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§16. 光子気体とプランクの空洞輻射公式
!16! "#$%&'()*+,-./01 ! 22345 6"#$%7+89:;,-./+01<=>52+01:; Wien +?@A5Stefan-Boltzmann AB&5!11 3CDE6Planck +FG7&4H+I J<EKLMNOP"#4QR)3SOT5UHe +$%V#3SOQR)&4 WXYZ[4\]^_P"#46"`a+V#7b:;5cdefghB+i jTkl&^OPmE5 6"#$%7+nV#o45pq[YO"#+rstu vTw2O+35xyz{E|y}&4^;^_P~+pq&"#$%&+• +€•‚ƒ„[Y…5†;‡<ˆ‰|y}&^O&G]OPUHe +$%V#o +YZ[5Š];{EV#oT‹Œz{O•34^_P 2+Ž+••345 6" #$%7+€•‚‘<’“”•–&Q—˜•–:;G]OP 16-1™"#+$% ! š[!11 3CDEYZ[5Planck +EK…›_E,-./œ•%./™žŸ ¡—¢a+0145 ! ! ! ! ! ! ! u(" ,T) = 8#" 2 h" ! ! ! ! ! ! ! ! !11-(3-5) 3 c exp(h" /kB T) $ 1 3SOP u(" ,T ) +£¤4 [J]" [m#3 ]" [(1/s)#1 ] 3S…5 u(" ,T ) 45(" + d" , " ) +¥¦ ! o§•+./"<G]E9¨+6©@%ª«…7 56©@¥¦o§•«E…7+ ./žŸ ! ! ¡—<j¬-OP,-./œ•%./™46"#$%7:;G]O 2&T3>OP! ®¯$‚+°Aœ±²*³´µ +¶J1™34·¸+¹º ¨»¼+½¾T¿8-O+35"#•[4eÀÁÂ4^Ã5 6"#$%74¾Ä QR)$%&Å^-2&TvÆO 26,76)P ! Ç:Ç5V#&ÇM+6"#7<G]O2&45 6"`a+V#7<G]O2 493 &b:;5cdefgh[ij-OklTSOP2{m3+Èh45ÉÊ+Z Ë[Ìefgh<ÍÎ[ÇM_O+356"#$%7+Èh42+Ï3Ð^OP cdefh+£+щ+1<ÒÓÔ{OPÕ|4V#+Ö¦`a[׌-Oq ‘[Ø-O135c <"`a5m0 <V#+ÙÚq‘&-O&5ÙÚÛ[fÇM V +`a3Ö¦-OV#+q‘45 ! ! ! ! ! ! ! ! ! m= m0 1" (V /c) 2 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! (18) 3SOPˆZ|‰+14cdefh3+V#+ÜžŸ ¡—<Š]O1 2 ! ! ! ! !! ! ! " 2 = c 2 ( p 2 + m0 c 2 ) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! (19) 3SOP2+Ñ1[ij-O&5 (18):;5 VÝ c +Þ5 m0 " 0 ^;ß m 4à ! á’&^LMÇmZ+35"#+ÙÚq‘4âã m0 = 0 &Ç^Ó{ß^;^_P ! äLM5(19)[ m0 = 0 <åÔÇM5 ! ! ! ! ! ! " = c p ! ! mE4! ! p = " /c ! ! ! (20) ! 3SOPÌefh3+q‘T m 3SOæçèV#+Ö¦žŸ ! ! ! ! ! "= ¡—&Ö¦‘4 1 1!2 mV 2 = p ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 2 2m (21) +Øé[SO:;5(21)4(20)&4ÜÃÐ^OP"#+YZ[6 m0 = 0 +V#7 ! 2 :;^O$%5SO_45(19)3 p 2 >> m0 c 2 3SO9¨+$%45 6êefhè ! $%7&ëß{Oìí5îï™P ! c <"`a&-O&5"+¥¦o( " )5ð¥¦o( " )5ñò( " )[4 ! ! ! " = c p = h# , " = 2#$ , c = " # $ ! ! ! ! ! ! ! ! ! ! ! (22) ! ! ! +ØéTSOP!11-4 3ÈhÇE6óôñ+óô¶õ&~+’>z7<öy! ! ! Oño÷*ø k 45¶õ4ñ+óô¶õ[|ùÇ5~+’>z45 494 ! k = 2"# /c = 2" / $ ! ! ! ! ! ! !11-4 (4-3) ! &yúÇEP2{&(22)[YO p = h" /c <ûZ&5 ! ! ! h ! # k = "# k ! ! ! ! p= (23) 2" ! &^OPüýþ45ƒÿóôñ[!"‰Ã6ñ¦è^Ö¦‘7<#-T52{ ! <6V#+Ö¦‘7&N^-P ! ˆZ|‰5"#T$%+QR)V#&Ð^O2&TSOP~ {46"#+n o7T%[Š];{E|y+}[4^_2&3SOP&]ß5QR)V#+UHe +9¨5|',-[(Ôz{{ß5~+V#+no4‹Œz{O&G]O2& T3>OPÇ:ÇP"#+9¨45,-+)<*-pq[rsz{E…5): ;uvz{E…-O:;5~+no4+,Š];{Eyo34^_P"#+n o4‹Œz{-5,-+)+pq&+•3+"#+.…/…50Ë5)+pq &"#1³&+•+€ƒ„[YLM2mO&G]ºß^;^_PŠ];{EœT5 V™+335$%+4 ! ! ! ! ! ! ( 56 7+89žŸ ¡—F T:;&^O<=, "F )T ,V # µ = 0 "N (24) T>?-O&G]Oœ2+Ï4!16@4 3ˆAh-O™P (24)+Bý46"#7 ! +C‚DE)FG +yú3ˆSO:;5 6"#7+C‚DE)FG Tï3S O2&j¬-OP!14-3 3CDEYZ[5 6QR)V#T”@ "r <HI-OƒJ V#o n r 745|K[5 nr = ! e"(# r " µ )/ kB T 1 = (# r " µ )/ kB T ! ! "(# r " µ )/ kB T 1" e e "1 ! !14-3 (26) &Q—˜•–Øo3Š];{OPÇ:Ç5(24)Y…5"#1³34 µ = 0 b:;5 ! ! 495 ! 6”@ "r <HI-OƒJV#o n r 745 ! ! ! nr = ! 1 " r / kB T e #1 Ý! ! ! n k = ! 1 " k / kB T e #1 ! ! ! ! ! ! ! ! ! (25) &^OP2234”@ r <5 œ22™5(23)+óôñ<§L-Oño÷*ø ! ! jÇM5‘#MN k &O,EP (25)+ÕÑ145Ö¦‘T ! k 3žŸ k [i ¡—T "k 3SO"#+ƒJHIoT n k 3SO2&<j¬-OP ! ! 16-2™Q—˜•–:;P;{O,-./žŸ ! ¡—¢a ! š[§11-4 3ÈhÇEYZ[5 (" + d" , " ) +¥¦o§•<G]E9¨5 ! 8#" 2 n(" )d" = 3 d" ! ! ! ! ! ! ! ! ! ! ! ! §11-4 (4-9) c ! T5©@%ª«…+®¯ñœ"#™+Qo5“R[45©@%ª«…+®¯ñ ! 3SO"#+‘#MNo <Š]OP®¯ñ4Sñ3SO:;5óô¶õ[TU ^ÿV+Ѷõ[•WÇEñ<G]O2&T3>OP2+X# 2 4§11-4 (4-9)[ š[Ym{M_OP n(" ) 8%45©@¥¦o§•«E…5©@%ª«…+®¯ ñœ"#™+‘#MNo<#-P2+YZ[5§11-4 (4-9)4©%ª«E…+}3 ! SO:;5,-+%ª< V &-O&5§11-4 (4-9) +Zý[ V <[ÓEˆ+T5 (" + d" , " ) +¥¦o§•[Ym{O®¯ñœ"#™+‘#MNo3SOP n(" ) # V # d" = V # ! 8$" 2 d" c3 (26) 2{[(25)+"#+ƒJHIo<[Ó{ß5 (" + d" , " ) +¥¦o§•[Ym{O ! >\]+"#o dN" &^OP ! ! ! ! ! ! ! ! ! ! 8$" 2 d" dN" = V # 3 # h" / kB T c e %1 496 (27) 22345Q—˜•–Øo[YO(25)+"#+ƒJHIo45 ! ! ! ! ! ! ! nk = 1 " k / kB T e #1 Ý 1 e h" / kB T &ÇEPz;[5"#ìQ+žŸ ! ! ¦o§•[Ym{O./žŸ #1 ¡— h" (= !# ) <[Ó{ß5 (" + d" , " ) +¥ ¡— dE " &^OP ! 8$" 2 h"d" ! ! ! ! ! ! ! ! dE " = V # 3 # h" / kB T c e %1 ! ! (28) 2+Zý< V " d# 3^LE!_45 6©@%ª«…75 6©@¥¦o§•«E…7 ! +./žŸ ¡—50Ë5(" + d" , " ) +¥¦o§•3+./žŸ ¡—¢a5& ! ^…5 ! dE 8#" 2 h" " u(" ,T) = = 3 ! ! ! ! ! ! ! ! (29) Vd" c exp(h" /k B T) $ 1 3SOP2{45!11-(3-5)+ Planck +,-./žŸ ¡—¢a1[|ù-OP ! 2+YZ[ÇM5Q—˜•–1&"#$%+G]¶:;5,-./žŸ ¡— ¢a[Ø-O Planck +01TP;{OP ! !11 3CDEYZ[5 h" << k B T +9¨45exp(h" /kB T) = 1+ (h" /k B T) + # # + `a:;, (29)4 ! ! dE " 8$" 2 # 3 kB T ! ! ! ! ! ! u(" ,T) = Vd" c (30) &5Rayleigh-Jeans +1&^OP2234 Planck yo h T?{^_P 8"# 2 /c 3 4 ! 6©@%ª«E…+¥¦ob+®¯ñ+Qo73S…5 kB T 4|‰+æçhè¥ ¦#œñ¦™+ƒJžŸ ! ¡—b:;5Zc+ª4©@%ª«E…5©@¥¦ o«…+¥¦ob+®¯ñ+ÜžŸ ! ¡—<j¬-OT5Planck yo h T^_ 2&452{Tæçh3+ñ¦žŸ ¡—3SO2&<j¬-OP 497 ! |¶5 h" >> k B T +9¨45exp(h" /kB T) >> 1, exp(h" /k B T) # 1 $ exp(h" /k B T) [ Y…5(29)4 ! ! dE 8$h" 3 " # % exp(&h" /kB T) ! ! ! ! ! ! ! u(" ,T) = Vd" c3 (31) &^OPWien +13SOP(31)[4'()*yo h 4Ym{M_Oˆ++5Planck ! +,-./1TÎdz{OeÍ[5Wien 45 u(" ,T ) = b" 3 # exp($a" /T ) +Øo¸ 3,-./+>fg—h<`a3>O2&<öiÇM_Eœ!11™P! (31)45 ! ! ! ! ! ! ! ! u(" ,T) # 8$" 2 % (h" )e(& h" / kB T ) c3 ! ! ! ! ! ! ! ! &j_Mˆk_P (h" )e(# h" / kB T ) = $ % e(#$ / kB T ) b:;5(32)4Q ! z¼OP!14@3@(28)3ÈhÇEYZ[5Q (32) 7±)•–<Äw 7±)•–3+6”@ "r <HI- ! OƒJV#o n r 745 ! ! ! ! ! ! ! ! ! ! !14!3!(28) 1 ! ! ! ! n r = e"(# r " µ )/ kB T = (# r " µ )/ kB T e ! 3SLE:;5lmï&ÇMG]{ß5(h" )e(# h" / kB T ) = $ % e(#$ / kB T ) 45 6ìV#žŸ ! ! ¡—7&6”@n<HI-OƒJV#o7+ª<j¬Ç5Q 7±)•–3 +V#žŸ ! ¡ — & Å ^ - 2 & T v Æ O P Ç : Ç 5 ¤ o45 (29) 3 exp(h" /kB T) >> 1&ÇE2&5 0Ë5 h" /kB T >> 1! [×OP2{45p 14@1 + Sq3 " = h# , µ = 0 &ÇMG]{ß5(" # µ) /kB T >> 1+<=&^O:;5r«45 ! Q—˜•–3ˆQ ! •–3SOP ! ˆÇ5Q $e (" nh# / kB T ) ! 7±)•–3ˆWX!_[^O&_Zj¬3+Q ! 7±)X#+nsœ•tØou™Tvs¥¦#+9¨+YZ[ 3Š];{O&-O&52+žŸ n ! 7±) 498 ¡—wx}4 ! ! < " >= % " j # e ($" j / kB T ) j ! ! ! ! = ! /% e ($" j / kB T ) j = % (nh& )e($ nh& / kB T ) / % e($ nh& / kB T ) n n (h" )e(# h" / kB T ) h" = h" / kB T ! ! ! ! ! ! ! ! ! ! ! ! ! ! (32’) (# h" / kB T ) 1# e e #1 &^…5(32) 4Q 7±)•–<yM“R[ Planck +./¢a1(29)&^OP ! 