_{n}$, Ї®«гзЁ¬
$$ <{\Phi}_{n}|F_{1}> \quad = \quad
<{\Phi}_{n}|({\hat {\cal H}}_{0} - E_{0n}){{\Phi}^{(1)}}_{n}>
\quad = \quad
<({\hat {\cal H}}_{0} - E_{0n}){\Phi}_{n}|{{\Phi}^{(1)}}_{n}>
\quad = \quad 0.
$$
’ ЄЁ¬ ®Ўа §®¬, а ўҐ­бвў 
$$ <{\Phi}_{n}|F_{1}> \quad = \quad 0
$$
ЇаҐ¤бв ў«пов б®Ў®© гб«®ўЁп а §аҐиЁ¬®бвЁ га ў­Ґ­Ё© ЇҐаў®Ј®
ЇаЁЎ«Ё¦Ґ­Ёп Є в®з­®¬г аҐиҐ­Ёо. Ћ­Ё ®ЇаҐ¤Ґ«пов §­ зҐ­Ёп Ї®Їа ў®Є Є
га®ў­п¬ н­ҐаЈЁЁ ў ЇҐаў®¬ Ї®ап¤ЄҐ Ї® Ї а ¬Ґваг $\lambda$:
$$ E_{1n} \quad = \quad <{\Phi}_{n}|{\hat {\cal H}}_{1}|{\Phi}_{n}>.
$$
ђҐиҐ­ЁҐ га ў­Ґ­Ёп
$$ ({\hat {\cal H}}_{0} - E_{0n}){{\Phi}^{(1)}}_{n}
\quad = \quad F_{1},
$$
Ґб«Ё ®­® бгйҐбвўгҐв, ®ЇаҐ¤Ґ«Ґ­® б в®з­®бвмо ¤® б« Ј Ґ¬®Ј®,
Їа®Ї®ажЁ®­ «м­®Ј® ўҐЄв®аг ${\Phi}_{n}$. ђ бЇ®ап¤Ё¬бп Ё¬ в Є, зв®Ўл
ўҐЄв®ал ${\Phi}_{n}$ Ё ${{\Phi}^{(1)}}_{n}$ Ўл«Ё ®ав®Ј®­ «м­л¬Ё ¤агЈ
¤агЈг. ќв® ®§­ з Ґв, зв® ¤®«¦­® Ўлвм бЇа ўҐ¤«Ёўл¬ а §«®¦Ґ­ЁҐ
$$ {{\Phi}^{(1)}}_{n} \quad = \quad
\sum_{m \not= n}{\Phi}_{m}C_{mn}.
$$
Џ®¤бв ­®ўЄ  нв®Ј® а §«®¦Ґ­Ёп ў га ў­Ґ­ЁҐ ¤«п ${{\Phi}^{(1)}}_{n}$
ЇаЁў®¤Ёв Є пў­л¬ ўла ¦Ґ­Ёп¬ Є®нддЁжЁҐ­в®ў $C_{mn}$:
$$ C_{mn} \quad = \quad
{<{\Phi}_{m}|{\hat {\cal H}}_{1}|{\Phi}_{n}>
\over E_{0n} - E_{0m}}.
$$
‡ ¬ҐвЁ¬, зв® Є®нддЁжЁҐ­вл $C_{mn}$  ­вЁбЁ¬¬ҐваЁз­л Ї® Ё­¤ҐЄб ¬ $m$ Ё
$n$. ‚ бЁ«г нв®Ј® ўҐЄв®ал
$$ {{\Psi}^{(1)}}_{n} \quad = \quad {\Phi}_{n}
\quad + \quad
{\lambda}{\sum_{m \not= n}{\Phi}_{m}{C_{mn}}}
$$
б в®з­®бвмо ¤® ўв®а®Ј® Ї®ап¤Є  Ї® $\lambda$ Ї®Ї а­® ®ав®Ј®­ «м­л:
$$ <{{\Psi}^{1}}_{m}|{{\Psi}^{1}}_{n}>
\quad = \quad
{\delta}_{mn} \quad + \quad O({\lambda}^{2}).
$$
€­ зҐ Ј®ў®ап, ЇаҐ®Ўа §®ў ­ЁҐ
$$ {{\Psi}^{0}}_{n} \quad = \quad {\Phi}_{n}
\quad \Longrightarrow \quad {{\Psi}^{1}}_{n}
$$
г­Ёв а­® б в®з­®бвмо ¤® ЇҐаў®Ј® Ї®ап¤Є  Ї® ${\lambda}$.

