2 semestr / lect14n.tex

\magnification \magstep 1

\centerline{\bf ‹…Љ–€џ 14. 12.09.2001}

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\centerline {\bf ЊЌЋѓЋ—Ђ‘’€—Ќ›… ‘€‘’…Њ›}

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—в®Ўл ®ЇЁб вм б®ЎлвЁп ў бЁб⥬Ґ, б®бв®п饩 Ё§ ¬­®ЈЁе з бвЁж,
­ ЇаЁ¬Ґа, $N$ н«ҐЄва®­®ў, $M$ Їа®в®­®ў, ... ­г¦­®

{\leftskip 1 true cm \noindent 1) ўлЎа вм ®ЇҐа в®ал
${\hat Q}_{e}$, ${\hat Q}_{p}$,... , ᮮ⢥бвўгойЁҐ
¤Ё­ ¬ЁзҐбЄЁ¬ ЇҐаҐ¬Ґ­­л¬ бЁб⥬л;

}

{\leftskip 1 true cm \noindent 2) гв®з­Ёвм бвагЄвгаг ЈЁ«мЎҐав®ў®Ј®
Їа®бва ­бвў , ў Є®в®а®¬ ¤Ґ©бвўгов нвЁ ®ЇҐа в®ал.

}

\noindent Џ®пб­Ё¬, зв® нв® ®§­ з Ґв. ЏаҐ¤Ї®«®¦Ё¬, зв® ¬л Ё¬ҐҐ¬ ¤Ґ«®
б бЁб⥬®© Ё§ $N$ н«ҐЄва®­®ў. Њ®¦­® ®ЇаҐ¤Ґ«Ёвм Їа®бва ­бвў®
б®бв®п­Ё© в Є®© бЁбвҐ¬л Є Є б®ў®ЄгЇ­®бвм Єў ¤а вЁз­® Ё­вҐЈаЁа㥬ле
дг­ЄжЁ© $\Psi(q_{1},...,q_{N})$:
$$ {\cal H} \quad = \quad \lbrace
{\Psi}(q_{1},...,q_{N}), \quad
{\int}dq_{1}...dq_{N}|{\Psi}(q_{1},...,q_{n})|^{2}
\quad \leq \quad \infty \rbrace
$$
б® бЄ «па­л¬ Їа®Ё§ўҐ¤Ґ­ЁҐ¬
$$ <{\Psi}_{1}|{\Psi}_{2}> \quad = \quad
{\int}dq{\Psi}^{*}_{1}(q){\Psi}_{2}(q).
$$
Ћ¤­ Є®, в Є®Ґ ®ЇаҐ¤Ґ«Ґ­ЁҐ ўл§лў Ґв б«Ґ¤гойЁ© ў®Їа®б: Ґб«Ё г
­ б ­Ґв Є Є®©-«ЁЎ® ЇаЁзЁ­л ¤«п ўлЎ®а  в®Ј® Ё«Ё Ё­®Ј® Ї®ап¤Є  ў
­г¬Ґа жЁЁ н«ҐЄва®­®ў, в® § ¤ ў дг­ЄжЁо ${\Psi}(q_{1},...,q_{N})$,
¬л ў ®ЎйҐ¬ б«гз Ґ ®ЇаҐ¤Ґ«пҐ¬ $N!$ дг­ЄжЁ©, Ї®«гз ойЁебп Ё§
ЇҐаў®­ з «м­®© Ї®б«Ґ ЇҐаҐбв ­®ў®Є ЇҐаҐ¬Ґ­­ле $q_{1},...,q_{N}$.
Љ Єго ¦Ґ дг­ЄжЁо б«Ґ¤гҐв ўлЎа вм?

Ќ  нв®в ў®Їа®б ў 1925 Ј®¤г, ҐйҐ ¤® ®вЄалвЁп ѓ ©§Ґ­ЎҐаЈ , ®вўҐвЁ«
Џ г«Ё, бд®а¬г«Ёа®ў ў {\bf ЇаЁ­жЁЇ ЁбЄ«о祭Ёп}. Џ®бЄ®«мЄг ў в® ўаҐ¬п
ҐйҐ ­Ґ Ўл«® Ї®­пвЁ© ®ЇҐа в®а®ў, ўҐЄв®а®ў б®бв®п­Ёп, ... Џ г«Ё
Ј®ў®аЁ« ­  п§лЄҐ Єў ­в®ўле зЁбҐ«. Ќ ¬ 㤮Ў­ҐҐ Ј®ў®аЁвм ® ЇаЁ­жЁЇҐ
ЁбЄ«о祭Ёп ў вҐа¬Ё­ е гб«®ўЁ©, Є®в®ал¬ ¤®«¦­  㤮ў«Ґвў®апвм дг­ЄжЁп
${\Psi}(q_{1},...,q_{N})$.

