karlov-kirichenko-kvantovaya-mekhanika-2016 (810755), страница 28
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2D # 5 '" ' !# U0 − U "-& hν0 ," -! -!' ! !" '"#!! ( 2! " ! -! "!# V + V ≈ 2V =! !M (V − V )V = h(ν − ν0 ).hD(( ' M (V − V ) "-& k = k !! !2π' *( "# "-kV = 2π(ν − ν0 ).?" A "4 " - " k = 2πν/c ;"( ' kV kV = kV cos θ =2πνcV cos θ.% '"#!! ' "- 5 "( - !!5Mν0Vν=≈ ν0 1 + cos θ .1 − (V /c) cos θcND! -! 5 " 5 "& 2"5 & "!.( -! θ = 0 !- '"-( ! ! ( !- ! *( 5 - !! 2" - ! ' 54! ω = ω0 + kV,OQ(Q( " " ! k = ω0 /c< ! 5 5( "( 22! ," "( "-('"#5* ! - " 0"#;* !& ( !-< !# !- ! " *! ( ! ! -! S .1& ( !#4 V ! !"# "0!& !5 S '.
! "( ( E "# ! 0'4! .! .!5 ! 0'4! ( * ! & 6"#& !5! -! & 2" 0'( +6 % - ! !,"( '"-( "! !# !# 26( !"#!!5 ?"!"# !(( ( '"-( *(1 ( 0J V U (T, V ) = ρ(T )V.I7& ' !# ρ ! !!5 "#' !.- !;MKIE =E − pV cos θ .1 − V 2 c2ABD # θ 3 " ! ! -! ! ! 2.! " ( !5 S ," "#' "41!;(ME = hν0 , E = hν, p = E/c = hν/c.AA< !"(( ! 5( N ';( "-; ( ! !.!"# ν "-ν = ν01 − V 2 c21 − (V /c) cos θ.AI=! 2" " V c *! 2" "-4 5;' 5* 0&% - ! "- θ = 0 ' AB "!1 + V /cν = ν0,AK1 − V /c-! !-! 05- 54 "( "# 22! ," . !- " 0"! ( "# (& (4.1& * C " θ = π/2 ! ' AB "!ν = ν0 1 − V 2 c2 .A =! 2" 5! -5& 22! ," !-41& "-4 " 5"! ( "( "4 ( !-.D - "#'( '5 ! &! !!4' !# "! ! '"-(G ; 8!5 3 ! "#!"(! ! - !65 1.
( !#4 υ = c <! " 2! ' (' "!. !#4 !;P = ρ/3.AKIKT dS = dU + P dV, S 3 !( P 3 " V 3 0J ? & !5 -!(U = U (T, V ) ' !# ∂U∂UT dS =dT ++ P dV. ∂T∂VVT<"( ' # dT = 0 ∂S∂UT=+ P.∂V∂VTT,"( "4-( ! ' ! ! "#' ( !.! ) "" "( 0& F = U −T S " ∂F∂F ! dF = −SdT − P dV <! S = −, P =−∂T V∂V T ∂S∂P= 9 0' "-! "! ! !T∂V∂P∂TT=V∂T∂UV∂V+ P.T< ! 4 5& A I "( "( P U '"-( ! 226"# 4T1 dρ3 dT1= ρ + ρ,3"dρdT=4ρT.G; "- ( ! ρ = aT 4 .N !! a ! 05!# " !"# ! !D! -! "( '! ' ; "- 3 2! . '"-( "! !# ! ! ( 2"&1q = nε υ,4O n 3 6!6( - !6 - !\ 3 ) υ 3 !# - !6 ε 3 .( *(1( ( - !6 "5 0 '-4! !! !- "4 - !6 ,"( 2! ! "- qQ(Q(! ! !#4 '"-( I < "# υ = c nε = ρ 3"! !# !cρcaI== T 4.AB7& !# !" !# !( " CP ,"( .
