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kneFf¹²º3TFo0xWXLe(]thLGd[LGR(N)_LGWaT(L³mXLQWXPSR#hLQN_OLGWXmae(T(to(LGN ±<WXT(d[LGme(TFo£bckno(WXLQTWXTdknN_OLGfWXfOPSRFLGT£·cggnNOo(WaT+k[_sLQT¹ » Jqrf)_o(WXL%KMLQxMLGthe(TFtµLGWaT(LGf¼Wao(Lk[mXWXfWXLQN_sLQTK2knmamXf&WXTLGWaT(LGNcz8k#|&½*e+knoFNsk[_OWXfOPSRFLGT zj{>|d*NOLGWafONe(T(o(LQTle(T(ozjP}| LGmamXWX_sWXfPSR(LGT^º3]Nsk[R(]²e(T(t^z8KWXmamXW¸k[NOo>|®¦,LGNthgo(WXfPSR«3¹©qrf_knmafOg¾wTFfOLG]µ{(mXLQ]^W`_O_sLQm+thmXLQWXPSRl¿®LGW`_s]Wa_O_OLGm ij(N%i8knf_ck[mXmXL©~©TFi8knTFthfOthLQfOPSR#xWXTFo(WXthd[LGW`_sLGTe(T(obcWaPSR#_se(T(tnLGT0¹ À JbLGPSRFT(LGTIUWaL ´ R(knfOLQT(NskneF]µhghmae(]LGTr¾wNR+knm`_se(T(tij(Nwo+knfÁ+LGo(LQNO LGT(o(LQmpT(knPSRZKLQ_ONsknPSR#_OLGTUWaL3o(knYGe o(WXL¾¼T#_rxWXPSd*mae(T(tloFLGfc¨©NOLQWXLQPSdf { (x , p ), (x + ∆ , p ), (x , p + ∆ ) }  BC8DFEGE u à JKMLGf_s[_sWathLGTÄUWaLn¼o+knff¢o(WXL N ro(Wa]^LQT(fOWaghT+knmaL¾¼WXT(RFLGWa_OfOd*e(thLQmM¦ri8knf)_TeFN^k[e(f^Å{ LGN)Æ>nPSR(L«I{pLQf_OLGR#_Z bcLQPSR(T(LQT0UWXL©o(knYGeoFLGTo(e(NPSR(fOPSRFT(Wa__smXWaPSR(LGT~3{(f_sknT(oLQWXT(LQf{ LGmXWaLG{(WathLGT ´ e(T(d#_sLQf%o(LGN¾wWaT(R(LQWa_sf, d*e(thLGmÇYQe(NÅ{ LGN)Æ>nPSR(LgoFLGNo(LGT£Wa__sLQmaxMLGN_nghT |r|~ kneFfGZ ULQR(LGTVUWXLKLGYQWXLQReFT(thLGTYGed[k[T(ghT(WÈ fOPSR(LQ]ªe(T(oI]^WadNghdknT(ghTFWXfOPSRFLGTI¾¼T(fOLQ]²{(mXL¹ É JKMLGfOPSRFNOLGWa{pLQTUWaL3o+k[fMÁ>knN{#hLGNm¸knei¸,¾pLQNOWX]LGT#_wij(NLGWaT³LQi8nÊ]Wa_wfLGR(NMthNOghÊnLGT^¾wmaLGd#_sNgoFLGTZ¾¼T#_ xWXPSd[LGmXT¶UWXLqrR(NL_sRFLGghNLQ_sWafOPSR(LµKLGfPSR(NOLQWX{(eFT(tT*e(N©{FWXfcYGe0oFLG] ´ e(T(d#_pknT0oFLG]ªLGf3LGWXTI]^k[_sR(L ]lk_sWXfPSR(LGfsË[T*e(]LGNOWafOPSR(LQf ´ Ngh{(maLG]ÌthLQxMghNOoFLGTÍWXf_G®o+k[fo+knT(T0¦rTeFNT(gPSR«thLQmX¤hf)_©xMLGNo(LGTÍ]²e(ffGZ zÎcWXT#xLQWXfQ¥>¨©LG{# hL~©{(fPSR(WXN]²e(T(t#| Ï JUdWaYGYQWXLGNLGT0UWXLoFWXLФhfOeFT(thLGTo(LGNc¨©W`Ñ®LQNOLQT*_OW¸knmathmXLQWXPSR*e(T(t n −x n 0 0 0 x 0 0 0 0 φ00 (x) = Coshφ(x) , φ(0) = 1 kneFijthNe(T(o½*e+knmaWa_sk[_sW`hLGNÒ~©Nthe(]LGT#_sL[hknT(knma *_sWafOPSR(LQN§Óc{pLQNOmXLQthe(T(tnLGTgo(LGN§T*e(]LGNWXfPSR(LGN¼Ð¤hfe(T(thLQTZ BC8DFEGE HÔJµÕÖ¶×ÙØ8Ú`nØÎ Û} XÖØ8 XÖ}ÜSÝ+ÚV 8ÞÚOnXÚàßÖGáqrTâLGWaT(LG] e(ma_ONskãNOLGmXk[_sW`WXf_OWXfPSR(LGTäWao(LknmaLGTå³knfd[k[T(Tâo(WaL NOLQm¸k[_OWaWXf)_sWXfPSR(L¾¼T(LGNthWXLrqr](e(mafONOLQm¸k[_OWXghT p E = |~ hLQNOLGWaTFi8knPSR#_cxLQNOo(LQT0YGe p| c + m c p Z E ≈ |~ p| c = |~ p|c vwLGNfOe(PSR(LQTäUWaLn&o(WXL¾¼T#_sNgh(WaL S = k T ln Ω(E, V, N ) e(T(oào+k[]^W`_¢¿®e(f)_SknT(oFfOthmaLGWXPSR*e(TFtÍWa] æ 2 2 2 4 2 2 B b3k[R(]^LQT^oFLGfc XçS}çnn¡®¡+ XÖ}ÜSÝ+Ú¡lÁ+ghNO]^knmXWafO]²eFfR(LQNOYGeFmXLGW`_sLQTZ3¤hNLGTlUWXLckneFi,hxMLGTFT3èjffPSR#xWXLGNWXt x WXNoZ cÐLGW`_sLGT¶UWXL²oFWXL¢¿®e(f)_SknTFo(fOfe(]^]Lnpe(T(o¶o+k[]^W`_o(WaLijNOLQWXL¾¼T(LQNOthWaL F = k T ln Z(T, V, N ) Wa] b3k[R(]^LQTo(LQf©çnn¡®¡+ XÖ}ÜSÝ+Ú¡Á>gnNO]^knmXWafO]²e(fwR(LQNOYQe(mXLQWa_sLQTZ¨knf%fOgnmXma_OL3TFWXPSR#_2YQefOPSR#xWXLQNOWXtfLGWaTFo(WaL o+knYQeT(¤[_sWXtnLGTqrT#_sLGtnNsknmaLdnLGTFT(LGTVUWXL{ LGNOLQWa_OfGZ H H+JµÕÖ 8ÞÚOnXÚ¢ßÖGáKLGf)_sWa]^]LGTUWaLijFNcLQWXTId*m¸knffOWXfPSR(LGfcWXoFLknmaLGfc³knfo(LGNcé&LQ]^ LGNOk[_seFN T o(WXLvwLQN) _sLQWXmXeFT(thfijeFT(d#_sWXgnT0o(LQNd*WXT(L_sWXfPSR(LGT;¼T(LQNOthWaLGT¶e(T(o¶oFLGN³LGfPSR*xWaT(o(WathdnLQWa_sLQT¶o(LGN©LGWXTFYGLGmaT(LGTÍIg[ mXLQdFmXLnZ ³LQ{pLQT£UWXLYGeF]LGWXTFLGTLGWaTILQWXTFi8k[PSR(LGfcR(LGeFNOWXf)_sWafOPSR(LQfc~3NOtheF]^LQT*_ce(TFo f_sknN_OLGT0UWXLo(LQfxMLGWa_OLGNLGT0]Wa_2o(LGNdknT(ghTFWXfOPSRFLvwLQN_OLGWXmae(T(thf)ije(T(d#_sWaghTo(LQf³LGfOkn]µ_sf) f_sLQ]^fQZ H u JµêM ¡äÚ ¡ëQÜSÝ+ÚSÖìÍGÞÚÞÚí&× ¡®ÚSÖOîÚ¡î}áM±<WaNµ{ LQ_ONsknPSR#_sLQTàLGWXTFLlmXWaT(Lk[NOLl~3T(ghNo(T*e(T(t£hgnTe(T LGT(oFmXWXPSRVWXLQmXLGT³kne(ff)ãnLGN)_sLGWama_OLGT¾wTFLGNOtnWXLGTFWahLGkne(f ε j = 1, 2, . . . ³LG{ LGTïUWXLLQWXT(LijghN]lknmaLGT ~©e(fo(NOeFPSdij(Nwo(WaL£ð ÚØrÚ Ùס¢ÞÚ2êM¡®ÚÎn ñÚ¡0ÞÚcX}çnnXÚ¡lì0 ¡+ ²knTZ¨©WXL3UW`_se+k_sWXgnTWaf_wij(NwLGWÈ T(LLQT(o(mXWaPSR(L~3T(YknRFm nghTò3W`hLk[e(f%T+knPSRFijgnmXthLQT(oo(LQ]^ghTFf_sNWXLQN_Z#±LGN2dknTFTF]¤hthLco+knf%¾¼NthLG{(TFWXf {(Wa__sLkneFPSR£knT+k[ma *_sWafOPSRe(T(o>Ëgo(LQN%T*e(]^LQNOWafOPSRkne(fxMLGN)_sLQTZ B j ó+ôÙ÷õ#öø÷S,ùM÷SùMú®ú®ûGõ*ûGõ*÷süj÷sý(üjýþÇÿû÷GüÙ,üÈ÷sùôÙüÙûÿ##õüÈ ÷sõ #÷ε Gû üÈùGû üÈù!rÿh¸÷ÿ#sõ#õ#önüÙôÙüÈõ#÷SôÈù¼÷ ÿ#õ"#ôÙõ##ô÷$û^ôÙõ#ôôûûGüÈûù%rö[*ôÈ÷s÷üÙüÈ÷Sù' &ôÈù(()*)û# ε j j BC8DFEGE ° H ,J + õûGü-hù,÷.#÷$/Çõ()*0#ôÈ÷1#÷SùûGüÙüø÷÷sôÙõ*÷sõö}ûGõ")õ#ôÈù((*÷sõ2/õ*ù,÷43#üÈ÷Sù S = −kB hln Pα i = −kB X Pα ln Pα ÿ#õ"657#÷sõrô98*ö}ûrô:)õ<;=)õ hE i ô%#÷ôÙõ#õ*÷,÷sõ#/õ*÷}ôÈ÷ U ÷û3>@?%hôÈ÷$@,÷sôÈ÷/õ*÷}ôÈ÷.#÷sõ6A¼ô:rôø÷sõ + ÿ*(ù r" ÿ ,ö F (T, V, N ) = −k T ln Z = U − T S. ÐLQWa_sLQTUWXLwk[T+knmXgntij(No+k[fÇknmXmathLG]LGWaT(LGTtnNOghÊ[ãdknT(ghT(WafOPSR(LQT¾¼T(fOLQ]²{(mXL¼]Wa_®o(LGNzjthNOgnÊ[rdknT(gnT(WXfPSR(LGT>| ¿®e(f_sknT(o(ffOe(]]L X α α kanonisch B e−β(Eα −µNα ) Zgrosskanonisch (T, V, µ) = e(T(oIo+knf¦,tnNOghÊhL_OR(LGN]^goF T+kn]WXfPSR(L ´ [g _sLGT#_OW¸knmÈ« R(LGN α J = −kB T ln Zgrosskanonisch J(T, V, µ) = U − T S − µ hN i B H ° ¨3WXfOd*eF_OWXLQNOLGT0UWaLqrR(NOL¾¼NxknN_Oe(T(t¢ij(N%o(WXLÁ&mXe(d#_Sk_sWXgnTo(LGNcKLQfOLQ_OYGe(TFt (nk − n)2 nk mXflÁ+e(T(d#_sWaghTÄo(LQfPSR(LG]WXfPSR(LGT ´ gn_sLQT#_sW¸k[mXfleFT(oMi8k[mXmXf¢]¤hthmaWXPSRwknmafÁ+e(T(d#_OWXghT o(LGN]Wa__smaLGNOLQT