{\leftskip 1 true cm \noindent Џа®бва ­бвў® б®бв®п­Ё© бЁб⥬л N
®Ўа §гов Ї®«­®бвмо  ­вЁбЁ¬¬ҐваЁз­лҐ ў®«­®ўлҐ дг­ЄжЁЁ
${\Psi}(q_{1},...,q_{N})$, 㤮ў«Ґвў®апойЁҐ б®®в­иҐ­Ёп¬
$$ {\Psi}(q_{1},...,q_{i},...,q_{j},...,q_{N})
\quad = \quad
- \quad {\Psi}(q_{1},...,q_{j},...,q_{i},...,q_{N})
$$
¤«п «оЎ®© Ї ал ЇҐаҐ¬Ґ­­ле $q_{i}$, $q_{j}$.

}

ђ бᬮваЁ¬ б«гз © ¤ўге н«ҐЄва®­®ў. Џгбвм
$\lbrace {\phi}_{a}(q) \rbrace$ --- Ў §Ёб ў Їа®бва ­б⢥ б®бв®п­Ё©
®¤­®Ј® н«ҐЄва®­ . Џа®Ё§ў®«м­л© ўҐЄв®а б®бв®п­Ёп бЁбвҐ¬л Ё§ ¤ўге
н«ҐЄва®­®ў ¬®¦­® ЇаҐ¤бв ўЁвм ў д®а¬Ґ
$$ {\Psi}(q_{1},q_{2}) \quad = \quad
{\sum}_{a_{1},a_{2}}{\phi}_{a_{1}}(q_{1}){\phi}_{a_{2}}(q_{2})
C_{a_{1}a_{2}}.
$$
ЏҐаҐ¬Ґ­­ п $q$ ўЄ«оз Ґв ў бҐЎп Їа®бва ­б⢥­­го з бвм, ${\vec r}$ Ё
ЇаЁ­Ё¬ ойго ¤ў  §­ зҐ­Ёп бЇЁ­®ўго ЇҐаҐ¬Ґ­­го $\sigma = \pm 1$. ‘।Ё
Ў §Ёб®ў ${\phi}_{a}(q)$ Ґбвм дг­ЄжЁЁ, а бЇ ¤ ойЁҐбп ­  Їа®Ё§ўҐ¤Ґ­Ёп
$$ {\phi}(q) \quad = \quad {\phi}(\vec r){\chi}(\sigma).
$$
‘ЇЁ­®ўго дг­ЄжЁо, ў бў®о ®зҐаҐ¤м, ¬®¦­® ЇаҐ¤бв ўЁвм ў д®а¬Ґ
$$ {\chi}(\sigma) \quad = \quad
b_{1}{\xi}(\sigma) \quad + \quad b_{2}{\eta}(\sigma),
$$
Ї®б«Ґ 祣® дг­ЄжЁп ${\Psi}$ ЇаЁ¬Ґв ўЁ¤
$$ {\Psi}(q_{1},q_{2}) \quad = \quad
f_{++}({\vec r}_{1}, {\vec r}_{2})
{\xi}({\sigma}_{1}){\xi}({\sigma}_{2})
\quad + \quad
f_{+-}({\vec r}_{1}, {\vec r}_{2})
{\xi}({\sigma}_{1}){\eta}({\sigma}_{2})
\quad + \quad
$$
$$
f_{-+}({\vec r}_{1}, {\vec r}_{2})
{\eta}({\sigma}_{1}){\xi}({\sigma}_{2})
\quad + \quad
f_{--}({\vec r}_{1}, {\vec r}_{2})
{\eta}({\sigma}_{1}){\eta}({\sigma}_{2}).
$$
…б«Ё дг­ЄжЁп ${\Psi}$  ­вЁбЁ¬¬ҐваЁз­  Ї® ЇҐаҐ¬Ґ­­л¬ $q$, в® дг­ЄжЁЁ
$f$ в Є®ўл, зв®
$$ f_{++}(2,1) \quad = \quad -f_{++}(1,2), \qquad
f_{+-}(2,1) \quad = \quad -f_{-+}(1,2), \qquad
$$
$$ f_{-+}(2,1) \quad = \quad -f_{+-}(1,2), \qquad
f_{--}(2,1) \quad = \quad -f_{--}(1,2).
$$
€§ Їа®Ё§ўҐ¤Ґ­Ё© ${\xi}{\eta}$ в Є¦Ґ ¬®¦­® Ї®бва®Ёвм бЁ¬¬ҐваЁз­лҐ Ё
 ­вЁбЁ¬¬ҐваЁз­лҐ Є®¬ЎЁ­ жЁЁ
$$ {\chi}_{11}({\sigma}_{1},{\sigma}_{2}) \quad = \quad
{\xi}({\sigma}_{1}){\xi}({\sigma}_{2}),
$$
$$ {\chi}_{10}({\sigma}_{1},{\sigma}_{2}) \quad = \quad
{1 \over \sqrt{2}}
({\xi}({\sigma}_{1}){\eta}({\sigma}_{2})
\quad + \quad
{\xi}({\sigma}_{2}){\eta}({\sigma}_{1})),
$$
$$ {\chi}_{1,-1}({\sigma}_{1},{\sigma}_{2}) \quad = \quad
{\eta}({\sigma}_{1}){\eta}({\sigma}_{2}),
$$
$$ {\chi}_{00}({\sigma}_{1},{\sigma}_{2}) \quad = \quad
