karlov-kirichenko-kvantovaya-mekhanika-2016 (810755), страница 17
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!& - !65 !5* !# 6" - " " "# ("(4! (iϕAO1* ! 3;! 34 D # ! !!# ( ! !! "5* 26&-!541* "- - !65 5 " ! ( 6."# '"- 0!"#5 ! C " 5&(' 55 ! !5* & .!"(! ( ! 226( ! " "! !& ! 0- !& !& *! !& '.! ( !5 0 - " - ! ! & 6& #?!! ! "( 26( - !65 ' ! ! .! ! ! & & sz Mψ = ψ(x, y, z; sz ).OAA< "# "- sz ("(! ( !& & 041&2s + 1 '-& ! 26( OAA !# 1 ! !# 2s + 126& ! % - ! "- s = 1/2 ( sz 0.! '-( ! & 26 ψ 0 !!# 26Ma = ψ(x, y, z, +1/2),OAIb = ψ(x, y, z, −1/2).<"4 "4 264 ! ' 5!# !"06a(x, y, z)ψ(x, y, z, sz ) =.OAKb(x, y, z)!5 a b !# "!5 (! ! *( - !65 5* !((* sz = +1/2 sz = −1/2 !! ! <"((! !# 0!# - !6 ! ! !- r !#dW = (|a(r)|2 + |b(r)|2 )dV = ψ + (r, sz ) ψ(r, sz )dV.OA D # 6( ! (( "( !"06 OAKMψ + = (a∗ , b∗ ),OA,- >- '(K: ' J :( $)/ 5 ` >-<- " * & " - ψ + ψ 1 !"(! ( 05-5 "( !6 "- 3 ! !"06C " - !6 *! ( " !( ! "4 264 !!# ,"( ! -!05 ' !# "#4 "4 264 "5 0' !'!# < ! "( !5 0' ."-AONψ1 = a(r)χ+1/2OA"( "-( sz = +1/2 "ψαβ (q1 , q2 ) =AOO1√2[ψα (q1 ) ψβ (q2 ) + ψα (q2 ) ψβ (q1 )] ,OIA1√2[ψα (q1 ) ψβ (q2 ) − ψα (q2 ) ψβ (q1 )] .OII "- 2ψ2 = b(r)χ−1/2OA"( "-( sz = −1/2 % !* 2"* "-5 χ±1/2 !# !"065 10χ+1/2 = 0 , χ−1/2 = 1 .OANC " 5 " - !6& "# ! "( 26(! ( "( 3 !& 26 ! % "- - !.65 s > 0 "( 26( ! 05!# !" !"06- " ! 2s + 11, E . 4 ! .3 ; 3< " !* '-& ! ! * ! !5* - !6<" -! 05- " '& ! ! !.!! >"#! !5 ! ' 5! ( Ĥ = Ĥ1 + Ĥ2 , Ĥ1 Ĥ2 3 "#!5 !"# & !& - !6 < !#- !6 A *! ( ! !( α - !6 I 3 ! !( β /0'-( "5 26 !* - !6 ψα (r1 ) ψβ (r2 ) ( "( - !-5* !65* !(&MĤ1 ψα (r1 ) = Eα ψα (r1 ),Ĥ2 ψβ (r2 ) = Eβ ψβ (r2 ).OAO7 ! ! (! !# ! 05!( -! "! *! ( !- (x1 , y1 , z1 ) !& 3 !- (x2 , y2 , z2 ) ".! (! ! ! 05!( - '4ψαβ (r1 , r2 ) = ψα (r1 ) ψβ (r2 ).OIBD # ! "! -! !# 5! !! 2! -! 5& "! *! ( & !- !& 3 !.& / "!5 ! !5 ' !# *.! ( & ' 5 '"-5* "! "#'( ".!# ' !!& ' "! * ! ! !- 3 #!!& ! ! ! 2"#& !- '( 26( OIB "!(! & ! ! OO " OAB ! .
"( ( - !6 ! !5*ψαβ (q1 , q2 ) =√% !* 2"* -5& !"# 1/ 2 ".41& ' !# 6 - !-5* "5*26& ψα (r1 ) ψβ (r2 )C " 05 5 " ! ' 0"#; - " - !6 ! "5 05"05 ' ! !! !414 !'64 ! 0' -!05 ! 5 - !6 "( 26( '" # !.-& "- 0' ! !-& "- 29# !"(! ! "-!# 6 <" % " " 2 *(! ( ! !( ! α = β ! ' OII " "!ψαα (q1 , q2 ) =1√2[ψα (q1 ) ψα (q2 ) − ψα (q2 ) ψα (q1 )] ≡ 0.=! '-! -! 1 !! * ! !5* 2 !505 *" # ! ! !( ! ' OII -! " 25 *(! ( !( s1z = s2z ! "( 26( !5 ψαβ (q1 , q2 ) 01! ( "# r2 → r1 ! q1 → q2 ) 5 " 2 . !( ! *!# ( & !- ! ! =! !!!# ! -! ! !.5 2 '! 22! !!" ('& ! *."0 05-5* "' OIA -! "( 0' 05 !( 0.!4!M 0'5 ! *!# ( 5* !5* !((* .! - 0"!# ( " - .! "5 !!"( (! !4! !1/ @.
2 .#' ' -! 0"( 6 <" *!'& !( * ! !5* 2 ' ! ! " . !5 9( ' !# '! " ! & ! !"& 26 ! !"# ! - !6 1 ! !.! " 5& "5& *! * ( "( " '& ! - !6 ! ! !!,- >- '(K: ' J :( $)/ 5 ` >-<- " * &%' "( - !6 0 "" * ! ! !#4 .'54! =! '& ! ("(! ( - !!5 22! ! * " - & *? !& "& " ! 0 '& ! .!"(!# "41 0' G ! "! ' !5* *! ( ! 6! !- r1 !& 3 ! 6! !- r2 5& "! ! ! ( ! !! < ! 5 '!# & ' * !*! ( 9 0' *! 55& 0 26.& ! ' !5 5 !( 0"4!# !5&! " 0"45 ("(C " "!5 *(! ( !( ! !"# 0"'!# ( " ! 6 <" 3 '! * 22!.
!!" "# 25 *(! ( "#; ! - ! '(5 0'5 "! !!- ( (! ( C " 5 "! !5 !"! 6 <" '1! "! 0"!# ( / !"! !!- ( ( * '& !( 5;! (' ' -! 0 '& ! (! 405-5* '& !( " ! "( - !6 ! !7& !5 "4-41& "!7 -! " 5 "! """#5 ↑↑) ! 5 . 0 ! !((* " 5 !"""#5 ↑↓) ! ! !(( ( '- " !5 ! !.(( '54! ! !"!5 s = 1, 2s + 1 = 3) "!5s = 0, 2s + 1 = 1)=( " '& !( "! '& ! !- "(! !( !5 1) <! "4 "4 264 !-414 " '-.4 !5 !!# '(IBBV (r1 , r2 ) =e2r12,OIK r1 r2 3 5.!5 "! r12 = |r1 − r2 | 3 !( < 01 " '- !& ! (2"&V = Ψ∗ (r1 , r2 )V (r1 , r2 )Ψ(r1 , r2 )dV1 dV2 ,OI "(2 26( Ψ(r1 , r2 ) "! ( & "|Ψ(r1 , r2 )| dV1 dV2 = 1 ?-!( "! '& ! "05 4 '& ! "! ( 5 0".