Џ®«Ґ§­® Ї®¬­Ёвм, зв® Їа®жҐ¤га  ўлзЁб«Ґ­Ёп б®Ўб⢥­­ле §­ зҐ­Ё© Ё
б®Ўб⢥­­ле ўҐЄв®а®ў ў ЇҐаў®¬ ЇаЁЎ«Ё¦Ґ­ЁЁ Ї® Ї а ¬Ґваг $\lambda$
ҐбвҐб⢥­­л¬ ®Ўа §®¬ а бЇ « бм ­  ¤ў  нв Ї :

1) б­ з «  ўлпб­п«Ёбм гб«®ўЁп а §аҐиЁ¬®бвЁ га ў­Ґ­Ё© ЇаЁЎ«Ё¦Ґ­Ёп;
ў १г«мв вҐ Ўл«  ­ ©¤Ґ­  Ї®Їа ўЄ  Є га®ў­п¬ н­ҐаЈЁЁ $E_{0n}$ ў
ЇҐаў®¬ Ї®ап¤ЄҐ Ї® ўҐ«ЁзЁ­Ґ $\lambda$;

2) Ї®б«Ґ нв®Ј® ®Є § «®бм ў®§¬®¦­л¬ ­ ©вЁ ўҐЄв®а б®бв®п­Ёп, Є®в®а®Ґ
®Є §лў Ґвбп бв жЁ®­ а­®¬ ЇаЁ гзҐвҐ нд䥪⮢ ЇҐаў®Ј® Ї®ап¤Є ,
бўп§ ­­ле б н­ҐаЈЁҐ© ${\lambda}{\hat {\cal H}}_{1}$.