{1 \over \sqrt{2}}
({\xi}({\sigma}_{1}){\eta}({\sigma}_{2})
\quad - \quad
({\xi}({\sigma}_{2}){\eta}({\sigma}_{1})),
$$
ђ §«®¦Ґ­ЁҐ $\Psi(q_{1},q_{2})$ Ї® дг­ЄжЁп¬ ${\chi}$ Ё¬ҐҐв ўЁ¤
$$ {\Psi}(q_{1},q_{2}) \quad = \quad
f_{++}({\vec r_{1}},{\vec r_{2}})
{\chi}_{11}({\sigma}_{1},{\sigma}_{2}) \quad + \quad
$$
$$ {1 \over \sqrt{2}}
(f_{+-}({\vec r_{1}},{\vec r_{2}}) \quad + \quad
f_{-+}({\vec r_{1}},{\vec r_{2}}))
{\chi}_{10}({\sigma}_{1},{\sigma}_{2}) \quad + \quad
$$
$$ f_{--}({\vec r_{1}},{\vec r_{2}})
{\chi}_{1,-1}({\sigma}_{1},{\sigma}_{2}) \quad + \quad
{1 \over \sqrt{2}}
(f_{+-}({\vec r_{1}},{\vec r_{2}}) \quad - \quad
f_{-+}({\vec r_{1}},{\vec r_{2}}))
{\chi}_{00}({\sigma}_{1},{\sigma}_{2}).
$$
‚ н⮬ а §«®¦Ґ­ЁЁ Є ¦¤®Ґ б« Ј Ґ¬®Ґ б®бв®Ёв Ё§ Їа®Ё§ўҐ¤Ґ­Ёп
ᮬ­®¦ЁвҐ«Ґ© ®ЇаҐ¤Ґ«Ґ­­®© бЁ¬¬ҐваЁЁ.
”г­ЄжЁЁ ${\chi}_{s{\mu}}({\sigma}_{1},{\sigma}_{2})$ --- нв®
б®Ўб⢥­­лҐ ўҐЄв®ал ®ЇҐа в®а®ў Єў ¤а в  Ї®«­®Ј® бЇЁ­  Ё Їа®ҐЄжЁЁ
Ї®«­®Ј® бЇЁ­  ­  ®бм $0z$:
$$ ({\vec S}^{2}{\chi}_{1{\mu}})({\sigma}_{1},{\sigma}_{2})
\quad = \quad
2{\chi}_{1{\mu}}({\sigma}_{1},{\sigma}_{2}),
\qquad
({\vec S}_{3}{\chi}_{1{\mu}})({\sigma}_{1},{\sigma}_{2})
\quad = \quad
{\mu}({\vec S}_{3}{\chi}_{1{\mu}}({\sigma}_{1},{\sigma}_{2}),
$$
$$ ({\vec S}^{2}{\chi}_{00})({\sigma}_{1},{\sigma}_{2})
\quad = \quad 0,
\qquad
({\vec S}_{3}{\chi}_{00})({\sigma}_{1},{\sigma}_{2})
\quad = \quad 0.
$$
’аЁ ўҐЄв®а  ${\chi}_{1{\mu}}({\sigma}_{1},{\sigma}_{2})$ ЇаЁ­ ¤«Ґ¦ в
{\bf ваЁЇ«Ґв­®¬г} б®бв®п­Ёо,   ўҐЄв®а
${\chi}_{00}({\sigma}_{1},{\sigma}_{2})$ --- {\bf бЁ­Ј«Ґв­®¬г}.
…бвҐб⢥­­® ў®§­ЁЄ ов ®Ў®§­ зҐ­Ёп
$$ {\chi}_{1{\mu}}({\sigma}_{1},{\sigma}_{2}) \quad = \quad
{}^{3}{\zeta}_{\mu}({\sigma}_{1},{\sigma}_{2}),
\qquad
{\chi}_{00}({\sigma}_{1},{\sigma}_{2}) \quad = \quad
{}^{1}{\zeta}_{0}({\sigma}_{1},{\sigma}_{2}),
$$
ќв  вҐа¬Ё­®«®ЈЁп ЇҐаҐ­®бЁвбп Ё ­  ЇҐаў®­ з «м­л© ўҐЄв®а ${\Psi}$.
”г­ЄжЁо ${\Psi}(q_{1},q_{2})$ ¬®¦­® ЇаҐ¤бв ўЁвм ў д®а¬Ґ
$$ {\Psi}(q_{1},q_{2}) \quad = \quad
{\sum}_{\mu}f_{a{\mu}}({\vec {r}_{1}}, {\vec {r}_{2}})
{}^{3}{\zeta}_{\mu}({\sigma}_{1},{\sigma}_{2})
\quad + \quad
f_{s0}({\vec {r}_{1}}, {\vec {r}_{2}})
{}^{1}{\zeta}_{0}({\sigma}_{1},{\sigma}_{2}).
$$
ѓ®ў®апв, зв® ваЁ ЇҐаўлҐ дг­ЄжЁЁ ᮮ⢥вбўгов {\bf ваЁЇ«Ґв­®¬г},
зҐвўҐав п --- {\bf бЁ­Ј«Ґв­®¬г} б®бв®п­Ёп¬.