'!# "4 264 Ψ(r1 , r2 ) 26& !& ".!5 -!4! ( 01 '& !41 ,"( ! -!05 !!# !04 264 -! -! " Ψ(r1 , r2 , s1z , s2z ) = χ(s1z , s2z )Φ(r1 , r2 ).IBAOID # χ(s1z , s2z ) 3 ( - !# " !#4 "(( '-(6& # z Φ(r1 , r2 ) 3 ! !( - !#7! (!# -! ( - !# "!(41( "4χ(s2z , s1z ) = +χ(s1z , s2z ),OI!-! ! !(4 ! s = 1 C " (- !# ! !-Mχ(s2z , s1z ) = −χ(s1z , s2z ),OI! 5 " !( 5& s = 0 ?'. !! ( -5 " - !# -! "- ! !(( 0 !5 ! 01 (! ".( ! "- !(( 5 !"""#5!( ! ! *!( 05 2"# ! '4 !(( !5," -! -! "( 2 5 ("(4! ( "!5 "( 26( OI " 05!# ! !-& <! ".- ! !(( ( - !# "& 26 !- ! !( - !# " 05!# ! !-&MΦ(r2 , r1 ) = −Φ(r1 , r2 ),OIN "- !(( ( - !# ! !- 3 .!-&MΦ(r2 , r1 ) = +Φ(r1 , r2 ).OIOC " 0'-!# -' ϕα (r) "!4 "4 264 !-! ' ' !#Φa (r1 , r2 ) =1√ [ϕα (r1 )ϕβ (r2 ) − ϕα (r2 )ϕβ (r1 )]2OKB1√ [ϕα (r1 )ϕβ (r2 ) + ϕα (r2 )ϕβ (r1 )]2OKA"( ! !(( Φs (r1 , r2 ) ="( !(( % OKB a 26 Φ '-! ! !.-4 ! !4 - !# OKA s 3 !-4 5 α β '54! !5 !(( "! ! !((1)(, , + 41 ,8 " ) D=1 +)B ) =),- >- '(K: ' J :( $)/ 5 ` >-<- " * &9# ! ' !# 5 "( & '&.!( "! <"( -! ( - !# "& 26 .
6 χ+ χ = 1) " OIK OI OKB OKA*1e2V =|ϕα (r1 )ϕβ (r2 ) ± ϕα (r2 )ϕβ (r1 )|2 dV1 dV2 = Iкул ± Iоб .0" ! #;4 !!"( ( * !(.4 0" ! "- !& "! 0" !( ! !! !!; " !& #;4 * & !- & % ! "( ( !5 ! ( ("(! ( !!- "# !( !'41 "" !(C " "!5 *(! ( !( -! !.! !! ! *(1& ( ! !( ! "!5 '5.4! ( ! ! ' 5 '5 ! !!" ( "! ( 3 !6"#& !! ( 0 .- " ! ! ! ' !! ((!- ( ( "! " !( !&U0" ! ( =! ! "-4 !5 '. ! 0'!# "" .! !(r0r9 0' (' !(. * ! 1 !! !"# "- " s = 0 "!.
!(( ! !( s = 1 " !"! .!( '5! ( !&-5D !# !6"#& 2 /) 5'& !( * ! . 1 +) ) D ! !(( r ( "( ,- @↑↑)!. !(& ' @↑↓) 7- r0 OA ""4 !41 ' +- 1 D% '"4- !& "5 & )4, )) 1 ! 541 .4 0 '& !( ".! < !# !5 ! "! !# ŝ1 ŝ2 ' ( ' ( 1 13- s21 = s22 =+ 1 = ," '! -!IBI2r12OKID + !-! !(4 s = 0) ' − 3 ! !(4s = 1) % OKI 5 0'-(ρα (r1 )ρβ (r2 )Iкул =dV1 dV2 ,OKKIобм =r12ρ∗αβ (r1 )ραβ (r2 )r12dV1 dV2 .OK ρα (r) = e2 |ϕα (r)|2 ,OKραβ (r) = e2 ϕ∗α (r) ϕβ (r).OK ' 5* 2" ( '& !( "! !! ' * - !&M Iкул Iоб %5 "( Iкул ! " - & !"(! 0& 05-4 4 " '& !(?" Iоб 0 "" ! ! !#4 - !6 '5! ( % " 4 4 *! 05-( "! !# '(OK % 04 4 *! "- OK !4 " '!# 0& "! !#4 '( C ! "( | ραβ (r) |2"(! "! !# (! ! *( * ".! & !- ! ! "( "-( 22!5 ! !. ! -!54! ( /- -! ! "- !"- ! "( ""5 26 ! "! 54! ( !* "(* -4! ("(!# ( 22!5 0 ""5 ! !.
