Лекция (3) (Лекции)

‹…Š–ˆŸ 8. 6.04.2001‘“ŒŒ€ ŒŽŒ…’Ž‚DZãáâì~l ¨ ~s = ~12 .= ~l + ~s:¥®¡å®¤¨¬® ¢ëà §¨âì ᮡá⢥­­ë¥¢¥ªâ®à뮯¥à â®à®¢J~2 ¨ J3 ¢ â¥à¬¨­ å ᮡá⢥­­ëå ¢¥ªâ®à®¢ ®¯¥à â®à®¢ ~l2 ; l3 ¨ ~s2 ; s3.‚ í⮬ ç áâ­®¬, ­® ¢ ¦­®¬ á«ãç ¥ ¯®«¥§­® ¤ âì ¢ë¢®¤ ᮮ⢥âá¢ãîé¨å ᮮ⭮襭¨© ­¥ á¢ï§ ­­ë© á ®¡é¥© ⥮ਥ© ª®íää¨æ¨¥­â®¢ Š«¥¡è -ƒ®à¤®­ ,¨á¯®«ì§ãï2~­¥¯®á।á⢥­­® ä®à¬ã«ã ᯥªâà «ì­®£® à §«®¦¥­¨ï ®¯¥à â®à  J . DZ®áª®«ìªã ®¯¥à â®àë ~l ¨ ~s ª®¬¬ãâ¨àãîâ, â®J~2 = J~2 + ~s2 + 2~l~s:…᫨ à áᬠâਢ â줥©á⢨¥ í⮣® ®¯¥à â®à  ¢ ¯®¤¯à®áâà ­á⢥ á ®¯à¥¤¥«¥­­ë¬§­ ç¥­¨¥¬ j , â® J~2 íä䥪⨢­® ᢮¤¨âáï ª3J~2 = j (j + 1) + + ~l~ :4â  ä®à¬ã«  ᮣ« áã¥âáï á ®¡é¨¬ ã⢥ত¥­¨¥¬, çâ® «î¡ ï äã­ªæ¨ï ᯨ­®¢®© ¯¥à¥¬¥­­®© ᢮¤¨âáï ª «¨­¥©­®©.

 ©¤¥¬ ¯à®¥ªæ¨®­­ë¥ ®¯¥à â®àëP = a + ~~l:‡ ¬¥â¨¬, çâ® ¢ ᨫ㠭¥ª®¬¬ãâ â¨¢­®á⨠á®áâ ¢«ïîé¨å ~l(~~l)(~~l) = ~l2 + i~(~l ~l) = l(l + 1) ~~l;¯®í⮬ã ãá«®¢¨¥ P 2 = P ¯à¨¢®¤¨â ª ãà ¢­¥­¨ï¬a = a2 + b2 l(l + 1);b = 2ab b2 :‚ १ã«ìâ â¥ ¯®«ãç îâáï ¤¢  ®¯¥à â®àël + 1 + ~~lP1 =2l + 1 ;J~P2㤮¢«¥â¢®àïî騥 ᮮ⭮襭¨ï¬Pi 2 = Pi ;P1 P2 = P2 P1 = 0;~= 2l l +~1l ;P1 + P2 = E;1lP1(l + 1)P2 =~~lŽ¯¥à â®à J~2 ¬®¦­® ¯à¥¤áâ ¢¨âì ª ª á㬬ã12¥âà㤭® á®®¡à §¨âì, ç⮠ᮡá⢥­­ë¬ ¢¥ªâ®à®¬ ®¯¥à â®à  J~2, ᮮ⢥âáâ¢ãî騬ç¨á«ã j+ ¡ã¤¥â à¥è¥­¨¥ ãà ¢­¥­¨ï P2 = 0. â® ãà ¢­¥­¨¥ ¯à®é¥ ¢á¥£® à¥è âì,¯à¥¤áâ ¢¨¢ ᯨ­®¢ë¥ ®¯¥à â®àë ¢ ä®à¬¥ ¤¢ãåà來ëå ¬ âà¨æ,J~2=j+ (j+ + 1)P1 + j (j=P2+ 1)P2;j = l :l l3ll+ l + l3 ;  ᮡá⢥­­ë© ¢¥ªâ®à { ¤¢ãåà來®£® á⮫¡æ :j+lM =Ylm (^r )C1 :Ylm0 (^r )C2…᫨ m0 = m + 1, â® íâ®â á⮫¡¥æ { ᮡá⢥­­ë© ¢¥ªâ®à ®¯¥à â®à  J3, ª®â®à멯।