ќв  б奬  ®бв ­Ґвбп ўҐа­®© Ё ЇаЁ гзҐвҐ ў®§¬г饭Ёп ўлбиЁе Ї®ап¤Є®ў.
ђ бᬮваЁ¬, ­ ЇаЁ¬Ґа, б奬㠢в®а®Ј® Ї®ап¤Є . ‚ н⮬ б«гз Ґ ў
га ў­Ґ­Ёп ¤«п б®Ўб⢥­­ле §­ зҐ­Ё© б«Ґ¤гҐв Ї®¤бв ўЁвм в ЄЁҐ
а §«®¦Ґ­Ёп б®Ўб⢥­­ле §­ зҐ­Ё© Ё б®Ўб⢥­­ле ўҐЄв®а®ў:
$$ E_{n} \quad = \quad E_{0n}
\quad + \quad {\lambda}E_{1n}
\quad + \quad {\lambda}^{2}E_{2n}
\quad + \quad O({\lambda}^{3}),
$$
$$ {\Psi}_{n} \quad = \quad {\Phi}_{n} \quad + \quad
{\lambda}{{\Phi}^{(1)}}_{n} \quad + \quad
{{\lambda}^{2}}{{\Phi}^{(2)}}_{n} \quad + \quad
O({\lambda}^{3}).
$$
…б«Ё ў Є зҐб⢥ ўҐ«ЁзЁ­ $E_{1n}$ Ё ${{\Phi}^{(1)}}_{n}$ ў§пвм
§­ зҐ­Ёп, ­ ©¤Ґ­­лҐ ЇаЁ аҐиҐ­ЁЁ га ў­Ґ­Ё© ў ЇҐаў®¬ Ї®ап¤ЄҐ Ї®
$\lambda$, в® га ў­Ґ­ЁҐ ${\hat {\cal H}}{\Psi} = E{\Psi}$ ЇаЁ¬Ґв
ўЁ¤:
$$ {\lambda}^{2}({\hat {\cal H}}_{0}{{\Phi}^{(2)}}_{n}
\quad + \quad {\hat {\cal H}}_{1}{{\Phi}^{(1)}}_{n})
\quad = \quad
$$
$$ {\lambda}^{2}(E_{0n}{{\Phi}^{(2)}}_{n} \quad + \quad
E_{1n}{{\Phi}^{(1)}}_{n} \quad + \quad E_{2n}{\Phi}_{n})
\quad + \quad O({\lambda}^{3}).
$$
ЏаҐ­ҐЎаҐЈ п Ї®б«Ґ¤­Ё¬ б« Ј Ґ¬л¬ ў Їа ў®© з бвЁ, Ї®«гзЁ¬ га ў­Ґ­Ёп
®ЇаҐ¤Ґ«пойЁҐ Ї®Їа ўЄЁ Є б®Ўб⢥­­®© н­ҐаЈЁЁ Ё б®Ўб⢥­­®¬г ўҐЄв®аг
ў® ўв®а®¬ Ї®ап¤ЄҐ Ї® ўҐ«ЁзЁ­Ґ ${\lambda}$:
$$ ({\hat {\cal H}}_{0} - E_{0n}){{\Phi}^{(2)}}_{n}
\quad = \quad F_{2},
$$
$$ F_{2} \quad = \quad E_{2n}{\Phi}_{n}
\quad + \quad E_{1n}{{\Phi}^{(1)}}_{n}
\quad - \quad {{\hat {\cal H}}_{1}}{{\Phi}^{(1)}}_{n}.
$$
“б«®ўЁҐ а §аҐиЁ¬®бвЁ нвЁе га ў­Ґ­Ё©, $<{\Phi}_{n}|F_{2}> = 0$,
®ЇаҐ¤Ґ«пҐв Ї®Їа ўЄг Є га®ў­о н­ҐаЈЁЁ:
$$ E_{2n} \quad = \quad
<{\Phi}_{n}|{\hat {\cal H}}_{1}|{{\Phi}^{(1)}}_{n}>.
$$
Џ®¤бв ­®ўЄ  пў­®Ј® §­ зҐ­Ёп ўҐЄв®а  ${{\Phi}^{(1)}}_{n}$ ЇаЁў®¤Ёв Є
д®а¬г«Ґ
$$ E_{2n} \quad = \quad \sum_{m \not= n}
{|<{\Phi}_{m}|{\hat {\cal H}}_{1}|{\Phi}_{n}>|^{2}
\over E_{0n} - E_{0m}}.
$$
ЌҐваг¤­® Ї®Є § вм, зв® в ЄЁ¬ ¦Ґ ®Ўа §®¬ ¬®¦­® Ї®«гзЁвм Ї®Їа ўЄЁ
«оЎ®Ј® Ї®ап¤Є  Ї® $\lambda$. Љ ¦¤л© а § Ўг¤Ґв Ї®«гз вмбп га ў­Ґ­ЁҐ
ўЁ¤ 
$$ ({\hat {\cal H}}_{0} - E_{0n}){{\Phi}^{(k)}}_{n}
\quad = \quad F_{k},
$$
ў Є®в®а®¬ дг­ЄжЁп $F_{k}$ ᮤҐа¦Ёв ўбҐЈ® «Ёим ­ҐЁ§ўҐбв­л© Ї а ¬Ґва --
§­ зҐ­ЁҐ н­ҐаЈЁЁ $E_{kn}$. “б«®ўЁҐ а §аҐиЁ¬®бвЁ га ў­Ґ­Ёп нв®в
Ї а ¬Ґва дЁЄбЁагҐв, Ї®б«Ґ 祣® ­ е®¤Ёвбп Ё дг­ЄжЁп
${{\Phi}^{(k)}}_{n}$.