223vs¥¦#TvMÃO2&[4¾9TSOP Maxwell ¶J13zCz{O ®{<Ym^_|,®¯945ào+|£¤vs¥¦#+}¨&6•‚è[4 ~•73SO2&T€•z{O:;3SOœ‚ƒ 11™P " = nh# <G]O2&45®¯9žŸ (n = 0,1,2,3,.....) ¡—+‘#C[«OPEbÇ5!11!5 3CDE YZ[5vs¥¦#+„ÏžŸ ! ! ¡—4à…Ç^Ó{ß^;^_P ! Rayleigh-Jeans +15Wien +15Planck +./¢a15+†c+‡_45(32) üý+ÕÑX#T5 ! ! kB T " (h# )e($ h# / kB T ) = % & e($% / kB T ) " (h# )e($ h# / kB T ) h# = h# / kB T ($ h# / kB T ) 1$ e e $1 &Ð^O2&[fˆÇM_OPkB T 4|‰¥¦#+žŸ ! ñ¦+žŸ ¡—3S…5æçhè ¡—+ƒJ}<#-P|¶5 (h" )e(# h" / kB T ) +9¨45 (" + d" , " ) + ! ¥¦o§•3+ìV#žŸ ¡—[Q 7±)X#œQ ! JV#HIo™<[ÓEˆ+352{<6ìV#žŸ &Å^¼ß5 6V#7+žŸ 7±)•–[×Oƒ ! ¡—wx}7+`a} ¡—&‰ŠvÆOP~+j¬35"œ./™4ñ ¦&V#+Z¶+gq<‹¼Œ‰&•LMˆk_P:•+ ˜tŽ•)F•h•)•–3+ìV#žŸ h" e h" / kB T 45Q— ¡—wx}3S…52{T>f} ! <A?-O./¢a1<Š]OPÇ:Ç5 (32’)[‘-YZ[5Q +nsœ•tØou™<vs¥¦#+YZ[G]{ß5 Q 499 #1 7±)X# 7±)•–3ˆ Planck +./¢a1TP;{OP ! Planck 45,-+)<6§LvÆOvs¥¦#7:;^OÛ&G]5)&, -./+€ƒ„34¥¦#+žŸ ¡—4 h" (= !# ) <©@&ÇMrstuvz {Oˆ+&’yÇœžŸ ¡—+‘#C™5Q žŸ 7±)•–<Â_M5,-./ ! ¡—¢a+1<=_EP2+1T5]zщWá3 Rayleigh-Jeans +1& Wien +1+Z¶<=>5:‰5,-./+“y}Ü%<“R[A?-O2&< Planck 4R”ÇEP6žŸ ¡—+•–Cm‘#C7<G]O—m…&^LEP 1900 ˜+2&3SOœ!11™P! |¶5‘#™š+Q—˜•–1<6V#<eÀ [§L3>^_7"#$%[›Â-O2&3ˆ5WX!h[œO2&TR”z {O+4 1924 ˜e•+2&3SOœ!16ž657™P ! ,-+)<6§LvÆOvs¥¦#ÝQ 7±)•–7&G]Mˆ5 6V#< eÀ[§L3>^_"#$%ÝQ—˜•–7<G]Mˆ5W|+!_TP;{ O + 4 | Å - O & ŸFÈ3SOPÇ:Ç5,-+)&"#$%+€ƒ„ ("F /"N)T ,V # µ = 0 (24) TÈh+ÍÎ[SO2&:;-O&5¨¾è^!h3S Oœ!16@45@5™P2+"#$%[f-O€ƒ„<=:;5’“”•–+•tØ ! o "(T, µ,V ) 45! 45“”•–+•tØo Z(T,V ) [¡O+35]3ÅEYZ [5'()*+./¢a14Q ! ¢£4!16@45@5 3Ah-O. 7±)•–<ûLMˆ=vz{Oî5ìï™P2+ ! 500 16!3!"#$%&'()*+,-Wein ./01& Stefan-Boltzmann 1 ! §11-(3-5).2345.'()*+,6 ! ! ! ! ! ! ! ! u(" ,T) = 8#h "3 $ ! ! ! ! ! ! ! ! §11-(3-5) c 3 exp(h" /kB T) % 1 76T83000K 9: 6000K .;<=6'(".>[email protected]@&CDEF&6 ! G 16-1 .HIJKLMNO*=PQ.'()*7RSL".>?@T! max!U !"""#$ %"""#$ &"""#$ '"""#$ 15.00 3 u(!,T) (Js/m ) /1E-16 20.00 10.00 5.00 0.00 0 2 4 6 8 10 !(1/s)/1E14 G 16-_M >?@AJ`YL Planck +,.ab'(cdefg)* u(A,T) /hMPQ^*.>?@T!max!UO* T .iQJj[iQYLM O*VWXYLJZ[6!max U\>?@]J/0YLMPQ^*.>?@T!max! 501 !"#$%&' u(" ,T) !()*+,-' "u(# ,T) /"# = 0 ./01234 ! ! ! "u(# ,T) 8$h 3# 2 (&1)# 3 h = ( 3 ) % { h# / kB T + h# / kB T % e h# / kB T % ( )} 2 "# c e & 1 (e & 1) kB T ! ! 2 8"h $ $ h h$ / k B T = ( 3 ) # ( h$ / kB T ) # {3 % ( h$ / kB T )( )e } = 0! ! ! c e %1 e % 1 kB T }9:56);<1234 x " ( (33)56.7$8%&'{ (33) h# ) ! .=$.'>9 kB T ! ?@&' A9BCDEF ! ! ! ! ! ! ! (3 " x)e x = 3 (34) .7$4>9G&HIG.,-A9JK%LM$4 ! h" ! ! ! ! ! ! x = 2.821 = max ! kB T (35) NOPQRS9TUHV!maxW&XS%YZ,-[\=$4 ! ! ! " max = 2.821( kB )T ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! (36) h ><&TUH]^_`ab9 Wein 9cde%fg=$4A%_$JK%' ! " = c / # 9hi]^'NOPQRS9jk&XS9lH%YZ=$4 ! §11-2 )&'TUH.jk9mn9cH)op,`PQqrstuvSF!f ! w%cx=$y%&z! ! ! ! ! ! ! ! ! ! ! ! ! u(" ,T) d" = u( #,T) d# ! ! ! ! ! ! ! §11-2(2-2)! .{/,-' " = c /# ]^9|}$?@'! ! ! ! ! ! ! ! ! ! ! ! ! ! ! d" = #(c /$ 2 )d$ ! ! ! ! ! ! ! ! §11-2(2-3)! ! !~K.•€`4,],'§11-2(2-3)& " = c / # .,-! ! d" = #(c / $2 )d$ §11-2(2-3)’ ! .