áâ ¢«ï¥âáï ¢ ä®à¬¥1l+032J3 =0 l3 21 ;1J3 j+ lM = j+ lM M;M =m+ :2DZ®áª®«ìªã ᪠«ïà­®¥ ¯à®¨§¢¥¤¥­¨¥á⮫¡æ®¢ á § ¢¨áï騬¨ ®â  à£ã¬¥­â  r^ í«¥¬¥­â ¬¨à ¢­®Z a(^r)a(^r)121 = b1 (^r) : 2 = b2 (^r) :h1 j2i = dr^ a1 a2 + b1 b2 ;â® ­®à¬¨à®¢ ­­ë© ­  ¥¤¨­¨æã ¢¥ªâ®à à ¢¥­j+lM =q10Ylm (^r ) +2M@A;q12=l+ ; MYlm+1 (^r ) 2M= m + 21 :‘®¡á⢥­­ë©¢¥ªâ®à J~2 ¨ J 3, ᮮ⢥âáâ¢ãî騩 ᮡá⢥­­ë¬ §­ ç¥­¨ï¬ j ¨ M =1m + 2 , à ¢¥­jlM=q0Ylm (^r ) 2M@qYlm+1 (^r ) +2M…᫨ ®¯à¥¤¥«¨âìtgr2 =1A;21212=l+ ; M =m+ : M;+Mâ® ¢ ᨫã ᮮ⭮襭¨©r+M2 ;r M=sin=222 ;ä®à¬ã«ë ¤«ï ᮡá⢥­­ëå ¢¥ªâ®à®¢ ¬®¦­® ¡ã¤¥â § ¯¨á âì ¢ ä®à¬¥ Y(^r)coslmj+lM = Ylm+1(^r )sin2 ;2 Y(^r)sinlm2j lM =Ylm+1 (^r )cos 2 :cos’ŽŠ€Ÿ ‘’“Š’“€ “Ž‚…‰ …ƒˆˆƒ ¬¨«ìâ®­¨ ­  â®¬  ¢®¤®à®¤  ¢ ¯à¨¡«¨¦¥­¨¨ ¡¥áª®­¥ç­® â殮«®£® ¯à®â®­  ¨­¥à¥«ï⨢¨áâ᪮£® í«¥ªâà®­  à ¢¥­2H0 = 21m p~2 + V (r); V (r) = = er :¥«ï⨢¨áâ᪨¥ ¯®¯à ¢ª¨ ª í­¥à£¨¨ í«¥ªâà®­  ¯à®¨á⥪ îâ ¨§ á«¥¤ãîé¨å íä䥪⮢.1.

DZ®¯à ¢ª , á¢ï§ ­­ ï á ¨§¬¥­¥­¨¥¬ á¢ï§¨ ¬¥¦¤ã ¨¬¯ã«ìᮬ ¨ í­¥à£¨¥©.— áâ­ ï ⥮à¨ï ®â­®á¨â¥«ì­®á⨠ãç¨â, çâ® í­¥à£¨ï ç áâ¨æë ¬ ááë m á ¨¬¯ã«ìᮬ ~pà ¢­ r2~p1 + mc:=“¤¥à¦¨¢ ï ¢ à §«®¦¥­¨¨ ª¢ ¤à â®­®£® ª®à­ï ¢ àï¤ ’¥©«®à  ⮫쪮 ª¢ §¨à¥«ï⨢¨áâáªãî ¯®¯à ¢ªã, ¯®«ã稬 â ª®¥ ¢ëà ¦¥­¨¥ ®¯à¥ â®à  ª¨­¥â¨ç¥áª®© í­¥à£¨¨ ~p2 21122T = mc +2m ~p2mc2 2m :Emc22. DZ®¯à ¢ª , á¢ï§ ­­ ï á ­ «¨ç¨¥¬ ã í«¥ªâà®­  ᮡá⢥­­®£® ¬®¬¥­â ª®«¨ç¥á⢠ ¤¢¨¦¥­¨ï.Š ª 㦥 £®¢®à¨«®áì ­  ¯à®è«®© «¥ªæ¨¨, ­ «¨ç¨¥ ã í«¥ªâà®­  ¬ £­¨â­®£® ¬®¬¥­â ~ = 0~s¯à¨¢®¤¨â ª ¯®ï¢«¥­¨î ã ­¥£® í«¥ªâà¨ç¥áª®£® ¤¨¯®«ì­®£® ¬®¬¥­â 1 (~v ~):d~ =c3­¥à£¨ï í«¥ªâà¨ç¥áª®£® ¬®¬¥­â  ¢ ªã«®­®¢®¬ ¯®«¥ ï¤à  ®¯¨á뢠¥âáï ®¯¥à â®à®¬2 2Hos = 2emh2c2 r13 ~l~s:3.DZ®¯à ¢ª , á¢ï§ ­­ ï á ­ «¨ç¨¥¬ ã í«¥ªâà®­   ­â¨ç áâ¨æë { ¯®§¨-âà®­ .DZ®á«¥¤®¢ â¥«ì­ ï ५ï⨢¨áâ᪠ï ⥮à¨ï í«¥ªâà®­ , ã⢥ত ¥â, çâ® í«¥ªâà®­ { íâ®ç áâ¨æ  „¨à ª  {   ­¥ª®â®à®¥ ®¡ê¥¤¨­¥­¨¥ ç áâ¨æë ¨  ­â¨ç áâ¨æë { í«¥ªâà®­  ¨¯®§¨âà®­ .

