北大郭鸿量子光学讲义.pdf-资料库

x1.1.1 L(~ri; _~ri; t) !" x1.1 Lagrange #$ L(~ri; _~ri; t) = T ( _~ri; t) V (~ri; t); (1.1) n 1 2 mi 78 qa(a = 1; 2; ; g) !";: 9[R]\^_`abcd>e>f (a = 1; 2; ; g) %'& T = _~ri _~ri")()*)+),)-).)/0#)$01 V (~ri; t)")-)2)/)#)$)1 ~ri = ~ri(q1; q2; ; qg; t) Pi=1 (i = 1; 2; ; n) 9 i 5678 "34 7080-0$09=<>: 7>8@?A">BC1ED>FGH>IJK>LM1ENO>+,QPARAS>T: $G@UA1WVM,NO+>X>"YZNO+ @ _qa Lagrange i0.0j0k09l_>mn 78mn _qa(t0) g0h !0" tAu qa(t0) ~ 4m,mn1W>>>`1V>>>F qa = qa(~qb; t), c0d0o0p0q0, L(~qb; _~qb; t), r0s0m0n g 4vwxyjkzX,>j>kz>1|{>} 7>8[m>n[[>[ . 2g 41W>/>i[.>j[k>,[>9RW>: g "-: 784 (1.2) cdop dt @L @L @qa (1.2) = 0; d g pa = @L @ _qa ; (a = 1; 2; : : : ; g) 2g 4m,mn9W: Hamilton #$ pa _qa L(qa; _qa; t); H(qa; pa; t) =Xa jk @H @pa @H @qa (canonical equation), (a = 1; 2; ; g): (1.3) (1.4) (1.5) jk1Wco _qa(q1; q2; ; qg; t) 9[UA1Wcd ceo Hamilton : ; _qa = 2g 4wjk ¡¢a£>, Hamilton c`> mp _pa = g 4vwjk9 g j0k0a mn 2g 4 (Qa; Pa) © padqa Hdt =Xa Xa (qa; pa) rd mn1§¨ KL 1 (canonial transformation) 009¥¤0¦0 >ª>« m,mn1 2g 4q, PadQa Kdt + dG; (1.6) 1 : 9 R g

2 dG")¹)º)»)x)y)9½¼ %'& #$ K(Pa; Qa; t) 9WcdÃ@Ä G¾)¿ pa = @G @qa ; ¢) Pa = ¯A°±³²´@µA¶·¸ ¬@A® ¢0m0n0Â0,0q0, Hamliton m)n01ÀU§M0Á0¿ @G @Qa ; (a = 1; 2; ; g) K(Qa; Pa; t) = H(pa; qa; t) + @G @t : ÅÆ g mnÂ1Wq,> j>k" @K @Pa _Qa = _Pa = ; (a = 1; 2; ; g) c¼Ç<, G Èo K 0 9W"V1WÉ Hamilton-Jacobi jk : @K @Qa (1.7a) (1.7b) (1.8) Hqa; @G @qa = 0 @G @t ; t + g 4>¹>º>Í> (qa; pa) ! (Qa; Pa) , 9>= >; (a = 1; 2; ; g); pa = @G=@qa = pa(a; a; t). @G @a = a a ,>Î> G = G(qa; a; t) mn1WÈ>o (1.9) j>kØÚ (1.10) >> ,>Ê>4>Y>»>>1|[[j[k c1 G ,c}¹ºÍ>[Ï>9|Ð G Ñ_ÒÓ K 0 9W¤dq, (1.8) ,>>4>Ë>Ì jk>ÔÕ>c>1| Qa = const = a Ø Pa = const = qa = qa(a; a; t) d>Ù Â>×>c 9WÎÛÜ1Wc¼ G " Hamilton Ý#$ %@& a = qa(t0) 1 a = pa(t0). R W = W (qa; a; t); pa = @W @a ; a = (a; a) ,0à0+09âá0M@ã 8f (qa; pa; t); F df dt = @f @t + ff; Hg; @W @qa ; t ,0ä0å01 qa; pa -00d0æ0Ê0, (1.11) c0d0Þß mn9 RA (qa; pa) T jk 0 (1:5), ç a Ö %

