lecture notes on qm8 - I bEX...

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Unformatted text preview: I bEX UvDQeBuhg`CXdxUB`WDC`dcsVjGDbSX|bEUQ`@whBdcsqf`W~dEwdvQUfjeQ Y W W W f B B Q y S c B [email protected] Q B Y H Y l S W Q Y C B Q Q c QdfbSDYhB)dzUWDQDeBX|ppuxwxwwtVPzQUfS`[email protected]`cVUCX`dWsqeYdceSdWXk`QUcX|H [email protected] S Q C B W Q y x u u r l [email protected] i YaEC c S Q B Q Qa Q xbx bEX`@W Y XD~UEVwv v B Q vC CDYoB`[email protected] E WC eQ c eQVXD8dldWbS`Yu~XkbSDYeBbWubgDYdfeQrctXdrEwbQdfS`YeBXddCUWQXhgdc`QGV~dEwGXUBeQj6~q WC B [email protected] Y B iECE y x [email protected] S Q c fq WqC f B Q vaE C W g dasqeYdceSdWXbWdWQUfS`YeBGdv`QeBuegVSGdSwX`|dlUWSXdSwVdS`YeWUf|DCXVVCXE Q Qa Y [email protected] pC W Wf YE Q v v BE YE Q vf W S BWCi f d!z k|!)kzw )! wV d z) x{ dxUWQdfbSDYhBhg~dEwdvbQdf8dWbQDYhBdcsV`CXjdEUgX [email protected] c B Q QC [email protected] [email protected] Q Q aE YUQUfXUWQ`YeBdcsVeCXtUldWbQDYeBUc`CXdc`Q`BGUldWQ`YeBUcVeCX|SwVDCkXbEXXDeBeWdcXXGX`VCqdYXedcS SEC [email protected] [email protected] [email protected] [email protected] B v [email protected] [email protected] Y v B C i YaE QaE Q B c f S eQdWQUfS`YeBTXDfrgoBDY~dEGaTGDVeCc{GE|xHCbfVCUWQDYhBdcsV`CGhgDceBwuegXxbSDY~dEGaTeQ c [email protected] Q B C B Q Y Q B v i Y f [email protected] [email protected] Q v v W Q y f B Q Y c UWQUfSDYhB}X`DUcXaeYXVdC`C|}XV{VC`[email protected]|WGD{rEzoxxwxwdwXDvGt1qolnQmxGdldWbQdfbSDYeBRrHB [email protected] Q B B B Q WE S B W QEC E B Qa Y B y u us r p x Y [email protected] C E B cC v C i WE S W W W C fq WqC C E Y [email protected] Q B Y W Qq E S YE `k`QVwRjGEDYhBWeYXVdCwdvDQhBuegfUWQXhgdceQXVf6CDYeBSXUa`QXX`6bEdUEdQ`QsdWSGS piX``WUWdQUSX`UYdSs`@X)dxwdv`QeBueghttsrqpiXVhbgGV2daeQdcXba`YXVUCTRPIGFDCA9 Q v v BE C S W f Q W Q y W W W gaC QEC fEC Q WE S Q H HCE B @ 3 ( % " #!210 )' &$ #! "87!265 4 3 tI p r iip {z w d6 x y w t w s w v r u p fw q t sp ir q E B [email protected] Q B aE E Q Q BC W Y W Q B Q Qq i [email protected] S Q VDCeCX}XD{XjVeCDcUBUSQUfGD|[email protected]}XDdEdQRrHUBsXEbYXXVUC}Xy nm E Q Q BC i [email protected] S Y c i Y W Q w u ov VeCDceBdSQUf}XDXEbYXXVUCbBG`qVC [email protected]}Xy i kjli fV`U`DekVGVeXy g h E C Y B S c B B q v C f C d Q 2 W v c Q B E CY B S c Q B EY i EY H Cff C Q B W EY B E C S y E C B C c @ a E 3w`UDVbDUDd`~kXb|x}GDfXb~VUAB zx VDeXX E Q C a E