‘ãé¥á⢥­­®, ¯à¨ ¯®¯ë⪥ ¢ë¤¥«¨âì ¨§ ç áâ¨æë „¨à ª  á®áâ®ï­¨¥ ᮯ।¥«¥­­ë¬ § à冷¬, ­ ¯à¨¬¥à í«¥ªâà®­, ¯®«ãç îâ ­¥«®ª«¨§®¢ ­­®¥ á®áâ®ï­¨¥, ¢ª®â®à®¬ ª®®à¤¨­ âë ç áâ¨æë ¬®¦­® ¨§¬¥à¨âì «¨èì á â®ç­®áâìîx c = h :mcDZ®áâ®ï­­ãî à §¬¥à­®á⨠¤«¨­ë c, £¤¥ m { ¬ áá  í«¥ªâà®­ , ­ §ë¢ îâí«¥ªâà®­ . DZநá宦¤¥­¨¥ íä䥪⨢­®© ­¥«®ª «ì­®á⨠«¥£ª®¯®­ïâì ¨§ á«¥¤ãîé¨å á®®¡à ¦¥­¨©: ¯à¨ ¯®¯ë⪥ «®ª «¨§®¢ âì í«¥ªâà®­ á â®ç­®áâì ª®¬¯â®­®¢®© ¤«¨­ë ¢®«­ë ¢®§­¨ª ¥â ­¥®¯à¥¤¥«¥­­®áâì ¨¬¯ã«ìá p h = mc;ª®¬¯â®­®-¢®© ¤«¨­®© ¢®«­ëcâ®¬ã ¨¬¯ã«ìáã ᮮ⢥âáâ¢ã¥â í­¥à£¨ï,¯à¥¢ëè îé ï ¬ ááã ¯®ª®ï ç áâ¨æë, ¢ í⮬á«ãç ¥ ¯à¨ ®¯¨á ­¨¨ ¯®¢¥¤¥­¨ï í«¥ªâà®­  ¯à¨å®¤¨âáï ãç¨â뢠âì ¢®§¬®¦­®¥ ஦¤¥­¨¥ ¯ àë "í«¥ªâà®­+¯®§¨âà®­".

