"!$#&%(')*#,+.-0/2 35454.! 6879;: <=?> @A BDCE"FGCIH JLK;MON,P*KQNSRUT$N*V K;RWTONYXOXZK\[ ]_^aẁmokqbdc*efegcihkjfxzlm]on*rwjfrpnk\v Ẁc*opR\q N*V Ẁjfc*Ẍ cybc*xzkqc*t5kopniẁc*ojxr y?cih Vc*XORWjx["KozẀMOKQ{OowRU`?c*T5KopNiẁc*o Ŝ i a j = a i+j a a 2 tgff a n v MZK MZK\oWKbdc*owK Ŝ a j = a +j JLKNYexRUcmMON,P*K coq Ŝ a n = 0v Ŝ a = a 2 = 0 a + a 2 + 0 a 3 Ŝ a 2 = a 3 = 0 a + 0 a 2 + a 3 Ŝ a n = a n y?cih hzk;egc_cn}ani` Ŝ 2 Ŝ = 0 0... 0 0... 0 0... Ŝ 2 a = a 3 Ŝ 2 a 2 = a 4 = Ŝ 2 a n = 0 ~
c Ŝ 2 = Ŝ 2 a n = 0 0 0... 0 0... 0 0... JLKVNYXDV cnx2ẁjfx OK ẁmzk;tzowc"v K\RwR ZT ẀcmẀMOK;c*T5KopNiẀcno Ŝ n jgx$nyefeg^ hik;mon,pnk Ŝ n = Ŝ n a = a n 0 0... Ŝ n a = 0 Ŝ n = y?cih hikkhnyx2`ẁcq{$xo[sn SNiẀowj owk\tzowk\rwkx2ẁnyẁjfc*x ẀcQẀMOK?c*T5KopNiẁc*o 0 jfopr ` ^*c RWMZcZex[ XZcYẁK ẁmzkqbdc*efefcihkjgxzlmtzowc*t5kow` ^ScYb ẀMZK;cnT$K\owNYẀc*ozhkMZjxVpM V\NYẌ ]$K[ZK\[OVK\[ORWjgXZl eŝi ẀMOK;K "TZefjfVj `kowktzowk\rwkx2ẁnyẁjfc*x cybẁmzk;cnt$k\ownyẁc*opr Ŝ n Ŝ i = Ŝn+i MZjxRkVN*X ]5KRUMOcihkẌ N*efRWcjgX ẀMZKQbdcnegefcihkjfXZlhzN,^ JLKVNYXDV c TONYowK ẁc Ŝ n Ŝ i = y?cih hzk;mon,p*k Ŝn+i Ŝ n Ŝ i a j = Ŝn a i+j = a n+i+j Ŝ n+i a j = a n+i+j RWjfXZlmẀMZK;TZowc*T5KoW` ^ScYbẀMOK;c*T5KopNiẀcno eŝi = + Ŝi + Ŝ2 i + Ŝ3 i 3! + + Ŝn i n! + hzkqmon,p*k eŝi = + Ŝi + Ŝ2i + Ŝ3i 3! + + Ŝni n!
y?c*ẁk0ẁmony` ẁmzkirwkowjgkijfr {OXZjgẀKz]5K\VNORWKvẀMOKowK0jxR cnxzeg^;n?{oxzjgẁkn czx2` X cybzcnt$k\ownyẁc*opr y?cih hzkqegc_cn} Ni`kK "TZefjxV jg`?p,n*ëok\rcyb i q = + eŝ Ŝ + Ŝ2 + Ŝ3 Ŝn + + 3! n! = + eŝ2 Ŝ2 + Ŝ4 + Ŝ6 3! Ŝn + + ( n ) 2! y?c*ẁkẁmoni`ẁmzksẁko Rhkj ẁm hkjgẁm Ŝ i i > n K "jxr ẁkx2` MZK;K "TZegjxV jg`?k "TZowK\RwRWjgcnX jxrzẁmzk\xtq hzkowka["owc*tzt5k\[l]5k\v\norwkẁmzk\^ln*owksxoc*x eŝ = eŝ2 = 0 0 0 3! 4! 3! 0 0 0 0 0 0 0 0 MZKQcYẁMZKopRkVN*X ]5KV cnxoruẁoov ẁk\[ jgẍ ẀMOKRWN mkqhn,^ BDCE"F ;H y?cih ẁmzk0]onnrujxrtpnk\v Ẁc*opR N*oWKI["K {OXOK\[;cnXN?V jfopv efk jfxov KQjg`?jfR ["K {OXOK\[ c*xdnv jfowvegk c*ojfxdl*kxokopnye a n+ = a a n+j = a j K K ]$K\o ẁmoni`hikimon,p*k n PnK\V Ẁc*opR MZK\oWKbdc*owK hik;vn*x TZowc"V K\K\[ ẀcRWN mk TZowc"V K\[ZoWKQN*Rz]5K bdcnowkq] "`khkjgẁm ẀMOjfRexN*RU` c"[ jg{$vnyẁjfc*x JLK;MON,P*Knq Ŝ a = a 2 Ŝ a 2 = a 3 Ŝ a n = a n Ŝ a n = a n+ = a