!#4 - !6% 0"#; ! "- 05& !" "!" <! .( ! !(( s = 1) '5! ( #; - ( !(.( s = 0) =!! 2! ! !*- 0 *!( "( - ! "-( *"!& !5 - !". " E ! !( "!& !5 " 05!# "#5/'5! ( 0 '& ! ! 1 !4"# 0' "!& (' ""* % - ! ! "" H2 C " "!5 '& !41* ! *(! ( !."5* 5* !((* -! !! !! !(4 "."5 ! ! 0"!# ( % '"#!! ! ! ("!5 ! *!# ( 0"#;& (! !#4 2( !&0" ! "! 0" % 4 -# 41 & !22ŝ1 ŝ2 =IBK412!(ŝ1 + ŝ2 )2 − ŝ21 − ŝ22 ./! 4 ( -! 0 !5 '-( ! ŝ1 ŝ2 !#11 113s1 s2 =+1 =s(s + 1) − 2 ·s(s + 1) − .22?!! ! !1212222[1 + 4ŝ1 ŝ2 ] ! 0 !5 '-((1 + 4s1 s2 ) =−1,s = 0,+1,s = 1.OKIB,- >- '(K: ' J :( $)/ 5 ` / - ' (! s 5414 0 '& ! "! !!# 1OKNV̂об = Iоб [1 + 4ŝ1 ŝ2 ] .
6 ),B ,1,?2/001 "-& !5 "4-41& '"# - " ".! !"(! !M1Jik+ 2ŝi ŝk ,OKOV̂об =i, k2 '! ( "! !5 %"-.5 Jik '54! ( 05 !" " Iоб ) /0! ! -! 5 OKN " OKO 5"(! 2 ! ! '& !( 5* !5* ! "!. % ! &- "- !( !! V̂об ('. !5 '& ! ("(! ( (" 6 <." D! -! 05 '& ! 0 "" ("2!' " ! "& % >&'0 5! ( "#! OKO@F : =A! *!'! ( "- ! ! .0 ';5* 1 !4 !5* !(& 3 !5*& D '! ! " *(! *5 ! " 5 (5 .. .3 % ANON ? & !!- '" 141 & !""!- "('5* !5* " 41* !"" ' !( "" -! !( " ! 0 !5& !"# !05 0!# ( ' ! '( e 141& ( & 0! z ABA % !& !6 '( '.! "# '"- "# - (! ( "#5&! "! d = er ! !"#.
( D! -! " '.z "! " z "(& " !0!5 '"- ! 4"('64 " ."( z 3 "&4!05 "4-!# "44 !.!- !!# ! 0"#;r- " "& "1* & " .e ! 9 '"#!! !2.6 " ! !"#5* "& .!"(41( '"-( " 2 . !B 8?8"( z -'! ,1 = @ =I?G ! ! < !# :1A! ' -! '"-( ."5 ! -! "#'!# ( 2" 541 1 "! ( <" "( !!5 -! "!*! ( & 0! r C " "( !# 1("! !# ω ! 6!0( " Fц.б. =me υ 2r= me ω 2 r.ABA,- .Z- @ K+ :(# +IB.Z-.- Q: : ! 2?" " !(( (e2Fкул =ABIr2;! 6!04 "Me2me ω 2 r =r2ABK./! 4 * ('# "& ! "! 0!5Me2ω=.AB 3me r<"( ( "! -! ABK E=me υ 22−e2r=me ω 2 r 22−e2r=−e22r.ABIB<0 (!# - ! 5 " 0 ! ! 2!! ! 2!L 8! ! (' !& "&/ !& " 5 ' -! ! -( " -! "!"#"( ' ! " ( "('6( % ! ?%.!*-!4! 0!!# "& "('& "& !& .1 !# " ! 7 ' !"# "&( 6."(( "('6( !4 ! !"(4! "&5' "(5 "('6 2' π/2% !& * 0 ' *5 "('6 0!# ".&5 6"(5 "5 !"5 !5 ABI ABI 9 "& "('( " ABI !# '6( * " & "('6& 0! .