¥¯à¥¤¥«¥­­®áâì ¢ ¯®«®¦¥­¨¨ í«¥ªâà®­  ¯à¨¢®¤¨âª ­¥®¯à¥¤¥«¥­­®á⨠¥£® ¯®â¥­æ¨ «ì­®© í­¥à£¨¨. ä䥪⨢­ë© ¯®â¥­æ¨ « ¡ã¤¥â ¢ë£«ï¤¥âì ¯à¨¬¥à­® â ª:Z00 0V (~r) =(~r )V (~r + ~r )d~r : §« £ ï ¯®â¥­æ¨ « ¢ àï¤ ’¥©«®à  ¨ 㤥ন¢ ï á« £ ¥¬ë¥ ¢¯«®âì ¤® ¢â®à®£® ¯®à浪  ¯® ª®¬¯â®­®¢®© ¤«¨­¥ ¢®«­ë í«¥ªâà®­  ¯®«ã稬 á㬬㠪㫮­®¢  ¯®â¥­æ¨ «  ¨®¯¥à â®à2HD = 8mh2c2 r2V (r):â  ¢¥«¨ç¨­  ¡ë«  ¢¯¥à¢ë¥ ¯®«ã祭  „ à¢¨­ë¬ { ¢­ãª®¬ ᮧ¤ â¥«ï ¤ à¢¨­¨§¬ { ¨ ­®á¨â ­ §¢ ­¨¥ ¯®¯à ¢ª¨ „ à¢¨­ .DZਢ¥¤¥¬ ¯®«­®¥ ¢ëà ¦¥­¨¥ £ ¬¨«ìâ®­¨ ­  í«¥ªâà®­  ¢ ªã«®­®¢®¬ ¯®«¥ ¢ ª¢ §¨à¥«ï⨢¨áâ᪮¬ ¯à¨¡«¨¦¥­¨¨:H = mc2 + H0 + H1 ;£¤¥2H0 = 2~pm + V (r);4H1 = Hm + Hso + HD :‚ᥠ᫠£ ¥¬ë¥, ®¡ê¥¤¨­¥­­ë¥ ¢ H0 ¨ H1 { ¢¥«¨ç¨­ë ®¤­®£® ¯®à浪 :£¤¥242h 2p~m i h er i me2 = 2 mc2;he2hc :=€­ «®£¨ç­® ®æ¥­¨¢ îâáï á« £ ¥¬ë¥, ¢å®¤ï騥 ¢ H1 :24hHm i mc1 2 ( er )2 = 2 me2 ;h0hHso i hHD i ¥§à §¬¥à­ ï ¯®áâ®ï­­ ï à ¢­ e8 mc2 h4=me42 2 :h1 ;= 137:035989®­  ¯à¥¤áâ ¢«ï¥â ᮡ®© á।­¥¥ §­ ç¥­¨¥ ᪮à®áâ¨ í«¥ªâà®­  ¢  â®¬¥ ¢®¤®à®¤ , ¢ëà ¦¥­­®© ¢ ¥¤¨­¨æ å ᪮à®á⨠ᢥâ :phe2i mcr=:h vc i = h mch c0‚¥«¨ç¨­ã ®¡ëç­® ­ §ë¢ îâ.¯®áâ®ï­­®© â®­ª®© áâàãªâãà뒅Ž…Œ€ ”…‰Œ€€-ƒ…‹œŒ€€ ‚ ‘‹“—€… Š€’›•‘Ž‘’‚…›• ‡€—…ˆ‰Ÿ¢­®¥ ¢ëç¨á«¥­¨¥ â®­ª®© áâàãªâãàë ã஢­¥© ¬®¦­® ¢ë¯®«­¨âì ®¡à â¨¢è¨áì ªâ¥®à¥¬¥ ”¥©­¬ ­ -ƒ¥«ì¬ ­ , ãâ®ç­¥­­®© ­  á«ãç © ¢ë஦¤¥­¨ï ᮡá⢥­­ëå §­ ç¥­¨©.DZãáâì ¤«ï ª ¦¤®£® n ¢ë¯®«­ïîâáï à ¢¥­á⢠Hns = nsEn; s = 1; :::; rn ;¯à¨ç¥¬ ¢¥ªâ®àë ns ®¡à §ãîâ ®àâ®­®à¬¨à®¢ ­­ë© ¡ §¨á ¢ ¯à®áâà ­á⢥ á®áâ®ï­¨©.