ẀMZK SNiẁoWj owktzowk\rwkx2`pniẁjfc*x jfr 0 0 0 0 0 0 0 0 0 0 0 0 Ŝ = 0 0 0 0 0 0 0 0 0 0 0 0 0 MZKQcYẁMZKopRkVN*X ]5KV cnxoruẁoov ẁk\[ jgẍ ẀMOKRWN mkqhn,^ 6879;: <=?>A MZKQ]ONnRUjxRP*K&V ẀcnowRN*oWK;NYl2NYjfX a a 2 gfg a n ˆL j a i = a i α a j J K;M$N,P*K ẀMONY` RWjfXZlẀMOK RwN KSẀK&VpMZXZj ZK NnRjfX ẀMZK TZoWK&V K&["jgXOl TZowc*]ZefK owktzowk\rwkx2ẁnyẁjfc*xsẁcẁmzk;cnt$k\ownyẁcnowr JLK;l*K` ˆLn j jgexnyoweg^ hik {OX$[ ˆL = ˆL 2 = α α α α... α 0 0 0... 0 0 0 0... 0 0 0 0... 0 0 0 0 α α α α... α 0 0 0... 0 0 0 0... 0 0 0 JLK VN*X ["csẁmzk RwN K;ẁMZjgXOlSbdc*o ẀMZK cyẁmoko cnt$k\ownyẁcnowrotẁc ˆLn c*t5kopniẁc*ozẁm$ni`kmon,p*kqk\jglnkx_pinyezk\r N*XO[ hkmzjfvpm jxr?["k {$XZK\[ ]_^ n α hik VNYX {$XO[GN SNYẀowj y?cih hikmon,p*kn*x n Ô = j= ˆL j + ˆM
zk\v\norwk jg`?monnrkjfl*kx_pinyezk&r MZK\X Ô = hzkqv\nyẍ hkowjgẁk n ˆL j = j= JLKQ{OXONYefef^aMON,P*K 6879;: <=?> aa BDCE"FGCIH JLK;hzN*Xn`ẁcmK "TONYXO[ hkmzk\owk ɛ 0 jfopr ` n α jg` VN*X ]5K;hkowj ẀẀKXNnR n α 0 0 n α 0 0... 0 0 0 n α 0... 0 0 0 0 n α... 0 0 0 n α n α α α α... α α n α α α... α α α n α α... α α α α n α... α α α α α... α n α ˆM n = Ô ˆL j = j= hik;vc*xorwjf[zko 0 α α α... α α 0 α α... α α α 0 α... α α α α 0... α α α α α... α f(x) = ( + ɛx) co x+ x ln f(x) J K;MON,PnK 0 MZK\X hzkqv\nyẍ owk\k "T5c*XZK\XnẁjfNYẀK ln f(x) = (co(x) + x) ln( + ɛx) = (co(x) + ) x) (ɛx ɛ 2 x2 2 + O(ɛ3 ) f(x) = e (co(x)+ x) = + (co(x) + x) (ɛx ɛ 2 x2 2 ) (ɛx ɛ 2 x2 + O(ɛ 3 ) 2 )
c BDCE"F ;H jgopru` hzkqruẁn*ou`k]_^ SN*}2jfXZlN N,^_efc*ozK "TON*XORUjfc*Ẍ cybczokk "TZoWK&RWRWjfc*Xtq ( + x ) n n (n )(n 2) = +x+ n n x2 + 3!n 2 x 3 (n )(n 2)(n j + ) ++ j!n j x j + "` ẁcẁny}nk ẁmzk;efj jg` X ẀMONY` VN*RWK n hzkqmon,p*k (n )(n 2)(n j + ) lim n j!n j = j! ( lim + x ) n x 2 = + x + n n + x3 xj + + + = ex 3! j! 6879;: <=?> BDCE"FGCIH Zcno ẀMZjxR K "KopV jxruk ^*c MON,P*K ẀcDV c T "ẀKmK "TZefjxV jgẁef^ V c mt "ẀK ẀMZK SNiẁoWj TZowc"[OV ` bdc*o?n*ege T5cnRwRWjg]Zjfefj ẁjgk&r MZKX ^*c hkjfege {$XO[ ẀMONY` σ x σ x σ y σ y σ z σ z σ x σ y σ x σ z gff v hkmzk\owk jfrzẁmzk;jx["kx2ẁjg` ^ SNYẀowj BDCE"F ;H σ j σ k = i ɛ jkl σ l + δ jk l JLKV\NYXRUMOcih ẀMZjxRowKexNiẁjgcnXORWjgXOl Vc T$cnXZKX2`pR SNiẁoWj jfowru` hzk;mon,pnk MZK\X σ A = σ j A j j (σ A)(σ B) = j,k hzkqv\nyxorwk ẁmzk;owk\nyegẁjfc*xdrwmzcihkx jfẍ TONYoW`?N y?c*ẁkzẁmony` σ j σ k A j B k RUẁN*XO[ZRbdcnoẀMZKjx["KX2Ẁjg` ^ JLK;l*K ` σ j σ k = i l ɛ jkl σ l + δ jk (σ A)(σ B) = jk (i l ɛ jkl σ l + δ jk )A j B k = j A j B j + i l ( A V ) l σ l = A B + iσ ( A B)