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\centerline {\bf ‚Ћ‡Њ“™…Ќ€џ ‚›ђЋ†„…ЌЌЋѓЋ ‘Џ…Љ’ђЂ}

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‘«гз © ўл஦¤Ґ­­®Ј® бЇҐЄва  Ј ¬Ё«мв®­Ё ­  ${\hat {\cal H}}_{0}$
вॡгҐв Ў®«ҐҐ вй вҐ«м­®Ј®  ­ «Ё§ . ЏаҐ¤Ї®«®¦Ё¬, зв® ®¤Ё­ Ё§ Ў §Ёб®ў
ў Їа®бва ­б⢥ б®бв®п­Ё© ®Ўа §гов б®Ўб⢥­­лҐ ўҐЄв®ал нв®Ј®
Ј ¬Ё«мв®­Ё ­ :
$$ {\hat {\cal H}}_{0}{\Phi}_{n \alpha}
\quad = \quad E_{0n}{\Phi}_{n \alpha},
\quad \alpha = 1,...,s_{n}.
$$
Џ®¤бв ­®ўЄ  ў га ў­Ґ­ЁҐ
$$ ({\hat {\cal H}}_{0} \quad + \quad
{\lambda}{\hat {\cal H}}_{1}){\Psi}_{n \alpha}
\quad = \quad
E_{n \alpha}{\Psi}_{n \alpha}
$$
ўла ¦Ґ­Ё©
$$ E_{n \alpha} \quad = \quad
E_{0n} \quad + {\lambda}E_{1n \alpha}
\quad + \quad O({\lambda}^{2}),
$$
$$ {\Psi}_{n \alpha} \quad = \quad
{\Phi}_{n \alpha} \quad + {{\Phi}^{(1)}}_{n \alpha}
\quad + \quad O({\lambda}^{2})
$$
ЇаЁў®¤Ёв Є ᮮ⭮襭Ёп¬
$$ ({\hat {\cal H}}_{0} - E_{0n}){\Phi}_{n \alpha}
\quad + \quad
{\lambda}(({\hat {\cal H}}_{0} - E_{0n}){\Phi}^{(1)}_{n \alpha}
\quad + \quad {\hat {\cal H}}_{1}{\Phi}_{n \alpha}
\quad - \quad E_{1n \alpha}{\Phi}_{n \alpha})
\quad + \quad O({\lambda}^{2}) \quad = \quad 0.
$$
ЏаҐ­ҐЎаҐ¦Ґ¬ б« Ј Ґ¬л¬Ё ўв®а®Ј® Ї®ап¤Є  Ї® $\lambda$ Ё ЇаЁа ў­пҐ¬ Є
­г«о Ї® ®в¤Ґ«м­®бвЁ б« Ј Ґ¬лҐ ­г«Ґў®Ј® Ё ЇҐаў®Ј® Ї®ап¤Є®ў Ї®
$\lambda$.
„ў  га ў­Ґ­Ёп
$$ ({\hat {\cal H}}_{0} - E_{0n}){\Phi}_{n \alpha}
\quad = \quad 0
$$
Ё
$$ ({\hat {\cal H}}_{0} - E_{0n}){\Phi}^{(1)}_{n \alpha}
\quad = \quad
E_{1n \alpha}{\Phi}_{n \alpha}
\quad - \quad
{\hat {\cal H}}_{1}{\Phi}_{n \alpha},
$$
Є Є Ё ў б«гз Ґ ­Ґўл஦¤Ґ­­®Ј® бЇҐЄва , ¤®«¦­л ®ЇаҐ¤Ґ«Ёвм Ї®Їа ўЄЁ Є
га®ў­п¬ н­ҐаЈЁЁ Ё Є ўҐЄв®а ¬ бв жЁ®­ а­ле б®бв®п­Ё©. Ћ¤­ Є®, ­  н⮬
и ЈҐ вॡгҐвбп ЇаҐ®¤®«Ґвм ҐйҐ ®¤­г ваг¤­®бвм. “б«®ўЁҐ а §аҐиЁ¬®бвЁ
га ў­Ґ­Ё© ЇҐаў®Ј® Ї®ап¤Є  Ї® ${\lambda}$,
$$ <{\Phi}_{n \beta}|E_{1n \alpha}{\Phi}_{n \alpha}>
\quad = \quad
<{\Phi}_{n \beta}|{\hat {\cal H}}_{1}|{\Phi}_{n \alpha}>,
$$
ў®®ЎйҐ Ј®ў®ап, ў­гв७­Ґ Їа®вЁў®аҐзЁў®, Ї®бЄ®«мЄг «Ґў п з бвм
а ўҐ­бвў  ўбҐЈ¤  ¤Ё Ј®­ «м­  Ї® Ё­¤ҐЄб ¬ $\alpha$ Ё $\beta$,  
Їа ў п з бвм ў б«гз Ґ Їа®Ё§ў®«м­ле аҐиҐ­Ё© ­г«Ґў®Ј® Ї®ап¤Є  ¬®¦Ґв
ᮤҐа¦ вм Ё ­Ґ¤Ё Ј®­ «м­лҐ б« Ј Ґ¬лҐ. —в®Ўл Ё§ЎҐ¦ вм Їа®вЁў®аҐзЁп,
бд®а¬г«Ёа㥬 Їа ўЁ«® ўлЎ®а  ўҐЄв®а®ў ${\Phi}_{n \alpha}$ б«Ґ¤гойЁ¬
®Ўа §®¬:

{\leftskip 1.5 true cm \noindent ўҐЄв®ал ${\Phi}_{n \alpha}$ ¤®«¦­л
㤮ў«Ґвў®апвм гб«®ўЁп¬

$$({\hat {\cal H}}_{0} - E_{0n}){\Phi}_{n \alpha} \quad = \quad 0,
$$
Ё
$$ <{\Phi}_{n \alpha}|{\hat {\cal H}}_{1}|{\Phi}_{n \beta}>
\quad = \quad
h_{\alpha}{\delta}_{\alpha \beta}.
$$
}

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

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‘дҐаЁзҐбЄЁҐ Ј а¬®­ЁЄЁ $Y_{lm}$ ¬®¦­® ®ЇаҐ¤Ґ«Ёвм ᮮ⭮襭Ёп¬Ё
$$ {\hat l}_{3}Y_{lm} \quad = \quad mY_{lm},
$$
$$ {\hat l}_{\pm}Y_{lm} \quad = \quad
\sqrt{(l \mp m)(l \pm m + 1)}Y_{l,m \pm 1}.
$$
…б«Ё ${\hat l}_{\alpha}$ ॠ«Ё§говбп Є Є ¤ЁддҐаҐ­жЁ «м­лҐ ®ЇҐа в®ал,
в® нвЁ 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«­ЁҐ ­ҐЇаЁў®¤Ё¬ле ⥭§®а®ў.