“à ¢­¥­¨ï ¤«ï ¢¥ªâ®à®¢ ns ¬®¦­® ¯à¥¢à â¨âì ¢ à ¢¥­á⢠hnsjHjns0 i = Æss0 Ens; Ens = Ens0 : hnsjns0 i = Æss0 :5‚ à¨ æ¨¨ ¬ âà¨ç­ëå í«¥¬¥­â®¢ £ ¬¨«ìâ®­¨ ­  ¯à¨¢®¤ïâ ª ãà ¢­¥­¨ï¬hÆnsjHjnsi + hnsjÆHjnsi + hnsjHjÆnsi = Æss0 ÆEns:DZ®áª®«ìªã ns { ®àâ®­®à¬¨à®¢ ­­ë¥ ᮡá⢥­­ë¥ ¢¥ªâ®àë ®¯¥à â®à  H, â®hÆnsjHjns0 i + hnsjHjÆns0 i = hÆnsjns0 iEn + EnhnsjÆns0 i = 0:’ ª¨¬ ®¡à §®¬ ¤®«¦­ë ¢ë¯®«­ïâìáï ᮮ⭮襭¨ïhnsjÆHjns0 i = Æss0 ÆEns:â  ä®à¬ã«  ¬®¦¥â ®ª § âìáï ¢­ãâ७­¥ ¯à®â¨¢®à¥ç¨¢®©, ¯®áª®«ìªã ¯à ¢ ï ç áâìà ¢¥­á⢠ ¤¨ £®­ «ì­ ¯® ¨­¤¥ªá ¬ s ¨ s0 ,   «¥¢ ï ¢®¢á¥ ­¥ ®¡ï§ ­  ¡ëâì ¤¨ £®­ «ì­®©.Ž¤­ ª®, ­¥âà㤭® ¯®ª § âì, çâ® ¢¥ªâ®àë ns ¢á¥£¤  ¬®¦­® ¢ë¡à âì â ª, ç⮡ë¢ë¯®«­ï«¨áì à ¢¥­á⢠hnsjÆHjns0 i = Æss0 hnsjÆHjnsi:„ã©á⢨⥫쭮, ¯®áª®«ìªã ᮡá⢥­­ë¥ §­ ç¥­¨ï ®¯¥à â®à  H ¢ë஦¤¥­ë, â® ¢¥ªâ®àë ns ®¯à¥¤¥«¥­ë á â®ç­®áâìî ¤® ã­¨â à­®£® ¯à¥®¡à §®¢ ­¨ïXns )ns0 Fs0 s ; = F^ + = F^ 1 :s0Œ âà¨ç­ë¥ í«¥¬¥­âë ®¯¥à â®à  ÆH ¨§¬¥­ïîâáï ¯à¨ â ª®¬ ¯à¥®¡à §®¢ ­¨¨ á«¥¤ãî騬 ®¡à §®¬:Æ H^ ) F^ + Æ H^ F^ :DZ®áª®«ìªã ¬ âà¨æ  ÆH^ { ª®­¥ç­®¬¥à­ ï íନ⮢ ¬ âà¨æ , â® ¢á¥£¤  ­ ©¤¥âáï ã­¨+^^^â à­ ï¬ âà¨æ , ¤«ï ª®â®à®© ¬ âà¨æ  F ÆH F ¡ã¤¥â ¤¨ £®­ «ì­®© ¯® ¨­¤¥ªá ¬ s ¨0s.‚ ¨â®£¥ ¯®«ãç ¥âáï ä®à¬ã« , ®¯à¥¤¥«ïîé ï ¢ à¨ æ¨¨ ã஢­¥© í­¥à£¨¨ ¯à¨ ¢ à¨ æ¨ïå £ ¬¨«ìâ®­¨ ­ :ÆEns = hns jÆ Hjns i:DZਠí⮬ ¢¥ªâ®àë ns ¤®«¦­ë 㤮¢«¥â¢®àïâì ãá«®¢¨ï¬hnsjHjns0 i = Æss0 En; hnsjÆHjns0 i = Æss0 ÆEns:Ÿ‚Ž… ‚›—ˆ‘‹…ˆ… “Ž‚…‰ ’ŽŠŽ‰ ‘’“Š’“›DZ®¯à ¢ª¨ ª ã஢­ï¬ í­¥à£¨¨, ®¡ãá«®¢«¥­­ë¥ £ ¬¨«ìâ®­¨ ­®¬ â®­ª®© áâàãªâãàë, ®¯à¥¤¥«ïîâáï ¯®á«¥ ¢ëç¨á«¥­¨ï á।­¥£® §­ ç¥­¨ï £ ¬¨«ìâ®­¨ ­  ¢ á®áâ®ï­¨ïå,®¯à¥¤¥«ï¥¬ëå ¢¥ªâ®à ¬¨ jn; l; j; mi.6DZਠãá।­¥­¨¨ ®¯¥à â®à  Hm 㤮¡­® ¯à¥¤áâ ¢¨âì ¥£® ¢ ä®à¬¥241 H0 V (r)2hHm i mc1 2 ( er )2 = 2 me2 ; = 2mc2h0¨ § ¬¥­¨âì ®¯¥à â®à H0 ­  ç¨á«® En { ᮡá⢥­­®¥ §­ ç¥­¨¥ í⮣® ®¯¥à â®à .