KLGfLQ_OYGe(T(t n ≡ hn i ijFN3KMghfOghTFLGTIe(T(oij(NcÁ>LQNO]WXghTFLGTZ &MôÙõA÷sôÈùý"ôÈ÷süÙüÈ÷sô:($#ôÙü9C$5D#õ*÷sõ<3(÷sôE#÷$M÷(#õÿ#õ>hôÈ÷F57#÷sõrôG k ∞ X m=0 m2 xm = x d 1 d x dx dx 1 − x x d dx (1 − x)2 1 1 d − = x dx (1 − x)2 (1 − x) 1 2 x(1 + x) − = x = (1 − x)3 (1 − x)2 (1 − x)3 = x H » ±LGmaPSR(LUPSR(NWa_O_OL]²(ff_sLQT UWXLe(T#_OLGNOTFLGR(]LGT ËxLQmXPSR(LÁ+ghNO]LGmaT WX](mXLQ]^LQT#_sWXLQNOLQTÇe(] ij(NLQWXT WXo(LGknmXLQ1f He+knT#_sLQT(t#knfhghT0Á+LGNO]WXgnT(LGTgo(LGNcKgnfOghT(LQToFWXL¨©WXPSR#_OLo(LQN%ijNOLGWaLGTI¾¼T(LGNthWXL F Ënò e(T(o o(LGNcqrT(T(LQNOL¾¼T(LGNthWXL U/N knmXfÁ+e(TFd*_OWXghTIo(LQNcéÒLG] LGNsk_se(N{ LGWijLGf)_sthLQR+knma_OLGT(LQN3éÒLGWamXPSR(LQT,¨3WXPSR#_sL P YQe0(magn_O_OLGT¶¹U*_sLGmamXLQT£UWaLoFLGT£~3{(m¸knei§dngnT(d*NOLQ_©o+knNQpk[e(PSR0xMLGT(TVUWXLo(WXLbcLQPSR n≡ hn i T*e(T(thLQT0T(WaPSR*_xWaNOd*mXWaPSRo(e(NOPSRij(R(NOLQT0]²FfOfOLQTZ 1 V k k BC8DFEGE » H À Jqr] ¼¾ WXTFf_sLQWXT(fPSR(LGT²go(LGmamho(LGNÒfO LGYQWJI+fPSR(LM± [NO]L§xWXNOoij(NÇknmamXL 3N ´ R(gnT(ghT(LQTµo(WaLGfOLQmX{ Lw¾¼T(LGNthWXL knT(thLQT(gh]]^LQT ε =ε ≡k T ³LQ{pLQTUWXL²oFLGTVknmamXthLQ]^LQWXT(LQT0~©eFfOo(Ne(PSd0ij(NoFWXL²fpLQYGWJI(fOPSR(L± nN]^Lk[TZ¨©WXfde_sWXLQNOLQTïUWXL²o(WaL ³NLGT(Yi8nmXmaLT(WaLGo(NWXthLQN%e(T(oIR(ghR(LQNcéÒLG] LGNsk_se(NLGTZ 0 ~k B 0 BC8DFEGE À H à J±LGmaPSR(L½*e+k[mXWa_sk[_sW`hL^é&LQ]^ LGNOk[_se(NOkn{(R+[T(thWXtndnLGW`_z T go(LGN exp(−T /T ), Z`Z`Z | LGN)x2k[N_sLQTäUWaL{pLQW T(WXLQo(NOWathLGT^é&LQ]^ LGNOk[_seFNOLGTijFNo(LQTKLGW`_sNsk[tµo(LQN2¾¼mXLQd*_ONOghTFLGTYGeFNfO LGYQWJI+fPSR(L± nN]^L3LGWXTFLGf2Á+LQf) _sd[¤hNO LGN%xMLGT(To(WXLQfOLGNLQWXTLQ_sknmXmÎFLQWXT©knma{(mXLQWa_sLQNGgoFLGNLGWaTIqrfghm¸k_sghN2Waf_3¹ ~©Nthe(]LGT#_sWaLGNOLQTUWXL(R# fOWadknmXWafOPSR(]^knPSR(LQTUWXLfWXPSRIT(WXPSR#_o(WaL(RFLn+vwghNi8k[d*_OghNOLQTYGeI{pLQf_OWX] ]^LQTZ x 0 BC8DFEGE à H É J¾¼WXTFLthLGNWXT(tnL·3ghTFYGLGT#_ONsk[_OWXghT nghTàÁ+NOLG]o+k[_Ogh]LGT WaT LGWaT(LG] ©knma{(mXLQWa_OLGNd[k[T(TYGeàfOgnthLGT+k[T(T#_sLGT U_s¤nNOf_OLGmamXLGT,ò3W`hLGkne(fij(R(NLGTo(WXLw{ LGW#o(LQNÒ¾wTFLGNOtnWXL LGWXTFWXthL æ*K ]Lãvàe(T#_sLQNOR+knma{oFLGfÇvknmXLQT(YG{+k[T o(LGfmaWXLGtnLGTZ§qr]y³LGthLQT(fsk[_OYYGeo(LQT0¿®eFf_S[T(o(LGTIo(LQEN%{pLxLQthmXWaPSR(LGT¾¼mXLQd#_sNOgnT(LGTe(T(oIФPSR(LGN%fWXT(o o(WXLU_s¤hNf_OLGmXmaLGTrò3W`hLk[e(fkn{ LGNNOne(]mXWXPSRfOgd*mXLQWXTpo+knfOf{ LGW¼¨3gh( LGmX{ LGfLQ_OYGe(T(tzÎU(WXT + e(T(o &deFNOYn¥ e(T(o |o(WXL^kn{(f)_sghffOLQT(o(L±LGPSR(fLGmaxWaNOd*e(T(t0T(WXPSR#_µnLGNT+knPSR(mXnfOfWXtn_xLQNOo(LQTd[k[T(T − fOghTFo(LGNT0LQ_rx↑k æ Lãv²>↓knmafO6g L +∞ ij(Ne(T(fOLQNOLµ¿pxLQPSdnL«S>{pLQWa_ONsntn_GZ ÐLGWa_OLGTUWXLco(WXL3KLGfLQ_OYGe(T(tnfije(T(d#_OWXghT eFT(o ij(NthLGtnLG{ LGT(L3é&LQ]^ LGNOk[_se(NMe(T(otnLGthLQ{pLQT(LGf PSR(LG]WXfPSR(LGf ´ gn_sLQT*_OW¸knm µ R(LGNQZ§± knNe(]yhnLQNOithWa{F_%fhnWXPSRidnLQWXT(LÁ+LGN]^WÈãvwLQN_sLQWXmae(T(tµ¹I± k[fx(NOoFLfWXPSR nT(o(LQNOTpxLQT(T¶o(WXLQfOLòcWanLkne(f3o(e(NPSRKMghfOghTFLGT£{ LGfLQ_sY_xLQNOo(LQTVfOghmama_sLQT£¹Ó3{ LGNNsknfPSR*_UWXLo+knf ¾¼NOthLQ{(T(WXf3¹ M ¡+Ú XÖ©îG×4 N©×ë##ÚGá%êÖ©ÝFn¡ ÞÚÙØwÖ ñÜSÝ¡+ ñÜSÝØwס+ ñÜSÝPØ OrÚSÜSÝ#Ö}ÚÙ¼ sç#Ú¡ ÞÚR¡ QpÚ XÜSÝ+ÚT¡ S S 1 2 1 2 ↑ U ↓ BC8DFEGE É H Ï JvwLQNOWJI(YGWXLQNOLQT£UWaLij(N zÎk#|MijFNc½*e+k[o(Nsk[_OWXfPSR(L¨©WXfpLQNOfOWaghTIe(T(o z8{>|ij(N%maWXT(LGknNOL¨3WXfO LGNfOWaghTo(WXL¿®e(f)_SknTFo(fOo(WaPSR#_sLGT WXT æ B +e(T(o U ¨3WX]LGT(fWXghT(LQTZ u ÔJ±<WaLGo(LQNOR(ghmaLGTµUWXL¼o(knfÒKghfOL,¾¼WXTFf_sLQWXTr·3ghTFo(LGT(fOk[_sWaghT+knNthe(]LGT#_ij(N æ B øe(T(o