•‚-ƒ„}>.);$4! §11-2(2-2)9…†&‡TUHˆ‰9PQqrst ! 502 !"#$%&'()*+,-./0123!"#45678$9:;<=>(? ./0123!"#@,ABC-DE=FG6HIJ-K=>(4L §11-2(2-3) MN@ (2-3)’ OCL d" PJ$'($ d" < 0 L d" PQ$'($ d" > 0 =>(?R OCLSTM=UV d" PJ$'($L§11-2(2-2)@! ! ! ! ! ! ! ! ! ! ! ! ! u(" ,T)d" = u( #,T) d# = $u( #,T)d# ! ! STWVL ! ! ! ! ! ! ! ! ! u( ",T)d" = #u($ ,T)d$ ! ! ! ! ! ! ! ! ! ! ! (37) =>(?XYZ[-\]! ! 8#" 2 h" ! ! ! ! ! ! ! u(" ,T) = 3 ! c exp(h" /kB T) $ 1 -^_` " = c / # Pab6NI-PcV, d` d" = #(c / $2 )d$ PefTgL./h ! -)*PDE$'(./0123!"#-FG(6)4iCT(?! ! ! ! ! ! ! !8"hc d# u( ",T)d" = #u($ ,T)d$ = 5 $ # exp(hc / #k B T) % 1 (38)! !./0123!"#Pjk`'(./h-)* ( "max ) @L(38)-lmPjn`' ! ! ()*=>(OCL ! ! ! ! ! ! ! " 5 {# $ [exp(hc / #kB T) % 1]} = 0 ! ! ! ! ! ! ! ! ! ! ! ! (39) "# $6Hop(S$4qr(? y " hc /( #k B T) $'($Ls-tuvw] ! ! !! ! ! ! y = 5 " [1# exp(# y)] ! -xy y = 4.96 z4L./0123!"#Pjk`'(./h-)* ( "max ) P{ ! |(?}~HL ! ! ! ! ! ! ! ! ! ! ! "max = ! ! 1 hc ( ) ! ! ! ! ! ! ! ! ! ! ! ! ! (40) 4.96 k B T 503 !"#$%&'()*+,-./01234&567)8#-.9:;< ( "max ) =>?@#$;<!ABC Wein :DEF!"#$ ! ! G 16-1 )HIC T=6000K :-./0234&JK=,LMNOPQR)S. IT+#-./0123JK)UVWI+$! XC,1964 Y) Penzias, A.Z Wilson, R. A. P[\]^IC_`a:bc-.d=,T=3K :efghij-. )k#$ _3K :`a:bc-.d:]^=, _lmnop`aqd5rs8#t u@vwxyZ@zC$ ! {|,}~pn:-.4&•€, ! ! ! ! ! ! ! ! u(" ,T) = 8#h "3 $ ! ! ! ! ! ! ! ! §11-(3-5) c 3 exp(h" /kB T) % 1 5,•‚ƒ„…R!†J8‡ˆ,-./01234&$‰Š,‹Ei†kCŒ ! :-./0123Z@#$•‡PŽ•%&:•‘)’“8#•Z=,Planck P -.4&•€5]^8#”), Stefan-Boltzmann FZIT•–‡T+C(—11)$ §11-(3-5)5•‚ƒ„…R!†JIC˜™5 u*=(U/V)Zš8Z,U =i†P V ! "#9›œi:•/01235•ž8#$ ! ! ! ! ! ! U 8"h % #3 u* = = 3 & d# ! ! ! ! ! ! ! ! ! (41) V c # = 0exp(h# /k B T) $ 1 ••!, x " h# /(kB T) ZŸ?Z " 3 d" = (k B T /h) 4 x 3 dx !Z@#¡–! ! U 8" (kB ) 4 T 4 $ x 3 ! ! ! ! ! ! ! ! u* = = % x dx V c 3h 3 x= 0 e # 1 ! ! # x3 %4 dx = Z@#$¢†J= $ x !"#¡–,Stefan-Boltzmann FZ@#$ 15 x= 0 e " 1 ! U 8" 5 (kB ) 4 4 T ! ! ! ! ! ! ! u* = = (42) V 15c 3 h 3 ! £zT,9›œi:•/0123U =, ! 504 ! ! ! ! ! ! U= 8" 5 (k B ) 4 V 4 T 15c 3 h 3 (43)! !"#$! ! ! ! 16-4) %&'()*+,-./012345! %&'()23456789:;<#:=>*+,-./0 "(T, µ,V ) 6?@ A#1BC$(43))DE=F)GHIJKLMF1N!O#$FF!=>*+, ! -./0 "(T, µ,V ) 6@CP>%&'()23456789:LQA#$ ! R9-3 !=>*+,-./0 "(T, µ,V ) N>ST9/U1VP> ! ! ! ! ! kB T ln"(T, µ,V ) = pV ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! R9W3 (58) ! !"X>YZ> ! ! ! ! ! pV = G " F = Nµ " F = Nµ " E + ST ! ! ! ! ! ! ! ! ! R9W3(56)! N[GA#F1K\]Z^E = U )_`!@CPC#a$FbJ)cdeX ! ! ! ! ! ! kB T ln"(T, µ,V ) = pV = Nµ " F ! ! ! ! ! ! ! ! ! ! ! ! (44) !"#$1FfN>%&'()g4hijklmn=o!"#IJ>N )p: ! eJq ! ! ! ! ! ! F = "k B T ln# ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! (45) 1r#$ n=o!, N NstubrCF1)_`=R16W5 !uJ:vwA#$ ! ! Sx>R14W2 !=>yz{|&:}A#*+,-./0=> ! ! "=# r 1 1$ e $(% r $ µ )/ kB T ! ! ! ! ! ! ! ! ! ! ! ! ! R14W2 (14) !"~Z$%&'(:Fb6•@Ab€>%&'()g4hijklmn•o ! !"#IJ 505 ! ! ! ! ! "=# r 1 1$ e $(% r $ µ )/ kB T =# k 1 1$ e $% k / kB T ! ! ! ! ! ! ! ! ! (46) !"