‚१ã«ìâ â¥ § ¤ ç  ᢥ¤¥âáï ª ¢ëç¨á«¥­¨î á।­¥£® §­ ç¥­¨ï ¢¥«¨ç¨­ë1 En2 + 2e2 En 1 + e4 :2mc2r r2DZ®á«¥ á।­¨å §­ ç¥­¨© à ¤¨ãá  ¢ ᮮ⢥âáâ¢ãîé¨å á⥯¥­ïå ¯®«ãç ¥âáï ç¨á«®e2 2 31 = En 2 31 :Em =2r0 n2 n3 4n l + 21n3 4n l + 12‘।­¥¥ §­ ç¥­¨¥ ®¯¥à â®à e2 h 2 1 ~Hos = 2m2c2 r3 l~s:¯à®¯®à樮­ «ì­®j (j + 1) l(l + 1) 34 ; l 6= 0~hnljmjl~sjmjlni =0; l = 0’ ª¨¬ ®¡à §®¬ ®¯¥à â®à Hos ¨§¬¥­ï¥â §­ ç¥­¨ï «¨èì ã஢­¥© á l 6= 0. …᫨ íâ®ãá«®¢¨¥ ¢ë¯®«­¥­®, â®e2 h 2 13 ):hi(j(j+1)l(l+1)Eso =2m2c2 r34DZ®¤áâ ¢«ïï ¢ ä®à¬ã«ã á।­¥¥ §­ ç¥­¨¥ r13 , ¯®«ã稬822e < (l+ 12 )(1 l+1) ; j = l + 21Eso =l 6= 0:2r0 n3 : l(l+1 21 ) ; j = l 21DZ®¤áâ ¢«ïï ¢ ä®à¬ã«ã ¤«ï ¯®¯à ¢ª¨ „ à¢¨­ 2HD = 8mh2c2 r2V (r):¯®â¥­æ¨ « V (r) = er2 ¨ ãç¨â뢠ï ä®à¬ã«ãr2 1 = 4Æ(~r);­ ©¤¥¬, çâ®r7 h2 e22:j(0)j222m c«¨èì ¯à¨ l = 0.=‡­ ç¥¨¥ ä㭪樨 (0) ®â«¨ç­® ®â ­ã«ïDZ।¯®« £ ï, çâ® l 6= 0 ¨áª« ¤ë¢ ï ç¨á«  Em, Eso, ­ ©¤¥¬ ¯®¯à ¢ªã ª ã஢­ï¬ í­¥à£¨¨ (¯®ª  ⮫쪮 ¯à¨l 6= 0):1 :e2 1 3Ef = 22r0 n2 n 4n j + 12DZਠ­ã«¥¢®¬ §­ ç¥­¨¨ ¬®¬¥­â  ¨¬¯ã«ìá  á¯¨­-®à¡¨â «ì­ãî ¯®¯à ¢ªã § ¬¥­ï¥â ¯®¯à ¢ª „ à¢¨­ .