Ќ ©¤Ґ¬ ®б­®ў­®Ґ б®бв®п­ЁҐ бЁбвҐ¬л Ё§ ¤ўге н«ҐЄва®­®ў. ЏаҐ¤Ї®«®¦Ё¬,
зв® Ј ¬Ё«мв®­Ё ­ бЁб⥬л а ўҐ­
$$ {\hat H} \quad = \quad
{\hat H}_{0} \quad + \quad V(1,2),
$$
Ј¤Ґ
$$ {\hat H}_{0} \quad = \quad
{\hat h}(1) \quad + \quad {\hat h}(2),
$$
  ®ЇҐа в®а
$$ {\hat h} \quad = \quad
{1 \over 2m}{\vec p}^{2} \quad + \quad V(r),
$$
ЇаҐ¤бв ў«пҐв б®Ў®© н­ҐаЈЁо н«ҐЄва®­  ў § ¤ ­­®¬ ў­Ґи­Ґ¬ Ї®«Ґ,  
Ї®вҐ­жЁ «
$$ V(1,2) \quad = \quad V({\vec {r}_{1}}, {\vec {r}_{1}})
\quad = \quad
{e^{2} \over |{\vec {r}_{1}} - {\vec {r}_{2}}|} \qquad -
$$
н­аЈЁо ®вв «ЄЁў ­Ёп н«ҐЄва®­®ў.
Џ®бЄ®«мЄг Ј ¬Ё«мв®­Ё ­ ­Ґ ᮤҐа¦Ёв бЇЁ­®ўле ЇҐаҐ¬Ґ­­ле, в® дг­ЄжЁо
${\Psi}(q_{1},q_{2})$ --- аҐиҐ­ЁҐ га ў­Ґ­Ёп
$$ ({\hat H}{\Psi})(q_{1},q_{2}) \quad = \quad
{\Psi}(q_{1},q_{2})E \quad -
$$
¬®¦­® ЇаҐ¤бв ўЁвм Є Є Їа®Ё§ўҐ¤Ґ­ЁҐ Є®®а¤Ё­ в­®© Ё бЇЁ­®ў®© дг­ЄжЁ©:
$$ {\Psi}(q_{1},q_{2}) \quad = \quad
{\Phi}({\vec {r}_{1}}, {\vec {r}_{2}})
{\chi}({\sigma}_{1},{\sigma}_{2}).
$$
‚ н⮬ б«гз Ґ ¬®¦­® Ј®ў®аЁвм (­ҐбЄ®«мЄ® Є®б­®п§лз­®) ® Є®®а¤Ё­ в­®¬
Ё бЇЁ­®ў®¬ б®бв®п­Ёпе Ї® ®в¤Ґ«м­®бвЁ. Џ®бЄ®«мЄг Ј ¬Ё«мв®­Ё ­ бЁб⥬л
ᮤҐа¦Ёв «Ёим Є®®а¤Ё­ в­лҐ ЇҐаҐ¬Ґ­­лҐ, ¬®¦­® ¤г¬ вм, зв®
га®ў­Ё н­ҐаЈЁЁ Ўг¤гв ®¤Ё­ Є®ўл¬Ё ¤«п ўбҐе бЇЁ­®ўле б®бв®п­Ё©.
Ћ¤­ Є®, нв® ЇаҐ¤Ї®«®¦Ґ­ЁҐ ®Є §лў Ґвбп ­ҐўҐа­л¬. Џ®бЄ®«мЄг
Ј ¬Ё«мв®­Ё ­ бЁб⥬л --- бЁ¬¬ҐваЁз­ п дг­ЄжЁп Є®®а¤Ё­ в, в® ҐЈ®
б®Ўб⢥­­л¬Ё дг­ЄжЁп¬Ё ¬®Јгв Ўлвм Є Є бЁ¬¬ҐваЁз­ п, в Є Ё
 ­вЁбЁ¬¬ҐваЁз­ п дг­ЄжЁп:
$$ {\phi}_{s}({\vec {{r}_{1}}}, {\vec {{r}_{2}}})
\quad = \quad
{\phi}_{s}({\vec {{r}_{2}}}, {\vec {{r}_{1}}}),
$$
$$ {\phi}_{a}({\vec {{r}_{1}}}, {\vec {{r}_{2}}})
\quad = \quad
- \quad {\phi}_{a}({\vec {{r}_{2}}}, {\vec {{r}_{1}}}).
$$
‡ ¬ҐвЁ¬, зв® га ў­Ґ­ЁҐ
$$ {\hat H}_{0}{\Phi}({\vec {{r}_{1}}},{\vec {{r}_{2}}})
\quad = \quad
E{\Phi}({\vec {{r}_{1}}},{\vec {{r}_{2}}})
$$
¤®ЇгбЄ Ґв аҐиҐ­ЁҐ
$$ {\Phi}({\vec {{r}_{1}}},{\vec {{r}_{2}}})
\quad = \quad
u({\vec {{r}_{1}}})w({\vec {{r}_{2}}}).
$$
…б«Ё Ї®вҐ­жЁ « ў§ Ё¬®¤Ґ©бвўЁп ¬®¦­® бзЁв вм ¬ «®© ¤®Ў ўЄ®© Є
®ЇҐа в®аг ${\hat H}_{0}$, в® ўҐЄв®а бв жЁ®­ а­®Ј® б®бв®п­Ёп бЁб⥬л
¬®¦­® бзЁв вм Ў«Ё§ЄЁ¬ Є ®¤­®¬г Ё§ ўҐЄв®а®ў
$$ {\phi}_{s}({\vec {{r}_{1}}}, {\vec {{r}_{2}}})