- " ! ! !5 505yD -! '"-( 0!5 (! ( < !dE =e22r 2? -! AB ! "# "! !!# e222L = me υr = me ω r = me r= e2 me r,AB3me r! -! ' 0!5 '5! (1e2 medL =dr.ABN "#'( AB ABN *dEe2== ω.3ABO2dLrme r< "# "( ( ! "# !5 "! ++ ! *(4! ( ! ! 4 ! ! "# ('# !5* "(! ( && 2"& ABO %"- ω ! ! - !!& '"-( "# "(! - !!'( "# ! '( "! ' ! -! ! . ! ( 6( ! & dE = ω 9 !! !ABO ! 0! !# ! "# dL = M ! "-0! 05!# 6( 0!"# ! Lz = m "! '"- 2!?' '5! ! -! "!- "# '"-. 4! ( 2!5 s = 1.kABdr.ABABakájxâ2 . +5 )1 U A +5E "A +5E A +5 @ϕ = ZrOSPAG ! 6"( "('5& ! %! 1! ( % "!& !& " 6"( "('& .! !! 1 ! " - ! !5* " ".& ! ! 1! ( ! " " 4 "5 ! " !<! ( -! 5 *! !*- !# 2!! '5! ( -! 2! "4 C " 05 2! 05"05 "4 ! ! 05 05!# "(' 5 " ' "' 22! ? 2! "- .!# s = 1 F "40& & - !65 A '52s + 1 = 3 6 '0 "M sz = +1, 0, −1F 2! " 0 !! -M ! *!# ( !"# * .5* !((* 6( " !(5 !"# sz = +1 sz = −1 =! & ! 2! 6! (- !#4 !& "!!& "5% " " ! - ! "!!5* " .
0 !& !6 5 ! !"! <! ! 65& "('6& "5 ! 1 ! ! !# !"# !6M ".4 !( "5 ! "4 " ! !IBN,- .Z- @ K+ :(# +.Z-0- e! ! :("(*/ ( )( <"('6 !( 0J! 0 *!.'!# !5 - " '55 "45 6 " "# M1λ = (sp).p−vaaABAAC " λ > 0 ! (! -! - !6 ! 4 " !4"# !# " λ < 0 "# !# '5! ( "& " ".!&,"( 2! !! ! ( '5 "('65 !(( "# !# ! !# '-(Mλ = ±1.vIBOABAI<0 !# & !* 2! * "(! ! &!& * '!!"# % ! ! "!# - !5 !( 8! ! ( !#4 ! ! -!' &! ! ! -! 05 " ( 7 ! '-!-! ' '"!# "5& "& ! 0!"#4- !# ! 0 !5& ! % (' ! ! -! 2! s = 1 '-! !"# ! -! ! '- !# "- ! ! "# ! ! ! 2! ," !-! 2!5 '4! '"#!! ( '( !(( !.5 - !6 '( ** "! !* *!- * & ?!! ! ! "# 2.! "(! ( ! ' ! "# !5';" '"#!! ( " "1(/ ! ( 0 " 26 !(& 2!,"( ! 0! ( ! (! -! !#5 I @ 2 ! <(! -! ! "(! & ! "& 26 0.' ! ?' - -! ! -! !# !."-! ( ! -! ! ! !"# ! - !6 !& 5" 51& " (' 2"& 6 <"< 0' 4! 0' '"#.