x1.1 èéê@ëìAí %@& Poisson Ìî fu; vg fu; vg =Xa @u @qa @v @pa c0d0ÃßÄ§1 Poisson Ì0î0ç0 Xa @u @v @pa ÔÕ1WFðñ, Poisson Ìîà+ @qa m0n0m1ï< @pa =Xa @u @v @qa @u @Qa @v @qa @u @pa : (qa; pa) T @v @Pa (Qa; Pa) "0 @Pa : @u @v @Qa 3 (1.12) m0n0M01 fqa; pbg = ab; óô Å0ø N0O (qa; pa) õ"ö à+ ÿ6,NO+, Hamilton #$" _0" fqa; qbg = fpa; pbg = 0: m,ó>ô>8÷ @ùAú,>4û61³ØüAýþÿ6,9³+ (a; b = 1; 2; ; g) (1.13) %ß& mi " 060(01 Ã@ÄA1 jk@R pi " (1.5) co"dqi i 2mi H =Xi p2 qi " .>1 = pi mi ; dpi dt dt + mi!2 i q2 i 78>1 = mi!2 i qi: (1.14) !i " 060, >9 Å0ø i qi = 0 d2qi dt2 + !2 co qi +d ¤d _0" N0O Ðpy !,_>`1|Î>Û#" $%,>>6å>k& +,-./012345#66879:;<2=> 1. ?@AB,C4@ÚAF,D Rayleigh-Jeans D (1900 E1 1905 E !i ",.1W ,ÿ ,N+@ùAú,ÿ616b,ÒÓ~ qi ei!it 1[R cdØ pi 9 Ð'>ä>à(>,)*>9 x1.1.2 ) %@& kB Í$1WVD ud = 82 c3 kBT d; FGHIHM>,JKLM>9 (1.15a) (1.15b) (1.16) " F ò h 9 & % % % & & & h h h

4 Stefan-Boltzmann N (1879 E1 1884 E u = aT 4; ) ¬@A® ¯A°±³²´@µA¶·¸ (1.17) (1.18a) (1.18b) (1.18b) FGWIHM Stefan Í$9 %@& Wein D 1 4 = ac " u =R 1 0 ud "-/OS1 QP 1893 E1 1896 ESR ud = 3 T d; ud / 3ec2=T d: c2 !"3vABÍ1VU KD = c= "ABIH1 %@& ,JKLM9 Plank D ) (1900 E ud = 82 c3 h eh=KB T 1 d (1.19) 8h c3 8h h hc kB 82 c3 1 T d: 3 exp eh=kBT ! 1 + h 82 XY1WF Wien D c3 3 exp kBT d = h Planck Í$9 ! 0 M01 Ì¢ Wien D [§, \} 1W\>" !>,>[1|[_^`_A_B[/>[[_a[a[,[1|[/[ h = kBc2=c 9 14 cl1 d>>6 ¤0d01 Planck D Ç,?@ABD Planck D ">¢>o>p ¢4>T3>vAB>ÍW>F>à,>Í>$1| 61 Planck Berlin d>b>Oe 1900 E 12 b 14 c (1900 E ) 9 ÄA1WVgcih 6,Ógc 2. ?@AB©jAB>6,>N>O+ R Couloumb kl1W r ~A = 0 þ, Maxwell jk1WF kBT 00F Rayleigh-Jeans kBT d ~ Rayleigh-Jeans D Planck Í$ d = c3 12 b (quanta) (1.20) r2 ~A 1 c2 @2 ~A @t2 = 0: h T Å h h h F D h h h h ] h 9 h U g Þ % D h f