\quad = \quad
C_{s}(u({\vec {{r}_{1}}})w({\vec {{r}_{2}}})
\quad + \quad
u({\vec {{r}_{1}}})w({\vec {{r}_{2}}})),
$$
$$ {\phi}_{a}({\vec {{r}_{1}}}, {\vec {{r}_{2}}})
\quad = \quad
C_{a}(u({\vec {{r}_{1}}})w({\vec {{r}_{2}}})
\quad - \quad
u({\vec {{r}_{1}}})w({\vec {{r}_{2}}})).
$$
…б«Ё дг­ЄжЁЁ $u$ Ё $w$ ­®а¬Ёа®ў ­л ­  Ґ¤Ё­Ёжг, в® ­®а¬ЁагойЁҐ
¬­®¦ЁвҐ«Ё $C_{s}$ Ё $C_{a}$ а ў­л
$$ C_{s} \quad = \quad {1 \over \sqrt{2(1+C)}},
$$
$$ C_{a} \quad = \quad {1 \over \sqrt{2(1-C)}},
$$
Ј¤Ґ
$$ C \quad = \quad
|{\int}{u({\vec r})}^{*}w({\vec r})d{\vec r}|^{2}.
$$
‘।­ЁҐ §­ зҐ­Ёп н­ҐаЈЁЁ ў нвЁе б®бв®п­Ёпе а ў­л
$$ E_{s} \quad = \quad {E + A \over 1 + C},
\qquad
E_{a} \quad = \quad {E - A \over 1 - C}.
$$
‚Ґ«ЁзЁ­г
$$ A \quad = \quad {\int}
{u}^{*}({\vec {r}_{1}}){w}^{*}({\vec {r}_{2}})
V({\vec {r}_{1}}, {\vec {r}_{2}})
w({\vec {r}_{1}})u({\vec {r}_{2}})
d{\vec {r}_{1}}d{\vec {r}_{2}}
$$
­ §лў ов {\bf ®Ў¬Ґ­­л¬ Ё­вҐЈа «®¬}. ђ®«м в ЄЁе ўҐ«ЁзЁ­ ў ⥮ਨ
ўЇҐаўлҐ ўлпб­Ё« ѓ ©§Ґ­ЎҐаЈ. ‡­ Є®®ЇаҐ¤Ґ«Ґ­­®бвм ўҐ«ЁзЁ­л $A$
¤®Є § вм ваг¤­®. Ћ¤­ Є®, Ґб«Ё дг­ЄжЁЁ $u$ Ё $w$ §­ Є®®ЇаҐ¤Ґ«Ґ­л, в®
бЇа ўҐ¤«Ёў® ­Ґа ўҐ­бвў® $A > 0$. Џ®Є  ­Ґ гзЁвлў Ґвбп ў§ Ё¬®¤Ґ©бвўЁҐ
¬Ґ¦¤г н«ҐЄва®­ ¬Ё, §­ зҐ­ЁҐ н­ҐаЈЁЁ ®ЇаҐ¤Ґ«пҐвбп «Ёим б।­Ё¬Ё
а ббв®п­Ёп¬Ё н«ҐЄва®­®ў ®в п¤а . ‚§ Ё¬®¤Ґ©бвўЁп н«ҐЄва®­®ў Ё§¬Ґ­пҐв
н­ҐаЈЁо бЁб⥬л. —в®Ўл 㬥­миЁвм н­ҐаЈЁо ®вв «ЄЁў ­Ёп н«ҐЄва®­л
¤®«¦­л ­ е®¤Ёвмбп ў в Є®¬ б®бв®п­ЁЁ, ЇаЁ Є®в®а®¬ б।­ҐҐ а ббв®п­ЁҐ
¬Ґ¦¤г н«ҐЄва®­ ¬Ё ¬ ЄбЁ¬ «м­®. ќв®¬г вॡ®ў ­Ёо ®вўҐз Ґв б®бв®п­ЁҐ,
ᮮ⢥вбвўго饥  ­вЁбЁ¬¬ҐваЁз­®© ў®«­®ў®© дг­ЄжЁЁ. ‘Ё¬¬ҐваЁп
ў®«­®ў®© дг­ЄжЁЁ --- бў®©бвў®, Є®в®а®Ґ Їа ЄвЁзҐбЄЁ ­Ґў®§¬®¦­®
Їа®ўҐаЁвм. Џ®н⮬㠢 Є зҐб⢥ е а ЄвҐаЁбвЁЄЁ б®бв®п­Ёп б ­ Ё¬Ґ­м襩
н­ҐаЈЁҐ© 㤮Ў­® ў§пвм бЇЁ­®ўго ў®«­®ўго дг­ЄжЁо. ЏаЁ ¬ ЄбЁ¬ «м­®
 ­вЁбЁ¬¬ҐваЁз­®© Є®®а¤Ё­ в­®© ў®«­®ў®© дг­ЄжЁЁ ¤®«¦­  Ўлвм
¬ ЄбЁ¬ «м­® бЁ¬¬ҐваЁз­®©. ‚ б«гз Ґ ¤ўге н«ҐЄва®­®ў --- нв® Їа®бв®
бЁ¬¬ҐваЁз­ п дг­ЄжЁп. …© ®вўҐз Ґв §­ зҐ­ЁҐ Ї®«­®Ј® бЇЁ­ , а ў­®Ґ
Ґ¤Ё­ЁжҐ, в.Ґ. ¬ ЄбЁ¬ «м­® ў®§¬®¦­®Ґ.
‚§ Ё¬®¤Ґ©бвўЁҐ н«ҐЄва®­®ў ­ҐбЄ®«мЄ® 㬥­ми Ґв Єа в­®бвм ўл஦¤Ґ­Ёп:
зҐвлॠб®бв®п­Ёп б а ў­®© н­ҐаЈЁҐ© ৤Ґ«повбп ­  ¤ўҐ ЈагЇЇл: ваЁЇ«Ґв
Ё бЁ­Ј«Ґв.