'"-( '0- & 6 P = ][^_` ("(! ( . '!- T = ][^_` "# ! ! !(( P = ρ/3 ! ρ = ρ(T ) =! '-! -! .6 dT = 0 ?"!"# CP = ∞ =!! '"#!! "-!#2"# ' !;( % " ; ! !;"( "-( P = ][^_` (∂U/∂V )T + P∂U∂U∂VCP =++P= CV +. NKI44? ! ! ' ?!2 : @"#6 I = σT 4 '."(! 5'!# !! a -' !(4 ?!2 : @"#6 σ Ma=4σcAA.∂TD - 7&! !" !# '"-(G ; "#' !- ! ! @ *.!# ' ' *( 2Aδ Q = dU + P dV.
' ! !" !# "(! ( !; δQCпроцесс =.dTпроцессI/ - ' ! ! ! 6 -! -! 6 )5 & !" !# * 6 *M '*- V == ][^_` '0- P = ][^_`G !( !44 4 264 !!5 0J. '; ∂U∂UdU =dT +dV.K∂T∂VVT? -! ! A : K * 5 "( !" !M ∂U∂UCпроцесс dT =dT ++ P dV.∂T∂VVT ,"( *( !" ! !( 0J " ' #dV = 0 ! -! '5! ( ∂UCV =.∂TV 05" ' '- !(( ( '".-( '41 0J V 41 !! T U = aT 4 V, a 3 !( !! 9 ' "! -!CV = VdρdT= 4aT 3 V.V∂VT∂TKIP(∂T /∂V )P,"( '"-( "-5 !(1 - "!" 4!"!"#5 '-( "( * "5* !! <'( (∂T /∂V )P !(1( '!" "4 "# !(" !! '"-( ' ! ! 0J "( !(( =! '-! -! CP = ∞D - <"-!# ('# 0J& "! !#4 . !" '"-( '"-!"#& 0 !#4 !"G ; ,"( "!5 '"( !"# !5"( 2!!5 1 Φ '5! ( "- ! '"-( . 6 [Φ] = %! %5" '"- - !! "& ! ! - !!5 ω 9 ! '"-( Φω !;dΦ = Φω dω.A<! dΦ "1 * ! !" dS !" 5& " dΩdΦ = B dΩ dS⊥ = B dΩ dS cos Θ.I%"- B '5! ( " 1 ? & (.' RMR = B cos Θ dΩ.K- Bω - !! ω "(.! ( !;dB = Bω dω.
dR = Rω dω "- !;! ( Rω !- !(1 ( !" '5! ( "! / " !"#( 0 !# ' ! ! ."( 5 " "1 & "1#4Q(KINQ(dS⊥ A '"-4! !! !" 5& " 5& !< !# ' +0! '"-! " ( * !# 0 Θ π/2) % ! "- !"#( 0 !# B ! !# R('5 !5 !;M<! ! 41& !" dΦ(пад) =dSB cos Θ 2π sin Θ dΘ = π B.Θ=0Aω< !# !" ! ! . Φ(пад) < ! !" "1!! Φ(погл) %"- A = Φ(погл) Φ(пад) '5! ( % !" G !( '"- - !! ω (погл)(пад) ! % Aω = ΦωΦω < !Φ(погл) = Φ(пад)dω.ω<& !# ;4 !"& '-G ! !( ( "141 !" '"-(< !# ρω 3 !"#( "! !# '"-( .- 0J %5" '! ω ÷ ω + dω % " ! '!.