U ão(WX]LGT(fWXghT(knmXL§U f) _sLQ]Z é%WX¥§·3ghT(YQLGT#_sNWXLQNOLGTUWXLcfOWaPSRkneFio+knfM±LGfLGT#_smaWXPSR(L²zÎknmafOgT(WXPSR#_2kneFie(T#xWaPSR*_OWXthL©Á>knd#_sghNLGT>|¼e(T(o hLQNxMLGT(o(LQTÍUWXL~3eFijt#kn{ L æ V Z BC8DFEGE Ï u H+J¿nRFmXLGTUWXLkn{hxWaLQWXLQmXL·3ghTWI(the(Nsk_sWXgnT(LGT¢o(LQf¼YQxMLGWXoFWX]LGT(fWXghT+k[mXLGTµqrfOWXTFt[,bWXT(tnLGf§]Wa_ N ´ m¸[_OYGLGT X + LGNWXgo(WXfPSR(LbcknT(o({ LGo(WaT(the(TFthLGT s = s Y xMLGmaPSR(Lµ¾¼T(LQNOthWaLR+kn{ LGT s = ±1 zÎk#|gnR(T(LL_sLQNOT(LQfÁ+LGmao E = −J P s s z8{>|2]Wa_%L_sLGNT(LG]ªÁ+LQmXo E = −h P s − J P s s ±<WXLWaLGmaZL L,¨©gh]^nT(LQT)³NOLQT(YGLQT«LQNxknN_OLGT0UWXLWa] ³NOLQT(YQi8knmam T → 0 FxWaLQWXLQmXLij(N T → ∞ ¹ N +1 1 N i=1 i i+1 N N i=1 i ij=1 i i+1 i BC8DFEGEIHÔ u u J\[]F^C_`ba@cdfe2ghijJj:k%KLQf_sWa]^]LGTIUWXLo(WaLlmDon®E*l¿®e(f)_SknT(oFfOfOeF]^]Ln(ijNLGWXL¾wTFLGNOtnWXLe(TFoIWaT(T(LQNOL¾¼T LGNthWXLij(No+knfLQWXT(oFWX]LGT(fWXghT+k[mXL%qrfOWaT(t[,goFLGmXmÒz8·LQ_O_OL}|w]^W`_ N + 1 ´ m¸_sYGLQTghR(TFL©L_OLGNOTFLGf%Á+LGmaoZ ³Wa{F_LGfLQWXT(LQT£0k[thT(LQ_OWXfWXLGNe(T(thf(R+knfLGT*({ LGNt#knT(tl¹ ±ïgWX]]LGN%UWXLLGfij(N%fWXT(T#hgnmXmR+k[ma_sLQT(d[¤hT(T(LQT¶UWaLoFLGTÓ3{ LGNOthknT(t N → ∞ ]^knPSR(LGT®Z êM¡+ØjÖ}ÜSÝ+Ú 8ÞÚ¡ ÞÚp QÇr q®áåÁ(R(NLGT UWXLVÐWXT(dËnKMghT(oñvknNOWXkn{(mXLQT t = s s = ±1 ]Wa_ s = Q LGWaTZ£U*_Sk[__ LrF{pLQN²knmamXL·3ghsT I+the(NOk[_sWaghT(LGT YGe fe(]]^WaLGNOLQT« s dn¤hTFT(LGT£tUWXLthLQT+kne(fg^tneF._ Lr({ LGNknmamXL·cghWT I+tne(Nsk[_OWXghTFLG{sT {s: i, t=: 1,i =. . 1,. , .N. .