ЌҐ®Ўе®¤Ё¬® ®в¬ҐвЁвм ҐйҐ ®¤­® ®Ўбв®п⥫мбвў®: Ґб«Ё Їа®бва ­б⢥­­лҐ
Єў ­в®ўлҐ зЁб«  ®в¤Ґ«м­ле н«ҐЄва®­®ў б®Ї ¤ ов, в®  ­вЁбЁ¬¬ҐваЁз­ п
Є®®а¤Ё­ в­ п ў®«­®ў п дг­ЄжЁп ®Ўа й Ґвбп ў ­г«м Ё ўл஦¤Ґ­ЁҐ,
бўп§ ­­®Ґ б бЁ¬¬ҐваЁҐ© Ј ¬Ё«мв®­Ё ­ , бўп§ ­­®Ґ б ­Ґа §«ЁзЁ¬®бвмо
®¤Ё­ Є®ўле з бвЁж, ў н⮬ б«гз Ґ б­Ё¬ Ґвбп. €¬Ґ­­® в Є ®Ўбв®Ёв б
 в®¬®¬ ЈҐ«Ёп: ҐЈ® ®б­®ў­®Ґ б®бв®п­ЁҐ ®ЇаҐ¤Ґ«пҐвбп Єў ­в®ўл¬Ё зЁб« ¬Ё
н«ҐЄва®­®ў $n = 1, \quad l = 0$. ‚ бЁ«г ЇаЁ­жЁЇ  Џ г«Ё ў®«­®ў п
дг­ЄжЁп нв®Ј® б®бв®п­Ёп а ў­ 
$$ {\Psi}(q_{1},q_{2}) \quad = \quad
u_{1s}(r_{1})u_{1s}(r_{2}){\chi}_{00}({\sigma}_{1},{\sigma}_{2}),
$$
Ї®н⮬㠣Ґ«Ё© --- нв® Ё­Ґав­л© Ј §.