{\leftskip 1.5 true cm \noindent ЌҐЇаЁў®¤Ё¬л© ⥭§®а а ­Ј  j -- нв®
б®ў®ЄгЇ­®бвм 2j + 1 ®ЇҐа в®а®ў
$$ {\hat T}_{jm}, \quad -j \leq m \leq j,
$$
㤮ў«Ґвў®апойЁе ЇҐаҐбв ­®ў®з­л¬ ᮮ⭮襭Ёп¬
$$ [{\hat j}_{3}, {\hat T}_{jm}] \quad = \quad m{\hat T}_{jm},
$$
$$ [{\hat j}_{\pm}, {\hat T}_{jm}] \quad = \quad
\sqrt{(j \mp m)(j \pm m + 1)}{\hat T}_{j,m \pm 1}.
$$
}
‘Їа ўҐ¤«Ёўл Їа ўЁ«  ®вЎ®а :

$$ 1) \qquad
<{\gamma}_{1},j_{1},m_{1}|{\hat T}_{jm}|{\gamma}_{2},j_{2},m_{2}>
\quad \not= \quad 0, \quad Ґб«Ё \quad m_{1} \not= m + m_{2},
$$
$$ 2) \qquad
<{\gamma}_{1},j_{1},m_{1}|{\hat T}_{jm}|{\gamma}_{2},j_{2},m_{2}>
\quad = \quad
<{\gamma}_{1},j_{1}|T_{j}|{\gamma}_{2},j_{2}>
<j_{2}m_{2}jm|j_{2}jj_{1}m_{1}>,
$$
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 ¬®¦­® 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 ­г«о. ‚ н⮬ б«гз Ґ бЇа ўҐ¤«Ёў  д®а¬г« 
$$
<{\gamma}_{1},j_{1},m_{1}|{\hat T}_{jm}|{\gamma}_{2},j_{2},m_{2}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j_{1},j_{2})
<{\gamma}_{1},j_{1},m_{1}|{\hat Q}_{jm}|{\gamma}_{2},j_{2},m_{2}>.
$$
‘гйҐб⢥­­®, зв® Ї®¤®Ў­лҐ ᮮ⭮襭Ёп бЇа ўҐ¤«Ёўл Ё ¤«п ¤ҐЄ ав®ўле
б®бв ў«ойЁе ­ҐЇаЁў®¤Ё¬ле ⥭§®а®ў.

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 .

„ҐЄ ав®ўл б®бв ў«пойЁҐ ўҐЄв®а  $A_{\alpha}$ 㤮ў«Ґвў®апов
ЇҐаҐбв ­®ў®з­л¬ ᮮ⭮襭Ёп¬
$$ [J_{\alpha}, A_{\beta}]
\quad = \quad
i{\epsilon}_{\alpha \beta \gamma}A_{\gamma}.
$$
’аЁ 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  ¬®¦­® 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
$$ A_{\alpha \beta} = {1 \over 3}{\delta}_{\alpha \beta}TrA
\quad + \quad
\Big(A_{\alpha \beta} - {1 \over 3}{\delta}_{\alpha \beta}TrA \Big).
$$
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 зЁб« ¬Ё Ё
ЇаҐ¤бв ў«пҐв б®Ў®© ¤ҐЄ ав®ў ­ҐЇаЁў®¤Ё¬л© ⥭§®а ўв®а®Ј® а ­Ј .
ЌҐваг¤­® ­ ©вЁ пў­лҐ д®а¬г«л, ЇҐаҐў®¤пйЁҐ ¤ҐЄ ав®ўл б®бв ў«пойЁҐ ў
б®бв ў«пойЁҐ ${\hat T}_{2,m}$:
$$ T_{2,0}
= -\sqrt{3 \over 2}T_{33},
\quad
T_{2, \pm 1}
=
\pm(T_{13} \pm iT_{23}),
\quad
T_{2, \pm 2}
=
-{1 \over 2}(T_{11} - T_{22} \pm 2iT_{12}).
$$
$$ T_{\alpha \beta} = T_{\beta \alpha}, \quad
\sum_{\alpha}T_{\alpha \alpha} = 0.
$$

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
$$ <{\gamma}_{1},j,m_{1}|\hat T|{\gamma}_{2},j,m_{2}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j) <j,m_{1}|{\hat E}|j,m_{2}>
\quad = \quad
C({\gamma}_{1},{\gamma}_{2},j){\delta}_{m_{1}m_{2}}.
$$













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