#$%"#&'()*+,-# . / 0 12345$67897:;<=* ! !>?(45)! 1@A-BC?! F = "k B T ln# = k B T % ln(1" e"$ k / kB T ) ! ! ! ! ! ! ! ! ! ! (47) ! ! ! k !"#$ ! ! (47)1DE#FG7&'() k 1H-#IJK"LM/?NO=P1H-#(0? Q)7RS1T-6!UVW#$! X1§16-2 VYZ45[\1?]R V 7^' _]V/? (" + d" , " ) 7NO=`a1bcB#defg^'h7&'()=/ n(" ) # V # d" = V # ! 8$" 2 d" c3 ! ! §16-2 (26) Vi#jk?(47)l9m7LM/?NO=P1H-#(0?Q)7n7RS!"#$ ! F = kB T $ ln(1" e"# k / kB T ) = kB T " k & 8#$ 2 ' {V " c 3 " ln(1% e% h$ / kBT )}d$ $=0 8# & 2 = kB T " V " 3 ' {$ " ln(1% e% h$ / kB T )}d$ ! ! ! ! ! ! ! ! ! ! ! g48h c $=0 ! ! RS/? x = h" /(kB T) !DoG?pSRS*q\!? ! ! % & {" 2 # ln(1$ e$ h" / kBT )}d" = ( "=0 =( ! kB T 3 % 2 ) & x # ln(1$ e$ x )dx h x= 0 kB T 3 # x 3 ) $ ( )'ln(1" e" x )dx h x= 0 3 # ! # k T x3 x3 e" x = ( B ) 3 {( )ln(1" e" x ) " $ ( ) dx} "x h 3 3 (1" e ) x= 0 x= 0 1 k T # x3 %4 k T = "( )( B ) 3 $ ( x )dx = "( )( B ) 3 3 h x= 0 e " 1 45 h ! ! !"#grs7tu/v12w4(20)VxX1qyGo#h$ g48h/?z{? ! 506 ! ! ! F=" 8# 5 (k T) 4 $V $ B 3 45 (hc) !49" #$%&'()*+,-./0123456789:S ; F <=>?%& ! #F 32$ 5 (k B ) 4 3 ! ! ! ! ! S=" = %V % %T #T 45 (hc) 3 !50" #$%&?@/A5BCD:U ;/ F = U " TS 4*+<=/! ! 8" 5 (k ) 4 ! ! ! ! ! U = F + TS = # V # B 3 # T4 15! (hc) !51"! EF%& !51";/Stefan-Boltzmann GEHIJ@(43)KLMN%&01234' ! ! OPEQ(;/RSTS/U 4UVWXE F 43YWX<=Z[4-\K>? %]! "U 32# 5 (k B ) 4 3 ! ! ! ! ! CV = = $V $ $T "T 15 (hc) 3 #F 8$ 5 (k B T) 4 P = "( )T = % #V 45 (hc) 3 !52" !53" ! #$%&! ^=K/!51"E!53"<= PV _`%E/0123#; ! 1 ! ! ! ! ! ! PV = U 3 !54" EF%&La/bcdef1gh#;! PV = nRT, U = (3/2)nRT !-./! ! 2 ! ! ! ! ! ! PV = U ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !55"! 3 ! #$%& !55";/i1-2(16')#jk23KlN% Bernoulli!mCn:o"4,E ! pq@r4#$%&!54"4*+;/0123stlud23#$%vE_w HIx%&La/ !55"4*+;ytlu23#$%vEK-%& !54"E!55" ;tlud23Eytlud23_z{N%|}F,#$%& !54";~•Q!0 Q"K*€H@|}F*+,#$%& 507 16!5"#$%&' N ()*+,-./0 ! (44), (45)1234567=89, #$%&' N (:;9)*+,-.<=>? 1+@1ABCD=EFGH<%IJ1K.5LMDH#$%&' N NO;C DPQ(-.<=>6R14-S9TU4GVWXYZ2'( " )N[\]^(-.< =1_(DHWXYZ2'`Z"9abcdDHGe46fg$h%i$jk1 ! aCDg$'lm%LMn>o%ppqr+,DH! ! st%/0uvw56R14-1) xi$jk%g$'lm=WXYZ2'y 1z{5LM|\H}4~•4G€EN•‚CD=6! ƒ„„„…†g$r@‡Dfg$h9>6ˆ‰%g$>Š‹9Œ-.•6Žg $(•%|\-i$jk1•DrNABCD<=>/0(-.H4r46Sg $%i$jk%n>‘‹’“=LMD%96o%|\-Sg$i$jk1”• r N–—6r ”˜%Sg$i$jkN™w5.Dg$'N nr =456™šg$' nr N r %›”1œ1•FGžK%'Ÿ6 ! ! ! ! ! {i} = (n 0 , n1, n 2 , . ., n r , . .) NLM6<,N R14 (1) .5g$h¡¢%i$jk{i}N£;CDH<,>fg$h¡ ! ¢%žK%i$jk{i}%g$'lm(number representation)9•DH¤¥¦§g $9>6…ž%Sg$i$jkN™wDg$'1>6 ! ! ! ! ! n r = 0, 1! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! R14`2" %¨©(•DHžn6ª«¬g$>6 ! ! ! ! ! ! n r = 0, 1, 2, . . . ., " ! ! R14 (3) %¡5%-®¯'N°D<=(cdDHg$h%¡g$'N N =46g$'l ! mN[\=6 508 ! ! ! ! ! N = n 0 + n1 + n 2 + n 3 + .... + n r + .... = " n r ! ! ! !14 (4) r "#$%&'()*+,-./0.1*23{i}.456789(r :;.<)* ! 1*23.45678= "r >?$>( ! ! ! ! E i = "0 n 0 + "1n1 + "2 n 2 + "3 n 3 + .... + "r n r + .... = # "r n r !14 (5) r ! "#$%/@(A.B)*+CD?