W Y W Q B a E x x W v C B S Q B B Q c Y H v B Q v C VeCDceBdSQUfqz 7X8 XEG`@fX`kX v S eWqzdc`QUEdQRXD`[email protected]`QDbB| XD~UEVwv c fiEf Y cC Q Bl XXDbBXeqVX`2 r 2 dQ GhBWX}VogDYdfeQX`[email protected] DY~dE`QXy SE Ya Q B c Q B EC Wa Q Q f B [email protected] Q ~v x u u e f u r E YE Y v Q B g XV`CbBfwAGD~q dEbgVWeidcXUEDDCXz~eWT8V`C`CGGDDCjxXXbEXDXf`Wv `CDceBdSbQdfQ Q Yi Y g Q Qf B B Q y x W v C E B [email protected] Q B B Qa Y Q B aE W C E Q B B c Qa Y Q y x W W c iC B W v B Qi cag Q v Yx Q B C E B X`DC`WUcQU`Q}hxXXbEGRdv`[email protected]`CXu~}XDB VCtyX`8CDYS oibYXXQXwveYdcXba`YXVUCXUBw bgXXFgXCDYeBSeYDceBUW`QsdlUWQdfbSDYhBbSDY~dEGaCdldcUgRxHA [email protected]@ [email protected] W W Q WE S W Q x [email protected]@ v WE c [email protected] B Q Y c Q Q C 2 af C 0 (' # i E H Cff C Q B l W Qf SY B c @ Qf q WY i EY B WY a c C l W y x W Q 3WGVX 1)%&$XbY|x}GDGUdU`eAX`bXVXbDebXVqdXdUX oif~EdVibdYQ|QiXEeYWAQX`BiXbEXXa~wadXbEBGCAeQUWQX~UEViUYAX`uplV x dWbQXegUc`QXVVogDYoBXXV" g Q f Ya gq Q Y q c fg Q Q Q B fq WqC Q Yaa cC V xs ! r xwxw r w [email protected]|UBXGteCX}XDEeB|`WV`CDUceQVXy S W Sa [email protected] Q B Y S c B [email protected] WC 7 5E qFR ( E gqi( gfWeT B `7 )' bG ' `7 YX' WC # R ( d c a G @P E I P p h C I 7 5E u P I G E C [email protected] 8 a v7 t# 3sFD7 )X' r7 @ qd ' G 7EG T R P I G E C [email protected] 8 7 5 0 FVUG # C ( S7 Q# 3HFD7 AA' 964 x r bX``B EY @ W f B C r XD~UEVwDDCB v B Q vC vf B E Yaa VbCDYbBGX VI Q F eyRw c s e p h ge d P b a ` P X V PR P H D Bu @ u ' IYTx%UvutrqifSURUP!cWICQWYWUFTSQ8IGFECsAa 9 2 2 1 2 1 1 2 xUdWbQ`YV`CUBdSeQ87DDc`[email protected]`Q|X`[email protected]~EDeBdcUQXX|X`}`CXXdQGE`YbQXqeY`[email protected] c B B c W Q B f BCEE S QECf S [email protected] B Y SE Q B B Qa S W f WC cUVXVbCXWAWX`dlUWdQXeBbWXSwVDCF~da`[email protected]`Q`QFUgGD!odldcUgRxHAbQXeWXVGEDYhBWXA QiE f CE Y Y B SE Ya Y v B gq B c W c Q B Y Q Q C x fq Yi Ya W daeQdcXba`YXdCsdTegXDUQXhBWXASXdC`S`CdcT~da`[email protected]`QfXVeCDceBdSQUfrHkGVCRW Q WE S Qq E S Q Bl SE Ya [email protected] W S v gq B c W c WE Q C B W iE f 5 3 d!z k|!) 