karlov-kirichenko-kvantovaya-mekhanika-2016 (810755), страница 20

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'"-( !"" & ! 65* *- ! '55* 55 " !"5 ?"! .'!# -! ! ! ! 0- " !#4 "!5 !.!- 5'!"# 4! !# ("(% !! ! !"! =&;!& (! !# 6.5* * 6 6"# "! ! ; "( - !"# !" ! !"#&0J& "! ! M(ин)w21(ин)w12= B21 ρω ,AI = B12 ρω ,AIA21 =ω2π 2 c3ω B21IKOAIB21 = B12 .AI,- .0- Q:+ +C + :(# +- @*+ :(K+.0-0- Q:)(!/ 8( :(# < " "( "-( ! ! 5( ! !.! !- & I A 5 '- AN ' '"?<5 !;( -# 5 / (! & .(! ! -! - !6 5 !(6 '"-( "1( 6"# ! .(! ! ! '"-( 226! =&;!& B21 9 '15 65 *5 ! 05!# ! '".-( 0! ! ! ! ! 05!# 6.

'"-(% !; (! !( ! 6'"-& ( *! 0 - !!5 '"-( AI <-* 5* "(* ! ' !# ! ! -! "# .! '"-( " - !!* " !- ' - ! "(! ( ' - !6 '0 !(!( "! '"(! 5- "!# !"# ." 2! ! '"-( *(1* ( ;&ΔE = A21 ! " ! '"-( '5! ( 41! '54 "6 2 ;& AIN+6 2 " "(! ( 2.2! NI BΔω012π (ω0 − ω)2 + (Δω0 /2)2G ! !"#& ; " 6 '."-!"# " "1!"# *% "* (!& ; !& 0 " ( - ! ;5( ( ' - !6 ! - ! ".!"# !# τ C ! ! "405 6 5 ;(41 # (! ;4 !"#5* "& !! !41* *9 0' " !# !(( ! ." ! - !!5 * & 1/τ %"- τ !"(! 0& *!5& ;!0 0*.

"( ! -!05 '0( ! !" 4 4 D-. τ "(! ( !( ! '"-( 0'5'"-!"#5*" 65* * !;41* '05& # . % ! ! ! ;* '& !& ! '"- ("(! ( !5 2! "(41 ( ' !(( <!#;( '( ! '5( ! !( ; " ."(! ( !#4 ! * A21 MAIN% " ! 6"# ! 226! =&;!& A21 0- !!5 ! !( ; 0" 1 ! 5 * - !!* /. 05- "( ! '"-( ; " * 0-# ! "#5* "(* !"!"#5 ".65 6 5 0" 22! 14! ( '9 0" 01 2!"#5 *' .-41 * ( ' - !65 '0 ("(! (! '"- ! " ! 0' !# !"#.4 ; !! !414 ! ! AIO.86( G(ω) "!(! !04 G(ω)dω = 1 ! "! ( -! ; " " Δω0 ω0 )95 !"#5 & ! ! '"-( C ! . !# - !! ' ?"!"# (! !# ! !5&! !5& !!# wω(спон) = G(ω)w(спон) = G(ω)A21 .

! ) 7Δω0 = A21 .G(ω) =I A< ! 0* -!05wω(спон) dω = w(спон) .AIABAIAA%5; 05" 5 AI ('541 (! ! ! 6 '"-& =! ('# '.-! -! (! !# 6 '"-( ! - !! ' ! !"#4 "! !#wω(ин) = G(ω)w(ин) = G(ω)B21 ρω .AIAI< ! "- ! '"-( 0* -!05wω(ин) dω = G(ω)B21 ρω dω = w(ин) .AIAKC " 641 '"- *!- ! !& !-. ' ! ( ρω = ρδ(ω−ω0 ) δ(x) 3 "#!.26( ,9 AIAK !w(ин) = G(ω0 )B21 ρ =2π Δω0B21 ρ.AIA ?1 ' ;(41 !"#5 " .65* * #;! (! !# !* * 6.5* ' 5 *!- '"- 0! 6."# ; "%5 AIA ! ";4 ! !! % ."# "- "- Δω0 "(! ( !"# ! !&,- .0- Q:+ +C + :(# +- @*+ :(K+.0-L- h& :N (* *$/;& " "! "!# Δω0 = A21 7 226!5 =&.;!& A21 B21 ('5 0& (& 6"# !#4<! ' AI ! &! 226! B21 M< !&; ;( ("(! ( ;0"4 !"# " ' '( " !22! ," !" ! ' D! -! ." ; "" " ! ' !( ! 2 " '- AO ' '" ?I IB21 = Δω0π 2 c3ω 3.< ! ! 5 AIA '( "5 0'("-w(ин)=2π c3ω 3ρ=1 N2π T,AIA T = 2π/ω 3 "0& "( λ 3 " "5 ' '"-( "- N = nλ3 3 - " ! "! 0. #4 " "5 n = ρ/ω ! 5 " - " ! !'"-( *(1* ( -5& 0J9 0' " !"#( "( 0"4 * ;.