"( '"-( "!(1( !" 5&dΩΘ !! !41(" dΩ !#4πdS2 ( , + (01dU = dSυ⊥ d tρω dω"! !# !" - !! dωdΩ ρω dω D ( d t *.4π ! "!! '"- " ! !( υ⊥ d t = c cos Θ d t I =( '"-( *(1. ( 6" 5 !& υ⊥ d t "1.#4 ( dS dΩ4π=cρω4πOdω cos Θ dΩ dS.dω cos Θ dΩ dS d t.ABNAA% !( ( "15& '"-5& ! 4!MdΦ(погл) = dΦ(изл) /! 4 "! -!2 dS ) dS⊥ D8 8 0v dt4πdΦ(изл) = Bω dω cos Θ dSdΩ.π/2=cρω<"1( ( 0'! ( '"-4 4MB cos Θ dΩ =Θ=0Qdt=dΦ(погл) = Aω dΦ(пад) .π/2R=dU=!! ! "1! ( !"MdSdSKIO"cρω4πdω cos Θ dS dΩ = Bω dω cos Θ dS dΩ,BωAω=cρω4π.AIAK<"- ! 5! )M !; .!"#& 0 ! !" "1!"#& 0 ! "( * !" ("(! ( "#& 26& - !!5 0 "4!&!!5 ,5& ' 05" !" ANO 9" "( ! Aω = 1 '5! ( e ". !#4 "1! 41 '"- 0!5( '"- !!& & !! !" <!B!; ω = Eω ! !"#& 0 !#4 0 "4!Aω- !" "( "& 2" AK ! ( 5 -'"! !# '"-(MEω =cρω4π.D! -! '"- 0 "4! - !"+0!A "! 'D - N 7&! !"#4 "! !# - " !(& ".!! '"-( " ! '5 !& 0"#; "5"5G ; 7& - " '"-5* 6""(!- !!ω ÷ω +dω G ! '"- 1 0"#;& ("#5&(1 0J V ! Lx , Ly , Lz <" '"-( ! (1 !"(! 0& '64 " * " *!'5* - !.!& ω "5 ! ?-!( -! " *! ( Q(Q(('# ω M ω = ck % (1 '"#5 ! 6" ! ! !# "41 '-(M% ! ! !; ! - " 41* '" " λ ÷ λ + dλKKBkx =2πLxNx ,ky =2πLyNy ,kz =2πLzANz , Nx , Ny , Nz = 0, ±1, ±2, .
. . /! 4 * -! - " ΔNx'"-5* '-& kx !" Δkx ΔNx = ΔNy =Ly2πΔky ΔNz =Lz2πLx2πΔkx "-.Δkz <" - " '"-5* '-&"5* ! 'kx ÷ kx + Δkx ,ky ÷ ky + Δky ,kz ÷ kz + ΔkzIΔN = ΔNx ΔNy ΔNz =Lx Ly Lz(2π)3Δkx Δky Δkz .K9# 0* - !# '5 "('6 2! -!! "- - " ΔN ( -! Lx Ly Lz = V 3 0J(1 "- -!"# "# ' - " 2! (1 "! 0J K. . ! !MdN = 2Vd3 k(2π)3 .< "# 5 " 5 '"- ! '.! ' 5( d3 k = k2 dk dΩ 5"(( ! "( !" " Ω = 4π "- ! dN = Vk2 dkπ2.76 *( ! "5* - " - !! 2" ω = ck* !;4 "( " - " '- !! ω ÷ ω + dω Mω 2 dωdN = V 2 3 .%"-π c1 dNV dω=ω2π 2 c3! !"#4 "! !# - " "!! "( -.
0J " !8" 0 "#'!# &( ! - !! " " 2" ω = 2πc/λ '( ' dω/ω → dλ/λ9 ! 0 !#dN = ndλ = V8π dλλ3 λ.ND - O %5 ! 2" <"G ; =( "!! "( E 2 + H2W =dV.8πKKAA< ! " '6 " * "ME = E0 (ω, k) exp[i(kr − ω t)], H = H0 (ω, k) exp[i(kr − ω t)].I9 " ! ! ! !"#5* " "( ( !05!# !" 5 & !"#5* "ME=Ei .K ? & " & "& !"(! ( - & 6""(! .( ! " !& * !! (M1En = n + ω.2=! '-! -! "!! " ! 05!# !" !& ! ! "0&D! -! " !' <" ! '"-! ( "1! (6( =! '-! -! * 6 * '& !( 1 !" 5 !! !"# ! 3 2! & ω "#.