+, N1}} fOe(]]WXLGNLGT«sZ 1 i i i=1 i i i+1 i+1 i 1 i GDFE #D >D VH>6 J Ç×#ÖØñn¡ ÞÖGÞn ñÜSÝØ,Ú t!uwv x yu{z}| o~%l ±LGmXPSRFL¨3WX]LGT(fWXghTR+k[_oFWXL¿®e(f)_SknT(oFfOo(WaPSR*_OL g(E) WaT n bckne(],¨©Wa]^LQT(fOWaghT(LQT¹ KLQf_sWa]^]LGTÄUWaLIoFWXL¿®e(f_sknT(o(fo(WXPSR#_OLij(N¢LGWaT z8k#|LGWXTe(T(o zj{>|YQxMLGW`ão(WX]LGTFfOWXgnT+knmXLQfU f_OLG] • e(T+kn{FR+nT(thWathLGN%éÒLGWamXPSR(LQT]Wa_%¨©WXfpLQNOfOWaghT ε(~k) = ck . Ó3{ LGN(NOijLGT¶UWX'L LQxMLGWamXf%LFFmXWXYQWa_GFgh{IqrR(Nc¾¼NOthLQ{(T(WXf%oFWXLNOWaPSR*_OWXthL¨©WX]LGT(fWXghTR+k_Z • • 4 u J jëÚ\Öq ñÚ ±<WXLWXLGm]lk[mxknR(NfOPSR(LQWXT(maWXPSR(LQN§WXf)_§LGfQ#]^W`_fLGPSR(fwthmaLGWaPSR(LGT²±<(N)ijLGmFkneFi ~©T(RFWXLG{µLGWXTFL2t#knT(Y%nS}ÖSÖ}Ú ØÎÖSÖ}Ú {1, 2, 3, 4, 5, 6} YQeIxFNijLGmaT0k[mXf2LQWXT(LQTInghmXmaLGTULGPSRFfOLGN+knfPSR {6, 6, 6, 6, 6, 6} Z • U PSR+_sYGLQTUWXLo(LGTVºcT#_sLGNfOPSR(WaLGoVYQxWafOPSR(LQTÍU_sNOknfOfLGTVe(T(o ´ knfPSR(LGTVthNgh{¶kn{0ijFN©o(LQT¶Á>knmXmÎ+o+knfOf æKK L ´ fLGe(oFg[ã±<(N)ijLGmÈ«©]Wa_~©eFijfPSR(NOW`i_sLQT æ B aZQZGZ æ*KK knTIU_OLGmXmaL©hgnTIULGPSR(fLGN)x(NijLQmXThLGN)xLQT(o(L_ xLQNOo(LQTàzÎò3nR(LQNOe(T(tnfijghN]^LQmXTfOWaT(oIghdk} F|Z ©k_3o(knf%x2knf]Wa_cU*_Sk[_OWXf_OWXfPSR(L ´ R* fWXdYGe_seFT zοpxLQW{(WXf%o(NLGW&UF[_OYGL[Zé%Wa¥WadNg*Ëh0k[dNg*| • • J6ÇÚ O7© ÛnÚOn×#ÖøÖØrÚ KLQf_sWa]^]LGT UWXL o(WaLà¿®e(f)_SknT(oFfOfOeF]^]L LGWaT(LGf¶LQWXT(YQLGmaT(LGT ¿pxMLGW`rò3W`hLk[e(f f_OLG]f¶]^W`_¶¾¼T(LQNOthWaL ò3W`hLk[e(f 0 eFT(o ε > 0 Z KLQNOLGPSRFT(LGTÄUWXLo(WaLlWXTFT(LGNL¾¼T(LGNthWXLlknmXfµ¾¼NxknN_Oe(T(thf)xLQN_¢o(LQN²¾¼T(LGNthWXLlo(LGf¢U* f)_sLQ]^fQZ¶º3T#_sLQN) • fOe(PSRFLGT^UWXLo(WaL%³NOLGTFYQi8nmamXLwR(ghR(LQNe(TFoT(WXLQo(NOWathLGNÇéÒLG]pLQNsk[_Oe(Ne(TFo²o(WXfd*eF_sWaLGNOL2UWXLo+knf§bcLQfOe(m`_Sk[_GZ zξ¼fcNOLGWaPSR#_ LQxMLGWXmafLGWXTUFk_sYnZ\| Á+LGN)_sWXtnLGTUWXLLGWaT(L©3knT(o(fd*WXYGYQL©hgnT U ≡ hEi knmafÁ>eFT(d#_sWXgnT^o(LQNéÒLG]pLQNsk[_Oe(NGZ§¨©WXL©Ud*WXYQYGL©dknTFT • thNOgn{r½*e+knmaWa_sk[_sW`ïfLGWXT®o(WaLKMLGfOPSRFNOWai_Oe(T(t