";# !5 '"- ! - " ! 6. '"-( 5* 6 & - !6& (6"# - " ! 641 '"-( *(1*.( "!5& 0J λ3 ! !! !41*"0& "(F; " 0 "" - !#4 ' !.(& ('5* !5 * '5! ( 55& ! *(1& ( !! !41 !( '"-! * * ' "4 ;& Δω 2.2! G(ω) ."- 5& ! *(1& ( !! !41 !("1! * ' * '"- ! ;& Δω !! ! !"#& ' !#4 G(ω) 7' .!# 4."0 "4 !"#4 ! ! G(ω).! " ! '(! ' 0"( 0 0.5* < ; ' ! ! 5 .!"#( ' !# G(ω) !# ( !"#( *! ! ! ! & * ! ' !& *! .! 6 ' ! " '& ! 0"#! *! 5 0' "( * !. 0"(< ;( ("(4! ( ; ! ! ; !"!"# '* "' ' '( , ." 3 ; = !"# 0"45 !"#.5 " ! (!# ( 0 !!5 " 0 '.;& '6& "#* ;5* "& % !* "(* ( - !6 " - !6 '"-! " "1! "* ";# - ! & !"# 0"4& " !( !"#( "( '5! ( ;& <-& ;( ! 05!# "40 0 !(!"# ! (1 '"-( "(* '"-( "1( "( '5* ."# 5* ! " 0"( - !6I K" # 8= 28 234!% " ! 226! =&;!& B12 = B21 6.5 *5 * ' '"- ' * "1 -! ! (!5 % 6" & !& ! 55* * *& * '5! ( 0"#; - 0!5* * !-! ' - !6 0"#; - * =! ! "14 .

; ' '"-( 641 !! !41*5,& !!"# ' ; "( '"-( -. 0J !& !5 "(! ( ' !#4 & '"-.5* "15* "#5* ** ' * < !# ! - 0J ! n ' 5* ! n = n1 + n2 5 A I ! (! ( !! ! * (9 '"-( 1 !# n2 w21 ω "1( 3 n1 w12 ω ?"!"# !! ! AIA !# '( "! ! "- 55* !(& A I dρdt= (n2 − n1 )B212ρπ Δω0ω.AIA=! "- n1 > n2 !6!"# =( ; "( "1.! ( < '! "1 !& " ! n1 n2 '(.4! ( (( ! 5 !# "1( <0( !"# "#5"( "( 1 ! 5 ! ( "* 05-& "'&"&& ! "( !& *! ' !# 226!"1( '"-( ! ! !,"( '"-( !(41 ( "5 01& .!#4 c " z 226! "1( "(! ( α=−1 dII dz.AIA.AIAN< "# I = ][^_` · ρ d z = cdt !α=−1 dρcρ dt9 ' AIA 5 "-α = (n1 − n2 )B212π cΔω0ω.AIAOI,- .0- Q:+ +C + :(# +- @*+ :(K+% !- ' " n2 n1 ≈ n α = nB212ωπ cΔω0.AIIB=! 54 !# 4 0" "(4 2< " "5* 0'& "#'; # 1 ' ('#4 AI 226! =&;!& ( ! " "5'"-( λ 5 ' !# 226! "1( α AIIAα = nσ.=! ' # ! ! ! '5 22! -.

σ '& !( - !65 ' 5 "!!5 "" ! - ' "1( *!'41 ".1!"#5 & ! - !65 =! "- '5! ( &σ=λ2 A212π Δω0.AIII% "# "- ! ! ;( A21 = Δω0 ! -!λ2σ= % "# ! *!5 '-( -( "1( '.2π ! ! !"# ' ! * !! "! -# ; ' 10−12 ÷ 10−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

! "'& 2'% ! ! ! - !6 t " '"-( '0! ( - !M !( - !6 " - " '"-( ?!! !. ( !& !5 !"(! ( ( ( !"#'(!5* - !65 "( '"-( "4 ( '& !( !6 5! ( "& 26& "!(41& .4 c ! *! '1 ! '&.!( !! !41& '& !(G ! "# '& ! "!- "(1E(t) = E0 cos ω t = E0 eiω t + e−iω tAIIK2"!!& "5 *& !& - !6& < !# ."#5& ! * ! ( 9 ('& !( - !65 " !"(!V = −dE(t).AII ,"( !!5 "#&; 0 -!!# !5 """#5< !# !& - !!5 "( ω ! !"# - !!5 * ω21Δ = ω − ω21 "MAII|Δ| ω21 , ω.AII=! !# " ! -! ! '& ! ("(! (' 5 / " !* ( '& !( "; " '! #; !- !(( (MV = dE ω21 .AII< ; ! "( '1 0 !5* - !! - !.65 !! ( '!5 (! ' !(! & "&5&5 " ,"( *; ';5* * ! d 5;4110−18 ?>?= A ,0& " AII -! '! ;!#("( 2 '"-( ! ! 1011 ÷ 1013 %! 2 "( "#!2"! 3 1013 ÷ 1015 %! 2 ! !! ( !(1 ( " !5 ! "'&2' ! 0* !# "41 0 !.(!"# ! ,"( *( ' *! '& !( .( !# "( '"-(E " 5;!# !4 (.