k 9 0' " !"(! 0& !# 2!2!5& ' "# 2! 0"! !& ."('6& " "# !#4 % " - ! "!!5*" "( " ! 1 !4! ' 5 "(.'6 "# ! !5 0 !# j = 1, 2=( 2! ' & 0'41* 2!M1E=ωk, j nk, j + , nk, j = 0, 1, 2, . . . ,k, j2 3 "& !< " !* '-& 5 -" " <" ."(41 - " '"( n "( 2! - !!& ω " # "!! " !!# !# 6""(! '5 - !! G ! .- & 6""(!0 !& - !!& ω C ( ! .!# '-( 5 2"& % !! ! "Q(Q(@"#6 - " - !6 & En !( ( ! ( 5.ENn = A exp − n ,<"- !; '5! ( ) 5 F ρω c/4 &( 6"- & - !! ν = ω/2π "- 5 "( !"#& 0 ! 0 "4! - !" " A2" AI A 3 -( !(( k Б 3 !(( @"#6 9(( ( 6""(! '5! ( &D - AB 7&! "- "! ! ! !" '"-(' " 1 ! !"1& L ! !!5 T 41 226! "1( αG ; % !! ! ' ?!2 : @"#6 !" !! T '! !" '"- !.
!#4I∞ = σT 4 .AG ! ! '"-( "# z < !# '"-! !#4 I ! "& 1 ! !"1& dz 7 ! !0! "1 - !# '"-( (KKIkБ TE =-Nn EnNnnωnω exp −n=0kБ T∞nωexp −n=0kБ T∞-1= ω +21= n̄ω +ω,2 0'-1n̄ω =exp (ω/kБ T ) − 1N.< " !; '5! ( 5 / ! - " 2! *(1* ( 6""(! - !!& ω " "(! 44 4 6""(!C " !# - " 6""(! 44 4 .6""(! E - !!& ω(k) ! 5 & 4 '"-(MdU = ω(k)n̄k dN,O(dI)1 = −αI d z.A = α d z.(2π)3,"-& 51& '- 2" ?!! ! !O "-dU = 2Vd3 kω(2π)3 exp (ω/kБ T ) − 1% 5 "( En 5 !0 " " ω/2 ! " "4 4 !5 ' ! ! !!5 !(! (! " ( -" ! -! C " !# - !# '!4 '"-( ! "! 5"!# !. '5 "( " ! -! !dU = Vω 2 dωωπ 2 c3 exp (ω/kБ T ) − 1AA.1 dU*./! 4 "( !"#& "! ! '"-( ρω =V dω 5ρω =1ω 3π 2 c3 exp (ω/kБ T ) − 1.AIcρ4= α d z σT 4 . % '"#!! " ' ! ! * "( !!dI = (−I + σT 4 )α d z.)5 "-" 226"# AB.K9 ' *2 !5& - ! 5 '! '."- ! !#4(dI)2 = Ad3 kI=! '-! -! "& ! "1!"#4 0 !# "- dN ! ( 2"&dN = 2VKKKdIdz= (−I + σT 4 )α,; ! -"# "I|z=0 = I0! I = σT 4 + (I0 − σT 4 )e−α z .% - ! "- I0 = 0 ! !! ( 0 ! '"-"( *I = σT 4 (1 − e−α z ),N 5* ' "( !"1& L ! !# !!I = σT 4 (1 − e−α L ).OQ(Q(9 0' !"# !!- !" !5 " 1 ! '4! '."- ! !# ! "! ' ?!2 : @"#6 A70* !!!# "41 % " '- "" #-! 226! "1( "( '"-( * - !!* α -! '"(" "#'!# 5 "( ! ! A % & !!"# ! 1 ! ! "1!# '"- .' '"-5* - !!* < !# '"- "1! ( .!"# ' Δω ! ! ." !( 05 ! ω kT ) @ -!!#-! ! ' 226! "1( "( * - !! α % ! ' 5 0 !!# *(1 '".- 9 ! !# !!# >2 ! "( "( * !! T1 T2 T2 > T1 ' AN ' ! -! - 5; !! '."-( ! 5; *! ( ρω % & !!"# ! ! !"# " 5* ρω (T1 ) ρω (T2 ) !!# !.