$EFGH Z 9(IJEK.LMCNO( ! ! ! ! ! Z = # exp(" E i /k B T) ! ! !14 (6) i "#$%A.P9(QR"S$)*+,-.1*23{i}.45678 E i .,T ! CUVWX$%AXC)*HYZ.!14 (5)=V[$>(\.N]CY^WX$_ .. ! ! ! Z = # exp(" E i /k B T) = i ! % % n0 n1 (n 0 + n1 + n 2 +....= N ) ! 1 % " " " " exp(# k T % $r nr ) B n2 r !14 (7) A.B`P"9(abH n r .cd$e (n 0 , n1, n 2 , . ., n r , . .) .,TCf'gTP= ! h$%ijT(B`P.kl.mCn['op! n 0 + n1 + n 2 + .... = N ! 9(,) ! ! *Hq N "#$A>=ZjTrO(i.B`PqstWX$u.vwop"# ! $%!14 (7)9IJEK.xy@Cz]EFGH"#$q(n 0 + n1 + n 2 + .... = N . vwopq#$'{(cd$abH (n 0 , n1, n 2 , . ., n r , . .) ! C0[T.P9(|} ! C~•"9d[%€•(‚ƒdq„(!14 (7).…†9( & ) & ) !) & #$ 0 / kB T n 0 #$ 1 / kB T n1 #$ 2 / kB T n 2 ( + ( + ( ) + , ( % (e ) + , ( % (e ) ++ , , , ! ! ! !14 (8) ! ! ! ! ! ! " ( % (e ' n0 * ' n1 * ' n2 * "#O(!14 (8)…†.N]d~•dP.‡.ˆC9‰Šd[%‹‹‹! Œ•= ! Žf•C!14 (7).B`P=suC•{$C9‘’=“?$%i.”.‘’>9( •–—B)*+CD?$)*HYZ˜>§9!2, 3, 4 "™š'•›IJ (grand! 509 canonical) !"#$%&'()*+,-.*/012!"#$3456&78) */9 ! ,+:3;<=>?5@(A, n 0 + n1 + n 2 + .... = N 5BCDEAFG/HIH I N J#)*K&L*MNAFG/+5O5BCDEAFG5-;i PQ52R ! "r 5ST$ n r A4UVUWXJYZ~[\0],^_)*,`abcG/deb; f14 (7)gh,f14 (8)ih&jk-'l+,3-m*/ ! ! % ( % ( % ( ! n0 n1 n2 ! Z = # exp(" E i /k B T) = '' $ (e"# 0 / kB T ) ** + '' $ (e"# 1 / kB T ) ** + '' $ (e"# 2 / kB T ) ** + + + i & n0 ) & n1 ) & n2 ) % ( % ( % ( ! ! ! ! ! ! ! ! ! ! = '' $ (e" n 0# 0 / kB T )** + '' $ (e" n1# 1 / kB T )** + '' $ (e" n 2# 2 / kB T )** + + + & n0 ) & n1 ) & n2 ) ! $ ' $ ' $ ' 1 1 1 =& * * *** "# 0 / kB T ) & "# 1 / kB T ) & "# 2 / kB T ) % 1" e ( % 1" e ( % 1" e ( ! % ( 1 (56) = "' #$ i / kB T * ) i & 1# e ! ,F*/+5'nA;012!"#$&oebpq(46)5ih,rs-.*/ ! ! ! ! ! "=# r 1 1$ e $(% r $ µ )/ kB T =# k 1 1$ e $% k / kB T ! ! ! ! ! ! ! ! ! ! ! ! (46) ! deb;(46)tu5vw-5012!"#$( " )A;12!"#$(Z) Jxm ! ya*+,3-m*/(56)3pzUq+,A;${|-`aq@(A (32’)-} ! ~q•K€•=,rs‚ƒ„512!"#$(Z),F…;Boltzmann !†‡zˆ Planck 5‰Š‹Œ•JŽ*+,&••)*/ ! +UAt‘5’“J”•–—˜™*/(46)tu5vw-A;‰Š‹Œ u(" ,T) & " = 0 ~ # -š!›qœ3oHUbG*5-;Y51]5#• ! ! 8# 5 (k B ) 4 4 ! ! ! ! ! u* " U /V = $ $T 15 (hc) 3 ! 510 # !"#$(41),(42)%&'()*" $ x= 0 x3 %4 dx = " x = h" /(kB T) ! !+,-./ ex " 1 15 0$123"! ! 8! # 5 (k B ) 4 4 8"h k B T 4 % x 3 dx ) # & x ! ! ! ! ! u* " U /V = $ $T = 3 #( c h 15 (hc) 3 x= 0 e $ 1 & & 8"h $3 = ' ( 3 # % h$ / kB T )d$ = ' u($ ,T)d$ e %1 $=0 c $=0 ! ! 456788%9:./);< Planck 9=>?@AB%C67 ! ! DEFGHIJ6KFGL%M"NO9FGMPQ%R5,7SHS"TF GUGVW9XMYQZ[4\]"^TFGUGVW$_N9FG<C6H% KFGLUGVW!`ab64c;`de"89fg5KFGLUGVWMhi $PQ%R6UGVWjHI"Boltzmann /kclm/ke%\]67nFG; N <opqr6stM"89 Boltzmann /kclm/ke$ Nuvw49xy< z{b69%"|9}}%MK~•<l€$•‚ƒ„5,7|9…$"n-9 N 9†4‡ˆ‰Š‹Œ•Ž•!+,-•lm/k!&g7SHS"‘G’“%M" ‡ˆ‰Š‹Œ•Ž•u”%•FG; N 9op–<J—S5,9%"KFGLU GVW$˜b6lm/kcBoltzmann /ke9™š%›Mœ•4567! Bose FG%C6‘G’“$"Boltzmann /k<ar6žŸ%C67 16¡2 %¢£(¤¥ ¦§¨©ªŽX«B%¬¢qr6c-®!¯}5,e°±-²sM"³;9v ´µ¶•·¸G9¹}º4»ˆ¼$M½¾%C684cz¿ 11eÀ4ÁÂtb67 511 16-6)! Bose !"