!(Mr → −rABAK ABK 8"# & ! ! 0'( ' 5! ( ".41 0' % ! P̂ !5& & !! ."ABA P̂ r = −r.aá2 . 1 @A 1 @"A =+ @4 +A?!! ! & ! ! ! "4 264 !! "41MABAP̂ Ψ(r) = Ψ(−r).=! " '"(! &! 0 !5 '-( ! P̂ <.(( 5 !! ! "& 26 5 "5 "-!#" ABA ! ! 0'MP̂ 2 Ψ(r) = Ψ(r).ABAC " 0'-!# " P 0 ! '- ! P̂ ! !.; ABA '-! -!P 2 = 1 " P = ±1.ABAC " ! !( ! ( P = +1 ! (! -! !( ("(! ( !( P = −1 3 < !# ! ( !( 26( ! f (r) C " ! 26( (! ( !* !5 ! ! 0'. ! '5! ( "( C " !! !"# ! (! ' 0' ! '5! ( "( "- 4! ( "(5 !5 "#5 !5 " !5 ! ! 0( 0'.(* 1( ! ! 3 05-5 !5 % ! ("(5 !5 *!'4! ( "5 " .! ! ! " ! '! ( " !.! ! 50 "!"# "( 1( " 0- < ! '" ABK "(5 !.5 (4! " !" "#5 !5 "( (4!8"# & ! ! P̂ "(5& "#.5& !5 ' 5! ( "41 0'MP̂ v = −v,P̂ a = a.ABAN,- .Z- @ K+ :(# +.Z-3- ?) #(/ $< "(5* ! ("(4! ( .! !- !"# ! ( ! "!- "( ! < "#5* ! ("(4! ( ! ! "# L = [r, p] !( ! ! "( ! ! ( < ! "( -! !# !5 ! 05!# !" P = PA PB (−1)l ,ABIIIAB5" I .7 47 @J7& -! !# - !65 ('4 6!"# "/'5! ( !6 !( "( 26( ! 05!# !" Ψ(r) = R(r)Ylm (θ, ϕ),ABAOYlm (θ, ϕ) = eimϕ Plm (cos θ).D # Plm (z) 3 5 "5 +MPlm (z) = (−1)m (1 − z 2 )m/2dmdz mPl (z),541 ( -' 05-5 "5 +MPl (z) =1dl2l l!dz l l 3 ! "# ! !"# ( - !& A B (−1)l 3-! !# "& 26 ! !"# (="!5 - !6 54! !44 -! !# "4.1 0'C " - !6 3 0' ! ' "4- 2! -! !# !!# " '- "(! ( ' ! !.5* *(! 1( - !6 5"(! ( ' *(-! !C " - !65 s = 0 0"4! -! !#4 P = +1 ! *'54! " -! !#4 P = −1 3 ! ,"( 2 !(( -! !# " '- "(* ! ( ! -! !# !.;4 !5 2!"#5 - !6 % - ! ! "( ".! ! &! ?!! ! "( "& 26 ! '-!Plm (cos (π − θ)) = Plm (− cos θ) = (−1)l−m Plm (cos θ),eim(π+ϕ) = (−1)m eimϕ ,ABIKP = +1.(z 2 − 1)l .% - ! ! Pl0 (z)≡Pl (z) <0 & ! !* " '!# ( "40 -0 !!- & 2' ,"( !!- ! -! ! "5 0"4! ! & ! -! Plm (−z) == (−1)l−m Plm (z)<0' ! ( 2- * !* 'θ → π − θ,ABIBϕ → π + ϕ.ABIAYlm (π − θ, π + ϕ) = (−1)m (−1)l−m Ylm (θ, ϕ) = (−1)l Ylm (θ, ϕ) .9 0' !(( "5 "5 ! 4! -!.
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