-" ! !'( (5& 26 ρω (T )% " !# & ! ! ( '"- !! T1 & 3 !! T2 - T1 < T2 < ! ! !! -' 2"#! 41'"- - !!& ω !" dω AKK B =c4(Δω)ABI∞ = BkT,ω 2 dωπ 2 c3KKT1T23 "- -!541( - " "4-5* !.& 05 "( ,"#&; -!5 "-5 5 5;% '"#!! "-I = BkT (1 − e−α L ).AAG !( '- ! ! !; !-.& ! 3 0 !! '4 !!5 '"-5*!" G'"-5 !5 4! 01 ( '"-5 '-( !.!5 % - ! ! "!# M !! 0 "4! - !" 41 !! "& ' ! !"#4 !- 4 ( !# Bω -! '- !"% !! ! 2"& AA ! !# '"-( I∞ = BkT '( 0 - !" !5 " ! 50 !"#. ' ! !#4 '"-( 0 "4! - !" C "05 5 " ' ! ! '"-( ! "( -&!"15 ! "-" 05 22!4 ( !4 !!MTяр = T (1 − e−α L ),AI!( !"-! ( ! T !"1 "& 1 ! ! !- 5 '"05 !4 !! ! ! 0" '- Tяр !!!" T D - AA "#'( '5 ! '!# -! .!"#( "! !# '"-( ! ' !! ! !!5 '"-( 5 " T2 > T1 "( "40&- !!5 ω ! ! ! ρω (T2 ) > ρω (T1 )G ; 05" ' " A !"#( "! !# '"-( ! ( 2"& <"Mρω =ω 31π 2 c3 exp (ω/kБ T ) − 1.AÑîñóä 1ÔèëüòðÑîñóä 22 %=) 1 ,0 )B 480 +,- +-0 ),0 T1 < T2 B -+ B ,?8 +,- ,+)) + ω ÷ ω + dω< "# "! !# ! '! '"-( dq == cρω dω/4 5& ! A dq = dq2 − dq1 = c[ρω (T2 ) − ρω (T1 )] dω/4.I?" ! -" ! 2" "' '.
'"#5& * !" ! !" ! !"0" ! 0' '( !(( * 5* !" =! '-!-! ! " 05!# " 41& #;4!! T1 ) ! dq > 0 ?"!"# ρω (T1 ) < ρω (T2 )% '"# ! 50 !"# ' ! . ! 5"(! ( "( * - !! ,' ! '-!! -! 5 ρω (T ) "( '5* !! 4! (D - AI %5 ! ('# !-5* !"& * 2.'- * "- !5 !5* !4!G ; < !# 2'- "-5 A B !5 -! !! !"(41* * ! !"- ! "(M[Â, B̂] = iĈ.A/!5 Â B̂ "5 05!# !5 "# !"(4! 2.'- "-5 ! Â+ = Â B̂ + = B )!"# i & - !Q(KKQ( ! -!05 0 -!# ! !# ! Ĉ Ĉ + = Ĉ <. -! '-( "- A B 5 "4MA = B = 0.IC " ! ! ! !!- ! !5 A1 = A − A B1 == B − B "( !5* !0 I 5"(! ( 7 -! '- "(! ( A = Ψ∗ ÂΨdV.,"( 1( ' !-5* 2" -! "4 2.64 Ψ &M Ψ∗ Ψ dV = 1% ! Q̂ = ξ Â + iB̂ ξ 3 !5& & !!"#5&! ?!! ! '"#!! ! (( % - ! " Â = x B̂ = p̂x = −i9 ' " "-ΔxΔp /2.=! !; " !& 2 %&"(<"- " !; (5 5- " <"-! - !6 ;! 2! ! -! "( 26(01! ( "# 0 - !M∞KAB|Ψ(x)|2 d x = 1, ! !"∞f (ξ) =dΨ 2xΨ(x) + ξ d x,f (ξ) = Ψ∗ (ξ Â − iB̂)(ξ Â + iB̂)ΨdV =' ( ' (= Ψ∗ (ξ 2 Â2 + B̂ 2 + iξ[Â, B̂])ΨdV = ξ 2 A2 + B 2 − ξ C .
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