#$%&'%()*'+,-. ! /01234/0156%7894()*%+,-. Satyendra Nath Bose (1894-1974)! :;<=>?S?N?Bose 9 1924 @:4AB%/0123%CD E:FGHIJ)KLMNOPQ?R%ST9UVW$XYZQ([\]%^_ Philosophical Magazine _:`abcQd4ef9ghbcQ?RiX4S? N?Bose 9R%jaN Einstein :kl4 mnopqdY>!rstc>ut-^ _:efbc>v#wxoyzoP{!|}:~o4Einstein %•€N•sQ 78‚? Bose ST%ƒ„…†‡ˆN,‰oQ Einstein 94R%Š‹jaNŒt*(•‹ :Žo4Zeitschrift für Physik :k•o4Bose ST9•‘R%*(•%-^_: ’“bcQ?”11•6 X–—oQv#:4Einstein˜1917‚94™6˜š™›6‚ œ•%žŸ t4Œ¡¢£!¤O¢£N¥¦o4/§•¨)LMN©Py Planck LMª%O«oyP>?Einstein 9 Langivin N¬-y®ZQ*¯°((de Broglié, 1924)%+„±%CD:²³N´µyPQ¶·‚? Bose ST9 Einstein %¸C%¹ º»:YZQ?Einstein Œ¼n S?N?Bose %E½N,¾¿À%23Á-:© =>STN 192441925 @:ÃÄo4/01ÅÆ()ÇÈÉ()ÊËNÌ"=> i!:u>¶·4Í΂?i%v#uÏÐ t4Bose 239 Bose-Einstein 23!n7 Ñc>?J)ÒÓÅ\ÔÇÕ•%Ö×Ø 17) X94“Bose statistics or Bose-Einstein statistics was introduced by S. N. Bose (1924) for light quanta, and generalised by Einstein.”!%ÙÚd~bcyP>?/01dIJ)KLMNOPQ23…ÛÜ %ÝDE94Þß56%23ˆNàC=>áâ!uZQ?Pauli %ãäj,%å æ9 1925 @4 Fermi 23d Fermi :vZyåçbc>%9 1926 @XY>?è E41925 @:9 Heisenberg %éêÁ-d41926 @:9 Schrödinger Eëªd4 512 !"#"$%&"'()*+,-+./012345 Bose 6789:; Bose-Einstein 67;'Pauli ,<=>?@()*+,-+./012ABC9D EFGH&"I:95 ! S. N. Bose ;'JKLMN,OPOQRSTU"VWXYZOPOQR Calcutta ;'2001 [\]^POR Kolkata F_`&"V2'aaS;b,Zcde4f5 !gI'h;OPOQR,ijklKmnoOjQkSpqdrsVtuf5ij klKmn@;'vw,JKL6xyz,{|}WBengal, Bombay, Madrasf, ~cS895 1857 [F;'v•F€9•‚pqƒ„2a,{|},XYOPO QR'…K†J'‡Lˆ‰FŠ‹&"V5a"],•‚pqƒ„2ijklK mnoOjQk@Œ•"'vw6xyz,JKLS,ZŽ•‚pqƒ„@•• V5‘’“+@,”•2e•"9a@–89]g:tuf5! S. N. Bose ;'ij klKmnoOjQk—˜™'OPOQR?š“+,›œ@••V2'•y, žŸ ¡S8•V¢£¤,¥+¦§;•\•Vtuf5g\g' h;OPOQR ¨©QOWBª;«K¬ˆl-m®f,|S¯°F±g:²?+d+³' Einstein –´µC9¶·.•¸¹dº9F»•V5š+,$¼F@•I'½.o².¾ ¿;À\FÁÂS892'!"€Ã–žŸÄŽ,Æ°@Ç*2ÈÉFÁÂ\ dÊCËÌS8Í45¢£¥+,¦§;•\•V2'1935 [F Î)ÏdGÐ gVÑÒÓÔ,ÕÖ–!4S895 ! JKLF;'JKLš+,×@Œ•"9–4ؽ,ÙZ•²?+ÄoT?+ Ä Jagadis Chandra Bose(1858-1937) 2:95…!Ú67, Satyendra Nath Bose (1894-1974)@ÛÜgI;:s•:5Ý2ÞbF‹:I:9“American Heritage DictionaryW1979 ßf”S;'…àKá),âãF Jagadis Chandra Bose 2äÕC 513 !"#$%& Satyendra Nath Bose '()*+,-!"Jagadis Chandra Bose &. /!01234567 2009 8&9:*+;2<=>?@AB$2 Bose 2CD&E -,FG-H!A'7<I!"Jagadis Chandra Bose JKLMNO2P'QR+ ,-!STUV@W%W%XYZ-[\<U!" ! ]^_`a&b!cKLM2NOdefg?'h/!ij7 2009 8 10 k&9: *+;"]^aJ@lmnopq%$rst<2@u'v,wxNO@yO, z O2{|<2q}~WKLM2NOd•€•v@KLM2‚ƒ„'…2†‡& E-,ˆ‰,-!"A2j<F@Jagadis Chandra Bose JŠ‹&ŒV^•$+, -!" J. C. Bose &Ž-,@•O'•OzC2P'*+! P. C. Ray (1861-1944), •‘yOd S. Ramanujan (1887-1920), 1930 8’“’”2•–—˜\™Oš›š d'W!cœ•LžŸe ¡d C. V. Raman (1888!1970), c¢£2¤¥¦e<H $+! M. Saha (1893-1956), §–¨n©2 S. N. Bose (1894-1974), ª2«¬-® &./!c¯°LMœ±²–˜³´e2µ¶d< 1983 8•–—˜\™Oš›š d S. Chandrasekhar (1910-1995)@·0¸¹º2»¼< 1968 8•–—˜š›šd 'W! H"G. Khorana (1922- )@2007 8•–—˜½¾š•›šv;c¿ÀÁÂ& ./!ÃÄÅÆǘ(IPCC)e2ÈÉ•ÊË! R"Pachauri (1940-)@…v,@Ìm •ÍÎ&©ÏÐNO@·0¸zOWÑ2{|<ÒÓ/!KLM[NOd;B2 1Ô7U!" ! ;Sv@IPCC '…2ÈÉ R"Pachauri a&E-,J@ÕÖ2סFØ1v, ÙI;-"]^a2ij9:2‹ÚkÛ2 2009 8 11 k@ cÜœKÝ–Þß–Þ àáef>e7âã;"A2àá<J@‹ä2 IPCC åæd$7@¿çÁÂè–é •cêëìíîïev@ cðñò¨&b!óôçõ•öe2÷øè–é&ùúv 514 !"#$%&'()*+#,-./0123456789:;<%=>%?