Ђ­ «®ЈЁз­лҐ б®®Ўа ¦Ґ­Ёп бЇа ўҐ¤«Ёўл Ё ў б«гз Ґ  в®¬®ў б Ў®«миЁ¬
зЁб«®¬ н«ҐЄва®­®ў. …б«Ё ЇаҐ­ҐЎаҐзм ५пвЁўЁбвбЄЁ¬Ё Ї®Їа ўЄ ¬Ё Є
н­ҐаЈЁЁ, в® Ј ¬Ё«мв®­Ё ­  в®¬  б $Z$ н«ҐЄва®­ ¬Ё Ўг¤Ґв а ўҐ­
$$ {\cal H} \quad = \quad
{\sum}_{i=1}^{Z}{1 \over 2m}{\vec {{p}_{i}}}^{2}
\quad - \quad
{\sum}_{i=1}^{Z}{Ze^{2} \over r_{i}}
{\quad + \quad}
{\sum}_{1 \leq i < j \leq Z}
{e^{2} \over |{\vec {{r}_{i}}} - {\vec {{r}_{j}}}|}.
$$
ђҐиҐ­ЁҐ га ў­Ґ­Ёп
$$ ({\hat H}{\Psi})(q_{1},...,q_{Z})
\quad = \quad
E{\Psi}(q_{1},...,q_{Z})
$$
¬®¦­® ЇаҐ¤бв ўЁвм ў д®а¬Ґ Їа®Ё§ўҐ¤Ґ­Ёп
$$ {\Psi}(q_{1},...,q_{Z})
\quad = \quad
{\Phi}({\vec {{r}_{1}}},...,{\vec {{r}_{Z}}})
{\chi}({\sigma}_{1},...,{\sigma}_{Z}).
$$
—Ёб«® а §«Ёз­ле бЇЁ­®ўле дг­ЄжЁ© а ў­® ў ®ЎйҐ¬ б«гз Ґ $2^{Z}$,
Ёе ¬®¦­® Є« ббЁдЁжЁа®ў вм Ї® §­ зҐ­Ё¬ Ї®«­®Ј® бЇЁ­  $S$ Ё Їа®ҐЄжЁ©
бЇЁ­  ­  ®бм $0Z$ --- $M_{S}$. Љ Є Ё ў б«гз Ґ ¤ўге з бвЁж
Є« ббЁдЁЄ жЁп б®бв®п­Ё© Ї® §­ зҐ­Ёп Ї®«­®Ј® бЇЁ­  б®ўЇ ¤ Ґв б
Є« ббЁдЁЄ жЁҐ© Ї® бў®©бвў ¬ бЁ¬¬ҐваЁЁ бЇЁ­®ўле ў®«­®ўле дг­ЄжЁ©.
„ «ҐҐ, Ј ¬Ё«мв®­Ё ­ Є®¬¬гвЁагҐв б Їа®ҐЄжЁп¬Ё Ї®«­®Ј® ¬®¬Ґ­в 
Ё¬Їг«мб  н«ҐЄва®­®ў
$$ {\hat L}_{\alpha} \quad = \quad
{\sum}_{i=1}^{Z}{\hat {l}_{i \alpha}},
$$
Ї®н⮬㠤«п Є« ббЁдЁЄ жЁЁ Є®®а¤Ё­ в­ле ў®«­®ўле дг­ЄжЁ© ¬®¦­®
ЁбЇ®«м§®ў вм б®Ўб⢥­­лҐ §­ зҐ­Ёп ®ЇҐа в®а®ў ${\vec L}^{2}$ Ё
${\hat {L}_{z}}$ --- зЁб«  $L$ Ё $M_{L}$.
’ ЄЁ¬ ®Ўа §®¬, бв жЁ®­ а­лҐ б®бв®п­Ёп ­ҐаҐ«пвЁўЁбвбЄ®Ј®  в®¬  ¬®¦­®
ЇҐаҐзЁб«Ёвм Єў ­в®ўл¬Ё зЁб« ¬Ё
$\lbrace {\gamma}, L, M_{L}, S, M_{S} \rbrace$. ‡­ зҐ­Ёп га®ў­Ґ© н­ҐаЈЁЁ
 в®¬  ®ЇаҐ¤Ґ«повбп Єў ­в®ўл¬Ё зЁб« ¬Ё
$\lbrace {\gamma},L,S \rbrace$.