x1.1 èéê@ëìAí %@& ~A pq#$ ç?@m1on`pqrst>K>LÐ ~A u" ~A(~r; t) =X q(t) ~A(~r); r2 ~A + 1 c2 !2 ~A = 0; r ~A = 0; ~A =r 2 V 0 ~e cos( ~k ~r); r 2 V 0 ~e sin( ~k ~r): ª« %@& AV1WÞ pqrstKL nx; ny; nz "Z$1 ~k cFv4 ~k = 2 L (nx; ny; nz) ~e() ~k ~k = 0; ~e 1W wv4xyj{z1V|! !2 = k2 c2: ª>« 1o}~ (1.20) 1 (1.21) Ù (1.22) co d2q dt2 + !2 q = 0; Vÿ78 q ,i.jk9WcdÃ@Ä1WV>M>Z>4+>,/>" # ; ZV 2X " dq dt 2 ["0E2 + 0H 2]d3~r = + !2 q2 1 2 1 %@& " %ç/,>" "0E2d3~r = 1 2ZV ; (1.24b) 5 (1.21) (1.22a) (1.22b) (1.22c) (1.22d) (1.22e) (1.23) (1.24a) 1 dt 2 2X dq 2X !2 q2 1 1 2ZV 0H 2d3~r = $%" 001? @ A B0,©j0+60,NO+, 1 $" + !2 q2 1 H =X 2p2 ; (1.24c) % Hamilton # U F

¬@A® ¯A°±³²´@µA¶·¸ 6 jk" _q = p; RA{" $%Fv4BC>,xy1V 1 = 2 Xnx;ny;nz X q: _p = !2 p nx ny nz: + d ,6-$" þÏ6-$9o>78>þ>1W nr =qn2 x + n2 y + n2 z = L 2 j ~k j= L = L c co 4nx4ny4nz = 4n2 rdnr = 4 L c L c d = 4 L32 c3 d; (1.25a) (1.25b) 1 = L3 82 c3 d 1 = 1 L3 L3 82 c3 d = 82 c3 d: cf1o46,>/>" U = R HeH=kBT dpdq R eH=kB T dpdq = kBT; X AV1o@{[A,6>$O>S>" 1 V X db, /y Åø /O>S" ¤d1Wco V Rayleigh-Jeans D Væ1 Boltzmann ud = 82 c3 kBT d 1884 E AB p = 1 3u ; %@& u =R 1 0 ud "-/OS1WÙNO>3>v N d(uV ) + pdV = T dS; R b % h 9 Ö

x1.1 èéê@ëìAí %@& S (entropy) 1 dS ,»xyop @T 4 T = 0; u @ 3 @ T @u V @1[R @v v @T u ) T ) u = aT 4: @u @T = 4 7 ò>N 1 Boltzmann Stefan-Boltzmann N V òN 1884 E 3. Planck @A£,?@ABD ç/T,y e@ b (Stefan ) 9 jSR (H U )2 = kBT 2 dU dT 1879 EJK 1R8d>bc>f>1W" dU 1 T = kB d ç Wien D %@& Í$co1 T U = exp c2 cT ; c c2 = ln U ; d dU 1 T = c c2U ; (H U )2 = kBc2 c U: j1Wç Rayleigh-Jeans D U = kBT; 1 T 1WF = kB U ; d dU 1 T = kB U 2 (H U )2 = U 2 9oyj~_">1| kBc2 (H U )2 = U 2 + U = c ; kB d dU 1 T d dU 1 T = kB U (U + kB c2 c ) = c c2 1 U + kBc2 c 1 U! ; ¤d Z Ö Z g Ö g h P h h

8 Zco 1 T U = ) ¬@A® ¯A°±³²´@µA¶·¸ U + kBc2 c = ln c c2 kBc2=c ec2=kB T 1 U = h eh=kBT 1 ; h = kBc2=c: %@& h 82 kB T 11 A exp h %'& h = 6:551034JS1 kB = 1:3461023JK19 G)<) N = 6:1751023mol19 }~ Planck D (1) Wien +$ ud =0 @ e@Adþà>+ c3 ; `p %@& (2) Stefan Í$ a = 85k4 15c3h3 ; Z 1 0 x3 ex 1 dx = 4 15 : = 1 4 ac = 25k4 15c2h3 : (3) `6^`1W^`AB6>,/> En = nE = nh; co = En exp En kB T U = Pn exp EkB T 1 exp En kB T Pn x1.2 E = h exp EkBT 1 : PAR @ã , Maxwell jk" r ~H = @ ~D ; @t r ~B = 0; @ ~B @t ; r ~E = r ~D = 0: 9>= >; (1.26a) Ö h &

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