@A BCD+#$EIPCC FG% R;Pachauri HBI"!JE C. V. Raman K S. Chandrasekhar L%MNO$PQBRSTUVJW"$XJYS; ! Z[%\]^% M. Saha $ _/`ab% S. N. Bose JcBEd0def%g hijkl/mdheiVno(pqErsJtuvwtxyBz{$|{( }q~•:€•WP‚ƒV„•:$…†HJ3‡!"#;ˆ‰EJagadis Chandra Bose JEŠk‹ŒeiCN•LŽ•‡: 1884 •BEd0def%ghijkl /mdheiB‘(’:“”NOV„#; 1897 •BJ•–Œ`%—˜™š›m œ•žŸB •9E\¡BI"!¢£¤¥$ž¦(§•:w%NOV„•:¨©E ª«¬ ;! -0® (Hertz, H. R., 1857-1894) B¯•!\°¡5±²89:%J 1888 •V„#•LEJagadis Chandra Bose 5³y%\°´N%µ¶·(¸•!":< $5¹#;G•º:•!d0def%ghijkl/mdhei%n»(¼½E ¾¿%ÀÁ%oÂBÃć:;ÅÆJ 1915 •VE<% 2 •ÀBÇÈ%™š›( ɘ‡EsƒÀÊ%™švË$‡:ƒ”NKƒ“N%™šBÌÍΨ©¬; ! M. Saha $ S. N. Bose %rsJEJ; C;Bose 5ÅÆ+#ÏÐ%Ñ%d0de f%ghijkl/mdheiBÒÓ‡:<$BW#;<%³yEP. C. Ray $ C. V. Raman JEÔ9Õ9EÖN׳n»E“”N׳n»V„•:ª«¬;S. Chandrasekhar JEC. V. Raman %ØV„#<$JÙ›•VÚÍÎ<$5„#5E M. Saha $ S. N. Bose 5P‚ƒV„•:<$J…†H%ÛÜVݽ!Þ•:; Bose $ßSà%rs%•kás“”NOEJagadis Chandra Bose $ Satyendra Nath Bose JE1900~1920 •Ñ%d0def%ghijkl/mdhei(âãB‡ :ä•káMNåæ%Sçèé%êVEë•Bì5•!"#; 515 !16"7#S. N. Bose $%&$'( ! S. N. Bose )*+,-.$/01233456789,:;<=>?@ABC ?D1E*)=FDGH7@:;IJ+KL=A,B Das and SenguptaMN#12 S. N. Bose $OPIQRS+,TUV2WXYZ[\]^_`\]ab$cd1 ef$ serendipity (!"#$%&'()*+,-./0)Vg,+,hdIij>k A,BC$lK$mn1\]o$pqrnVg,s2tu%v$:;=C?) wAB 1924 x)yz{?| S. N. Bose $ 2 }$TUD Einstein $~ 1 TUIR S>k=2S. N. Bose %vs•€ab$•‚Vg,ƒ„f…€$†‡ˆ‰Š‹$ qŒ)•>kA|D1Ž•@AD0•kA,B ƒ„f…€$†‡ˆ‰Š‹s0• u?,$12Einstein $~lTUVg‘2’“ 78#=2ƒ„f…€$†‡ˆ‰Š‹ 1 Einstein $~lTUV1”•k'–{?,D/>kA,B—˜™š mƒ•€ ›rœ•‹Mž#$!252p.319 )12Bose ab)Ÿ+, $/0sg,¡ !¢Planck 1£¤¥¦D§¨©ª>kA,p«$¬I-®€$¯°‘Dd@>2 ±$²®I•€³+,CD)´Fk2C$µ¶Planck .123#)·•>|B ¸¶Bose#$´¹)¥¦º%vI»¼•€³+,CD12•€›r$y½¾)¿) Debye sÀÁ2¶Planck .1234563#¶žÂZÃÂ#IÄÅ+,C D)ÆÇ>kA,BÈÉÊË$ÌÍUÎ1ϪnsÐÑ>|B ! Das and SenguptaMN#=—˜sÒÓ>| Debye TUMÃ#)ÔAk'–>2 $´ ¹)0•kA,¡! S. N. Bose sƒ„f…€$†‡ˆ‰Š‹IÕ†+,CDÖ> )ƒWXYab$cd‹)×F|$12ØÙ2S. N. Bose 1¢Planck $¥¦ÚÛ µ12Planck $-®€ÜÝÞV=2Einstein $…€ÜÝ޶ߕ€#V=2àr 7)1„”Vg,)áA@AÈD»â7)ãä>2{u)2±$»âI Debye 516 !"#$%&'(%)*+,-./*0!Debye !"#123456789:; <=>?@ABCDEFGAHI-JKL8M2&N567OPQ)A,-. /*07RS%&/5TU%&; S. N. Bose AVW,AXYZ[\],A^A _`#a$%&/5T “ I had no idea that what I had done was really novel . . . . Instead of thinking of the light quantum just as particles, I talked about these states. Somehow, this was the same question which Einstein asked when I met him (in October or November 1925), how had I arrived at this method of deriving Planck’s formula.” <b]cdeG,-.f;g-:<b]chiXYj\ZXYdeG.A,k 57/lAO;Das and SenguptamnoAJ!A9l,k5T ! 76p,;6AqrAst-;S. N. Bose O<=>?@ABCDEFG8u/ &vw.xB%*.*y)7z{%&/5T%*%;|16-2;16-4 ,}~)9 l8;•€7•‚ƒ8k5„…†‡-;bˆ‰ŠY‹Œ,•Ž&•b]c‹Œ ,•Ž&•=•JK7.5A,;64-ggL‘4Az{,k57’-“lT ”•A'(A–—8˜y&•Ž4™.š›+A67,kplmœoT•žŸ žtA¡¢£ g• -;UA•žt#¤¥%¦/T§¨;=§‹©Aª«A¬-t -;®¯A°±A²³#´%fµ¶,·5¸¹8k5Tª«Aº»7-;6A 9l.¼½¾5quAF¿#À8ÁÂ7’-“lT%*%;)7Ž;S. N. Bose A!"O<ÃÄL“•GAÅ•,ky&•;Einstein #Æ·Ç*¾678.y)T È);UAJK; <b]chiXYj\ZXYÉÊGAË_8Ìy)T64Í2 ,;S. N. Bose -΋8ÏЛ458Ѿ57’-“lTXYÒ•žAÓ/Ô-; ÕÖ,×Ø#12)Ù§«Ú (J. C. Bose)*+;Ù§«Ú89y&XYÒ,ØÈ 4)Ùq«Ú(S. N. Bise) Û7;ÜÝ8ÞßYhàá]›4)A,k5T! 517