„ «м­Ґ©и п ¤Ґв «Ё§ жЁп ®ЇЁб ­Ёп бв жЁ®­ а­ле б®бв®п­Ё© вॡгҐв
¤®Ї®«­ЁвҐ«­ле ЇаҐ¤Ї®«®¦Ґ­Ё©.

ЋЄ §лў Ґвбп, зв® Є ўЇ®«­Ґ 㤮ў«Ґвў®аЁвҐ«м­л¬ १г«мв в ¬ ЇаЁў®¤Ёв
ЇаҐ¤Ї®«®¦Ґ­ЁҐ ® ⮬, зв® б зЁб« ¬Ё ${\gamma}$ ¬®¦­® бўп§ вм
Єў ­в®ўлҐ зЁб«  ®в¤Ґ«м­ле н«ҐЄва®­®ў $\lbrace n, l \rbrace$. —Ёб« 
$l$ бўп§лў овбп б ¬®¬Ґ­в ¬Ё Ё¬Їг«мб  ®в¤Ґ«м­ле н«ҐЄва®­®ў,   зЁб«®
$n$, Є®в®а®Ґ Є Є Ё ў ⥮ਨ  в®¬  ў®¤®а®¤ , ­ §лў ов ${\bf Ј« ў­л¬
Єў ­в®ўл¬ зЁб«®¬}$, Ё Є®в®а®Ґ ЇаЁ § ¤ ­­®¬ $l$ ЇаЁ­Ё¬ Ґв §­ зҐ­Ёп
$n = l+1, l+2,...$.
ЋЄ §лў Ґвбп, зв® н­ҐаЈЁп  в®¬®ў (Ї® Єа ©­Ґ© ¬ҐаҐ  в®¬®ў, ­ е®¤пйЁебп
ў ­ з «Ґ в Ў«Ёжл ЊҐ­¤Ґ«ҐҐў , Ј¤Ґ ५пвЁўЁбвбЄЁҐ Ї®Їа ўЄЁ Є н­ҐаЈЁЁ
¤®«¦­л Ўлвм ¬ «л¬Ё) ЇаҐ¤бв ў«пҐвбп д®а¬г«®©
$$ E \quad = \quad E(\lbrace nl \rbrace,L,S)
\quad = \quad E_{0}(\lbrace nl \rbrace)
\quad + \quad E_{1}(\lbrace nl \rbrace,L,S),
$$
ў Є®в®а®© ўҐ«ЁзЁ­г $E_{1}$ ¬®¦­® бзЁв вм ¬ «®© Ї®Їа ўЄ®© Є $E_{0}$.
‡ ўЁбЁ¬®бвм н­ҐаЈЁЁ $E_{1}$ ®в бЇЁ­  ®ЇаҐ¤Ґ«пҐвбп ЇаЁ­жЁЇ®¬ Џ г«Ё:
в.Є. Є®®а¤Ё­ в­ п ў®«­®ў п дг­ЄжЁп ¤®«¦­  Ўлвм ¬ ЄбЁ¬ «м­®
 ­вЁбЁ¬¬ҐваЁз­®©, в® бЇЁ­®ў п дг­ЄжЁп ¬ ЄбЁ¬ «м­® бЁ¬¬ҐваЁз­ ,
®ЇаҐ¤Ґ«пп ¬ ЄбЁ¬ «м­® ў®§¬®¦­л© бЇЁ­.






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