北大郭鸿量子光学讲义

§11..1c£.2•d£Re£C£ayf%D%leD£i&%ghg%'£-JRdE£eaÚeF£nsp%G%h D¡&%h h '£2 h H£(I¡19P¡00Q£i%R¡I S51T£90TV5 iU£)W¡X¡Y£`¡P¡a£b
7d8
kB %Í%6%I£h
h
uν dν
=
8πν c3
2
kB
{u, v} =
a
∂u ∂qa
∂v ∂pa
−
∂v ∂qa
∂u ∂pa
.
(qa, pa) (Qa, Pa)
(1.12)
Ô%Õ%Ip%ð%ñ%D Ì%î%à%C a
∂u ∂qa
∂v ∂pa
−
∂v ∂qa
∂u ∂pa
Poisson
=
a
∂u ∂Qa
∂v ∂Pa
−
∂v ∂Qa
∂u ∂Pa
.
ÿ£ò £à%¢%CHT%h4 ÅHD%%øx%ó%{qxHay%ô ,pyC£(b}q¨%8a=,%pD aδ)adbH%,ùeamõ%ú%i4%ltDb{o%qn%a,ö%5%qR%b6%}û%=4 T%{m%I³pa%%,pD%Øb}óbüe=ôbý%0.V%%÷þ%ÿ¡(W a,£b ¢%=T%1,D£2,·¤¡··¥%,gW³)%C£¦%(1£.1§3)
uνdν ∝ ν3e−c2/λT dν.
(1.18b)
c2 £e£f (1.18b) £p£q££s£r
` %w%I %Ip h 2 ££s£r
uν dν
=
8πν2 c3
hν ehν/KB T
− 1 dν
ν £ h Wien
(1.19)
uν dν
∼=
8πh c3
ν
3
·
exp
−
hν kB T
pa
=
∂G ∂qa
,
Pa
=
−
∂G ∂Qa
,
(a = 1, 2, · · · , g)
(1.7a)
Å%Æ m%n%Â%Iq%Db jbk%4K(Qa,
Pa,
t)
=
H (pa ,
qa,
t)
+
∂G ∂t
.
(1.7b)
%g %¼%Ç%`%D È%o W4%%IÉ%% j%k Q˙a
T
dν,
£p£q£r¡s¡r%wbD¡t£u¡v£wbW
(1.16)
4
¬d­e®
0£x%2 i%I i • Stefan-Boltzmann
(1879
1884 )
¯e°%±³²%´dµe¶%·%¸
7d8 4%E%G% %%I 4 Í%6%W u =
∞ 0
uν
dν
h 2 %I h i • Wein
(1.15a)
¤%8 Уd 0%Hqip£4 %ÅH1%%øy£C£xH2£¦%y3%d D%8 ωD%bi 4££%£!%I|%ÿ©ÎbD£"%Û%¢%D£¡F%#£%5ddI$%42tq2x%i6¡+Cd7%qiωDbùe∼i2qú%bie=±D%T%iω0iÿ£åbt I£k¡¢%8 %T%8 %I&У%%d£9b%%äbT%£à££%@b'%ØD¡D% A£Ò%pBbiÓ%WW (~©1.(£15)b)
Rb¹bºbÍb
(qa, pa)
→
αa DbÎb
(Qa, Pa)
D%G

=
m%Gn%(qaI, αÈba,o
t)

Qa = const = αa

Pa
= const
=
−Βιβλιοθήκη Baidu
∂G ∂αa
= βa
(a

=
1, 2, · · ·
, g),
Ö Âb×b Ø dbÙ bb jbkØÚ qa = qa(αa,βa,t) %WÎ%Û%Ü%I%¼ 4 Ý%5%6 G Hamilton
7ß8 ÃdÄeI
7mi
4© ¢H THj%@Hkd% I
H=
4pi
(1.5)
%7 o% 4