oFLGN~%PSR(fOLQT e(T(o LGWXTFL0knTFthLGo(LQeF_sL_sL0¾wWaT#_sLGWamXe(TFtÍoFLGN ~%PSR(fOLQT£WXf)_ckn{ LGN2xWXPSR#_OWXt(Z ³LG{ LGT0UWXLLQWXTIKLGWafO(WaLGmLQWXT(LQfc¿pxLQW`,òcWanLkne(f) f_sLQ]^fQZ • • 1 ° J¾¼T(LQNOthL_sWafOPSR(L·3ghf)_sLQTo(LQN3U(WXTFpghmXknNOWafOWaLGNOeFT(t ¨©WaL©¿®eFf_Sk[T(o(fOoFWXPSR#_sFL q S q ¡mXkneF_sL_w{pLQdknT(T#_smaWXPSR¢ij(N¼ijNOLGWaLc¾wmaLGd#_sNghT(LQT^]Wa_wd*WXTFLQ_sWafOPSR(LQN¾¼T(LQN) • thWXL ε(k)~ = (oZ RZ+]Wa_%½*e+knoFNsk[_OWXfOPSRFLµ¨©WafO LGNfOWXgnT ~2 k 2 2m √ g3D (E) = √ m3 E 2π 2 ~3 3º ] xWaLQWXLQmcRF¤hR(LGNWXf)_o(WXL0³LQfskn]_sLGTFLGNOtnWXLn§xMLGTFT N éÒLGWamXPSR(LQT knmamXLo(LQT(fOLQmX{ LGT U(WXT R+[__sLGT hLQNOthmaWXPSR(LQT]^W`_o(LG]!e(T( ghm¸k[NOWXfWXLQN_sLQT¿®eFf_Sk[T(oWXTloFLG] N/2 ¾wmaLGd#_sNghT(LQTo(LGTIU(WXT s = +1/2 e(T(oIo(WXLk[T(o(LGNLGT N/2 o(LQT¶U(WaT s = −1/2 R+k[{pLQT£¹ » JTQ ÚZ q+ÚnØÎ×#WÝ n¡n `ç#Ú ØwÞÚSÖÜSÝ+Ú XÖ}ÜSÝ+Ú¡ ØrÚ¡+ØÎ 8n\Ö KLQf_sWa]^]LGTUWXL©ijFN2LGWaTIU f_sLQ] e(T+k[{(R+nT(tnWXthLQN2é&LQWXmaPSR(LGT]Wa_MvwghmXe(]LGT V e(T(o]Wa_2d[ghT(f)_SknT#_sL • ¿®e(f_sknT(o(fo(WXPSR#_OL X oZ RZ g(E) = gθ(E) ]Wa_oFLGNU(Ne(T(tnijeFT(d#_sWXgnT θ(E < 0) = 0 θ(E > 0) = oFWXL]Wa__smXLQNOLé&LQWXmaPSR(LGT(YGknR(m%k[mXfÁ+e(T(d#_OWXghT oFLGf¢PSR(LG]WXfPSR(LGT gn_sLQT*_OW¸knmaf Z§KLQ_ONsknPSR#_sLQT 1Y UWXLo+kn{ LGWT*e(Nµo(knfµòcWXLQo(LGNo(WXPSR#_sL,bLGthWa]^L[ÇWXTÍo(LG] UWaL^LQWXT(L^LGWX´ Ti8knPSR(L^Kghmaµ_OYG]^<kn0T(TñvwLGN)_sLGWamXe(TFt knT(T(LQR(]LGTIdn¤hT(TFLGTZ KLQf_sWa]^]LGTUWXLe(]thLQdnLGRFN_¢ij(N²o(WaLGfLGf^U f_OLG] µ = µ(N/V éc|Z§¨©WXfde_sWXLQNOLQT UWaLd*e(NOYo+k[f • x2k[NOe(] µ YGeIgo(LGNkn{FT(WX]]µ_%]Wa_%YQe>Ënkn{FT(LGR(]LGTFo(LG] T N/V g Z z z
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