“ç¨â뢠ï, çâ® ¯à¨ l = 0 ¯®«­ë© ¬®¬¥­â ª®«¨ç¥á⢠ ¤¢¨¦¥­¨ï à ¢¥­1 , ­ ©¤¥¬, ç⮠⮫쪮 çâ® ¯®«ã祭­ ï ä®à¬ã«  á¯à ¢¥¤«¨¢  ¯à¨ «î¡ëå §­ ç¥­¨ïå l.2ED‘‚…•’ŽŠ€Ÿ ‘’“Š’“€ “Ž‚…‰ …ƒˆˆŒë ⮫쪮 çâ® ­ è«¨ §­ ç¥­¨ï ã஢­¥© í­¥à£¨¨ á â®ç­®áâìî ¤® 4. “â®ç­¥­¨¥íâ¨å १ã«ìâ â®¢ âॡã¥â §­ ª®¬á⢠ á ®¡é¥© ⥮ਥ© ¢®§¬ã饭¨©, çâ® ¡ã¤¥â ᤥ« ­®¯®§¤­¥¥. ‡¤¥áì ¬ë ¢ëïá­¨¬, ª ª ¢«¨ï¥â ­  áâàãªâãàã ã஢­¥© í­¥à£¨¨ ª®­¥ç­®áâ쬠ááë ¯à®â®­ . DZ®áª®«ìªã ¬ ááë í«¥ªâà®­  ¨ ¯à®â®­  à ¢­ëmeâ® ®â­®è¥­¨¥;= 0:51 Mevc2mpmemp= 938:27 Mev;c2= 0:55 10 3¯à¨¬¥à­® ­  ¯®à冷ª ¡®«ìè¥ §­ ç¥­¨ï2 = 0:53 10 4 ;¯®í⮬㠢«¨ï­¨¥ ª®­¥ç­®á⨠¬ ááë ¯à®â®­  ¢ ¯¥à¢®¬¯®à浪¥ ¯® ®â­®è¥­¨î ¬ ááá2áãé¥á⢥­­¥¥ íä䥪⮢ á«¥¤ãî饣® ¯®à浪  ¯® .—â®¡ë ¯®«ãç¨âì ï¢­ë¥ ä®à¬ã«ë, ãç¨â뢠î騥 íâ®â íä䥪â, ®¡à â¨¬áï ª £ ¬¨«ìâ®­¨ ­ã, ®¯¨á뢠î饬㠯®¢¥¤¥­¨¥ í«¥ªâà®­  ¢® ¢­¥è­¥¬ ¬ £­¨â­®¬ ¯®«¥:2e2H = 2m1 p~ ec A~ (~r)2B ~sH~reDZ®áª®«ì«ã ¯à®â®­ ®¡« ¤ ¥â ᮡá⢥­­ë¬ ¬ £­¨â­ë¬ ¬®¬¥­â®¬ M~ , â® ®­ ᮧ¤ ¥â¬ £­¨â­®¥ ¯®«¥~ ~rMA~ (~r) =r38㤥ন¢ ï ¢ £ ¬¨«ìâ®­¨ ­¥ «¨èì «¨­¥©­ë¥ ¯® ¢¥ªâ®à­®¬ã ¯®â¥­æ¨ «ã á« £ ¥¬ë¥,¯®«ã稬 ®¯¥à â®àH = H0 + Hhf ;£¤¥e212;H0 = 2m p~re~Hhf = 2me c ~pA~ (~r) + A~ (~r)~p2B~sH:e‡ ¬¥â¨¬, çâ®11~~~~~pA + A~p = ~p(M ~r) 3 + 3 (M ~r)~p :rrDZ®áª®«ìªã~ ~r) = (M~ ~r)~p = M~ (~r ~p) = h M~ ~l;p~(Mâ® ¯¥à¢®¥ á« £ ¥¬®¥ ¢ Hhf à ¢­®~~2B Mr3lDZਠ¢ëç¨á«¥­¨¨ ¢â®à®£® á« £ ¥¬®£® á«¥¤ã¥â ãç¥áâì ¢ ­¥¬ ᨭ£ã«ïà­®áâì ¢ ­ ç «¥ª®®à¤¨­ â.

DZਠ¤¨ää¥à¥­æ¨à®¢ ­¨¨ íâ® ¯à¨¢®¤¨â ª ¯®ï¢«¥­¨î Æ-ä㭪樨. ‚ १ã«ìâ â¥ £ ¬¨«ìâ®­¨ ­ ᢥàåâ®­ª®£® ¢§ ¨¬®¤¥©áâ¢¨ï ¯à¨­¨¬ ¥â ¢¨¤:~~~~~s)Æ (~r )g:Hhf = gB nf rI3l + 3(I ~rr)(5 ~s~r) Ir~3s + 83 (I~‚ í⮬ ¢ëà ¦¥­¨¨ n { 拉à­ë© ¬ £­¥â®­, g { £¨à®¬ £­¨â­®¥ ®â­®è¥­¨¥ ¯à®â®­ .“஢­¨ â®­ª®© áâàãªâãàë ¯à¨ ãç¥â¥ í­¥à£¨¨ Hhf à á饯«ïîâáï, ¯®áª®«ìªã ¢í⮬ á«ãç ¥ §­ ç¥­¨ï í­¥à£¨¨ ­ã¬¥àãîâáï ç¨á« ¬¨ n; l; j; f , £¤¥ ç¨á«  f ­ã¬¥àãîâᮡá⢥­­ë¥ §­ ç¥­¨ï ª¢ ¤à â  ¯®«­®£® ᯨ­   â®¬ ,F~ = ~s + I~:9.