c @ ;7 =7 9 = vcdnyefe ["jx[ P*K\oW^ShzKefe c*ẍ ẀMOjfRkNnRWRWjglnX KX2` 6879;: <=?> OR `?N bdkh XZc*ẀK\RkN*]$c "`?XZc*ẁNiẁjgcnXtq NYXO[ NYowK vcmnyowkin*rw}*k&[qẁc {OXO[ẀMZK SNiẁoWj Qbdc*o ẀMZK&RUKzc*T5KopNiẁc*opR jfx ẁmzk;l*jfp*k\x S i e Si ]ONnRUjxR MZKowKSNYowK` hzc hn,^"r ẀMONY`Q^ncLVNYX K "TZowK\RwRQN SNiẁoWj MZKm{OopRU` hn,^ ORWK\[ jfx ẀMOK RWc*e "Ẁjfc*X RWK `pr jxr?ẁc RUj TZef^D[ZjfRWTZexN,^ ẁmzk SNYẀowj MZKmRUK&V cnxo[dhzn,^ jxr?ẁc l*jfp*kn bdc*o OefNbdc*ozẀMZK SNiẁoWj akefk K\Xn`pR Oc*okN l*jfp*kx c*t5kopniẁc*o NYX$[a]ONnRUjxRzP*K\V ẀcnowR Ô a ẀMOK NYẀowj KefK KX2ẁRkN*oWKQK _TOoWK&RWRWK\[ N*R\q i ẁmzk a m Ôa SNiẁoWj KefK K\Xn`kjfXDoWcih NYXO[DVc*e X X n = `mjxr cyb ẀK\X V cnx_p*kxojgk\xn`ẁc [ZK {OXZK MZK\X ["K\XZcYẁK\RẀMZK M mn a m Ôa n Ô c*t5kopniẁc*o NYXO[ [ZKXZc*ẀK\RzẁMZK SNiẁoWj K\egK mk\x2ẁrcybẁmojfrkcnt$k\ownyẁcno c mkqt5kcntzegk M jglnm2`kv\nyefe j ` mn hkmzjxvpm jxrknyexrwc cn} NnRzegcnXZlSN*Rz^*c ["cnx `lnk ` j "K&[OT ]5K ` v hzkk\x ẀMOK;c*T5KopNiẀcnozN*XO[ j O mn `pr SNiẁoWj owktzowk\rwkx2ẁnyẁjfc*x bdc*o K ZN TZefK jgb^*c N*XORUhzKo ẀMZjxR ZK&R ẁjgcnx ]_^lnjgp_jfxzlgnyx K "TZowK\RwRUjfc*Xbdc*o ẀMOKX;^*c M$N,P*KXZcY`N*V `ONYefef^ NYXORWhzKowK\[ ẀMZK ZK&R ẁjgcnx exzc*`kn Si a i jxr NkRUẁNYẀK!iP*K&V Ẁcno"i}*K` e Si a i SNiẁoWj K\egK mk\x2` cs}*k\kt jgx jgx$[ q c*t5kopniẁc*o XZc*` NmX ]5Ko Ô = RUẁNYẀK!iP*K&V Ẁcno"i}*K` XZc*` NmX ]5Ko Ôa i = # NYẀowj KefK KX2` a m Ôa NmX ]$K\o n XVN*RWKQ^*c NYowKQjgX2ẁKowK\RUẀK\[ ẀMZK SNiẁoWj K\egK mk\x2ẁrkc*b jgẍ ẀMOjfRkTOoWcn]ZegK NYowK*q S i a m S i a n = a m a n+i = δ m,n+i MZjxR Vc TZefK ẁkef^ RUT5K\Vj {OK&R ẀMZK SNiẀowj ]5K\V\NORWKjg`QẀK\egexR ^*c KP*K\oW^ SNiẁoWj KefK KX2` cihik\p*ko RUc K Ẁj K\RQjg`jfR cnowkorwk bzeẁc["jfrwtzexn,^dẁmzk SNYẀowj RWcD^*cLVN*X RWKKhkM$Ni`Qj %$ `;efc2cn}"r efjg}nk N*R jxr [Zc*XZKmjgX ẀMZKSRWc*e "Ẁjfc*X$R '& owcn]zegk bdc*oqk "N TZegK jfr $VpM K&N*RWjgK\oIẁcaRUcnegPnK jgb^*c efc2cn} Ni`ẁMZKQbdc*o cybẁmok NYẀowj 6879;: <=?> y?c*ẁk ẀM$Ni`?^*c N*oWK;NnRU}nK\[SẁcK "TONYX$[ jfẍ T$cihzKopRzcYb ɛ XOcY`?T5cihIK\owRzcYb (