i
F%bI
4 7 U%VbI p2i
2mi
+ miωi2qi2
qi
(1.14)
ωi 4©¢HTHD£bW£
%%%o
dqi dt
=
pi mi
,
dpi dt
=
−miωi2qi.
1λ = 2
∆nx · ∆ny · ∆nz.
λ
nx ,ny ,nz
þ£ £%Ï£¢%T%E%6%Wo%¡bU%VbþbI¡
nr =
n2x
+ n2y
padqa − Hdt = PadQa − Kdt + dG,
a
a1
(1.6)
2
¬d­e® ¯e°%±³²%´dµe¶%·%¸
798 5%6
dKG(P4Aa,¹AQºAa,»At)xAWyA%W½d%¼ Ãd0 Ä G ¾A¿
0
¢A

mAnHIÀ §wHÁH¿
0
¢HmHnHÂHDHqHD
Hamliton
(1.1)
,14%q2U%E%, ·V%X · · R1 , qg
,
t)
d ∂L − ∂L = 0, (a = 1, 2, · · · , g)
(1.2)
gHqR%tea(hum%t03H)%X ~4D%1 m%qLU%˙aan%(V%gt0rI)am%n%gn e%2iHWgbFHR%dbtjH%%kH%∂b q˙WlIa%(bb1I.2m%Gb)∂n% qbad%q%ao%i=bpFbq%aj(gbqbkbR%,p tD)v%,2Hbw%g R%dHWx%oHm%y%pH%bj%qHD%Xk%Dm%1z%n%%UbL(WDbVqb%jbmb,q˙bk%n,tzb)X , I|rHb1 {bsHF%}mH gn g
1 2
[ε0 E 2
+
µ0H
2]d3r
=
1 2
λ
47 ££
dqλ dt
2
+ ωλ2qλ2
,
(1.24a)
6£7%4
1 2
V
ε0 E 2 d3 r
=
1 2
λ
dqλ
2
,
dt
(1.24b)
H6%H4 Ic©d©e©fHD£££%©£jH12%V%µC£0H¦£2d£3r¢%=TH12D%x%λ y%ωλ2C£qλ2¨%D£©£%I 7
1893
u = aT 4,
£y
σ
=
1 4
ac
1896 i
Stefan
(1.17)
Å u£h h
uνdν = ν3φ
ν T
dν,
(1.18a)
7d8 4 %I 3%4%Q%v %Í%%I %w λ = c/ν £e£f£s£r
D %W £t£u£v£w
h 2h • Plank
i (1900 )
%e%o j%k Hamilton
a
(canonical equation),
m%r%u%n%%v I§dHa%m2g¨iltR%o(nq%a,w%paj%)jHqk%˙akH%=2g ¡%∂∂R%%pH¢%ag,%%£bm%bD p%˙agp= R%−m%(Qv%∂∂n aHqw%,aP(.j%caa)k%n© oW(na2iag=l R%t1r,aq%2n,sD%·fo·r·m,ga)tm%.ion%)D%Hm%Hn%W¥¤HI b¦H(ªb1.«5)
pa = ∂G/∂qa = pa(αa, βa, t).
(1.9)
7d8 I W = W(qa, αa, t), αa = qa(t0) βa = pa(t0).
(1.10)
HdHÞß 7 H DHàHCHWâáHwdã DHäHåHI (qa,pa)
βa
=
−
∂W ∂αa
,
(αa, βa)
∇2A
−
1 c2
∂2A ∂t2
=
0.
(1.20)
§1.1 è%é%êdë%ìeí
5
ç%£c£d£m%Ion%£p¡q£r¡s£tbubv%Ð A )£u%4
A(r, t) = qλ(t)Aλ(r),
(1.21)
7d8
λ
Aλ £p£q%5%6
∇2Aλ
+
1 c2
ωλ2 Aλ
=
0,
∇ · Aλ = 0,
(1.24c)
5 Hamilton
H=
1 2
p2λ + ωλ2qλ2
,
λ
6 j%k%4
¬d­e® ¯e°%±³²%´dµe¶%·%¸
q˙λ = pλ,
p˙λ = −ωλ2qλ.
e{46£7%p£v%R%f%gbD£x¡y¡¢%I£¡¡% ν p ν +dν D£¢%T%E%6%4