!# "( Eат. ∼ e a2 = 5 · 109 %/ " 0"!# ( &M,- .0- Q:+ +C + :(# +- @*+ :(K+.0-9- K/K &E Eат .AIIND # "; !# -! e = 4,8 · 10−10 ?>?= 3 '( "! a = 0, 53 · 10−8 3 & 0 & 0!5 ! . <"4 "-5 5 · 109 %/ !! !! ! !# "&"(' '"-( ( 3,5 · 1016 %!/ 2 <! " *!( " AII '5! ( 0" !< 5" "& AII : AIIN c .541 ! '& ! ;! ( ' .0" '5 ! 0" 141& ( "5< !# ( ( !"# '(!5* - !65 "( '"-( 5! ( "#! Ĥ0 9 "( Ψ.26 - !65 "'"-( 5 < !# ( Ψ.26( !"(! 0& '64 *5*"5* 26& - !65 ' (1 ! 226!MI i∂Ψ∂t= (Ĥ0 + V̂ )Ψ,AIIO ! '& !( V̂ " - !6& !! !! '& !( AII =! '-! -! "# 0".

! '& !( ! ( ! "# !* dˆM1ˆAIKBV̂ = − E0 eiω t + e−iω t d.2 ! "# ! * dˆ !- !-5"!5ˆ 1 d x d y d z = ϕ∗1 dϕˆ 2 d x d y d z = d21 ,d12 = ϕ∗2 dϕAIKA-!54! 62- 4 !& !& - !65 '& ! " - "!!5 " ?!65"5 26 ϕ1 ϕ2 ' (! !"# ! ! !5* ! "!(( ( !6 4 cMAIKIĤ0 ϕ1 = E1 ϕ1 , Ĥ0 ϕ2 = E2 ϕ2 ," !#4 *!'4! !(( A I !& !& .!5 E1 < E2 )9 - !! 6""(6& '1( AIKB 0"' 0 !.& - !! * ω21 = (E2 − E1 )/  !""#$ !""%$ & ' ! 28 @:!D (1 ! c AIIO ' 0" "( *& - !65 ! ( ( "("! "5* 26& ! !(& - !65 <".- ! (Ψ = a1 (t)Φ1 + a2 (t)Φ2 ,I AIKKΦ1 = ϕ1 (r) exp (−iE1 t/), Φ2 = ϕ2 (r) exp (−iE2 t/),i ' (1 !"# ! ! !5* ! 26 ϕ1 ϕ2 !#!65 "5 26 !(& A I ;& *&!& !5 "!"# "!(4! ( AIKI< ! AIKK AIIO -! AIKI F( "-;.( "- ϕ∗1 ! "- 3 ϕ∗2 -" !( ! !5 5 -!! ! "5* 26& * (M!da1i 1 = − a2 eiω t + e−iω t e−iω21 t E0 d12 ,dt2AIK !da1i 2 = − a1 eiω t + e−iω t eiω21 t E0 d21 .2dt% !* (* "#' " AIKA !-5* "!! "# ! *F ! !# ( AIK -!( " ' ! AII ! 05 ! 6""41 -"5 exp [±i(ω + ω21 )t] ≈≈ exp [±2iω21 t]Miida1dtda21= − E0 d12 a2 e−i(ω21 −ω)t ,21= − E0 d21 a1 ei(ω21 −ω)t .AIK2dt< ! !# " (41 ( 226!5a1 a2 !- Ma1 = ψ1 exp [−i(ω21 − ω) t/2],a2 = ψ2 exp[i(ω21 − ω) t/2].AIK,"( "! * 6""(6& ψ1 ψ2 ' AIK "-4! ( ! (iidψ1dtdψ2dt11= − (ω21 − ω) ψ1 − E0 d12 ψ2 ,221122= (ω21 − ω) ψ2 − E0 d12 ψ1 ,AIK!5 ("(4! ( 5 ( ' 0"(I N,- .0- Q:+ +C + :(# +- @*+ :(K+* ;!; #2dt !"(( '"#!! ' * " "-!# "( 26 ψ2 ! ( ! '5! ( -.

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