¢¡¢£ ¤¦¥¦§©¨©©©
§1.1 ©©©!¦"
0%1%#%2 $%&%'%(%) 3%4 5%6 §1.1.1
L(ri, r˙i, t)
Lagrange
UH6%794%VHqdQ%8 EHR eT 6HI=i %WaS%i=n`bw%T%1 12XD%U%m1ix%V r˙iUby%·r(˙iiVdCb4A=ceb@A14b,4BA2,f%CA·%L·g%(DA·%r,iIiEA,nx%r˙hb)iFA,y%Wtp%GH)C =q%q5AaW(rbT6Ha(s%r=I˙i%,Vt%1t),%(ub2r−i,%,v%·Vt·)%·w%(4Ar,%igIi,EA)t%x%)3%PA,%ybGA4YdbC%5AXebD6A1 f eI U%reiV =bI%ri(Xqg1
(1.22a)
ª%« %u%vAλ =
2 V
0
eλ
cos(kλ
·
r),
p£q£r£s£t
2 V
0
eλ
sin(kλ
·
r).
(1.22b)
7d8 4%%6%I nx,ny,nz p
kλ
=
2π L
(nx,
ny
,
nz
)
(1.22c)
e%IÞ 0 k %p£v%R eλ I%£w£ev%(kλ)R£· kx£λy£=¢%0,j{zI|¡3% ªb«
% %j%k%I%o W e%I%d 0%1 5%6 pa
=
∂L ∂q˙a
,
q˙a(q1, q2, · · · , qg, t)
(a = 1, 2, . . . , g) Hamilton
(1.3)
H(qa, pa, t) = paq˙a − L(qa, q˙a, t),
(1.4)
=
∂K ∂Pa
,
P˙a
=
−
∂K ∂Qa
.
G
K ≡0
(a = 1, 2, · · · , g) Hamilton-Jacobi
H
qa,
∂G ∂qa
,
t
+
∂G ∂t
=
0
(1.8)
Db%K%Êb≡IRb0 bWG¤%»bD%d%b%q%I|}%D%¹%º%jÍ%j%k bkb(1Ô%.Ïb8Õb)W|Db%Ð bbGRbI|ËbÑ% %Ì Ò%gÓ
(1.22d)
ωλ2 = kλ2c2.
%%I I Ù %o o}£~ (1.20) (1.21) (1.22)
(1.22e)
%%ÿ%%U%V qλ D%i%F%j%k%W%dd%d2tqÃd2λ Ä+Iωλ2bqλwb=b0,R%C¡¨bD%Gb%4
(1.23)
7d8 %ç%G%%D b4V
dν
=
8πh c3
·
ν3
·
exp
−
hc kB
·
1 λT
dν.
%p 7d8 Í%6%W c2 = hc/kB, ` h HwHI wHI p HHp h©r©s©r ©© λ
h Planck
ν →0
ehν /kB T
→
1
+
hν kB T
Rayleigh-Jeans
2. 4bT%fc£Id£¢bÄe¤He£I1ob 9dHÇ% g0f%p C0ID£0%o©£iiPuc£P1h%j%lloal1d£uan¢%©£2nmce£kc%e£bk%hf£Rbf£k£hT£1hh4u%¢bl%h'%νIhldQbT%IHD%νbIW v£DbH≈Óg4¡e¡xbRHP∇82¡lπfby%caËb·3νn(3bA1ÍC¡2cÌ%9kDb·=b0¨¢ 1%00pb+iWþ%BI|à%khieh1BeDrννDb2Tnlin−ÍbhMdb¡a16%h1xd4bw~I|νey¡e=lRl)efj%a8WPyπcgGblk%l3νaei2nÞIgc·hk7pk-JBÍ%heTa6 ndhνshdb¡h =bh kTD%BcI|§(2q/D©ucaW%Gnt}a)
pa
=
∂W ∂qa
,
t
m%en% W j%k ç p (1.5), ∀f(qa,pa,t),
qa,pa EHHdHæHÊH%%D
df dt
=
∂f ∂t
+ {f, H},
(1.11)
7d8§1.1
è%é%êdë%ìeí Ì%î Poisson
0%1
{u, v}
4
3
pHdHÃßħI ÌHîHçH mHnH%%m%%Iï%` 4H mHnHwHI Poisson