64 2 0 !1 ) { x (CDYzv GEDYhBdWeQdcDQ~bEXXdS`Yu~X}X`UWDQ`[email protected]|H 'C E BC v x xS i B YEf [email protected] Q B B c W Y Vkyx p eu u " & r p pu 8 u k " u %$DGXDdWUQXXu`C`cbE}XVC B WE Q B Sa B Y QE x u " pu u k # r e pu u k " u u E SE H Q B cC" B W Q B c VbCDYUBGXQ xgXDfq!UxdWDQeB}X`fVC k u UWDQoB`Y|{X c H aE V!x r r UWDQbEX`atUCVogDYdf`QXnv|dWUQXXu`CDc~bEGVGdl`Y~bEVC`YefXD`cfWY dQXbEY B Y cC S Q B c aE x xS Sa B Y QEC BE cg Y E B WE B S x k p 8 e r u u dW`QhB{jdaXG `@bWV`C )UEVVeCXu}X`8 Asr dQ`@hBdcsqbfA B W gq QEE W Y v B Qi cag Q B C S W Q Y Q xi f W QE [email protected] Q B B W f Q Bl v Q Wf cCE Q B B B BE S C B c Q Xy XXE`Y`[email protected] dcsu~GDDCfhB}X`XjG`DceBdSs`@TevVX}X``CDQGXbq`Y`cdCwrH|eBeWXy 1 s I V8bx u ( %Vp )" VpV n " $r u 2 u VpV " 2 B ' %Vp " B ' r %Vp " ' u u 2 Q S QY @ SY c B Q v v g WY B E E a E Q S QY @ SY c B Q v v g W C B EY a Q W C @ v C S Q a Q q E S u oUdbXeDewue`~FGddbXb`DewueD~jdeVwVUdXsjf%VV " DDeW QBB dXk X UWX~dEidYX` ~xsox `C{GEXeaVdCUSS VegDUcXdEdiXbaVC RrHB S Q W y x aE Q fg Q Q Q B W B i Y c x S Q Q Qa f iC W W f Q E C B B E Q v B C E Q C a Y B Q Q C x Q Y i Y a Q q B C E E S u ugbHXVfDd|XdWuXWGDqdldcUgRxHAz UaX`bWXVXbEDYhBWXsXXd%V V " aE u XVpV " dW`QhB6GDAplUWQdfbSDYhBhXDVbCDYoB`YVs`@[email protected]`Qc `bB|)SeYDceBwuegW B W Q B [email protected] Q B C E WC WE B B Q YH Qvv eQ6SX|H UWQXhgdc`QGVukQGq`YeWVs)[email protected]`[email protected]`QGwGDdQXS bEY c Y f q W q C Y g f W C @ f E C Y B W c i C B C E B c c @ Q B xE YE Y v Q B f Q GVDCoBfwX`[email protected] i dWUlVDC`dcsV)VbCDYDGkevdcsXD)`bB|`QGkwVdCUWQUfSDYhBbSDY~dEGa}dWbQXegUc`[email protected] Q c B [email protected] E B B [email protected] Q B Y H B v v S [email protected] B Q Y C fq WqC @# W Wbx jv VpV & r u u u W f |xHCbfVCR x `bB|{X Y H aE ~vbx u ! 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r ( n " dQXbE`YA~dEwXVCUfVegXwvDYVGV`CX`VX` Wbx DCXXa UEUv`QXewv S W B Q v @ Q Q a Q B i c B c C E B Q B Q c W Q i c Q E Qq Q QE E S B W W S C S c S v v W Q Bl cC Q c VGV`CX`BFdaiXS|cUXDoBdYXjDQDeBX`W dcDQeBD{eg`cUBwQ uegX` X`QevdcX`eBX" bx ( n tir r [email protected] c CRrH|GDqC|[email protected]`!hBdcQsqbfYARUBXGateCXXDskCDYoB`YVswVdCUSXGXVVVX`eBVUWXauxg`CG8~8ox Xy B Q B S W S [email protected] Q BC E [email protected] v QafECiC cC E Q Y [email protected] W VFybx r " B " # r " B B " # )" n " # r ( G fECiC cC c E [email protected] v Qa Q BC B [email protected] v S C B Q VVX`eBVeQ{VC`YbBeYVwVUCdSX}X`8|`WUEXVsdCwrH}Xy Ibx ( " n " r r )" ( n " `bB|H Y I s V x ( " V IV cC u u B n " I V cC u u B i Y Yi q Q Qa [email protected] v S c S W Q XbEubgV~g dasqeYdc`SUWXog df`QUBQGVUC`QdWUQ`@`QUWXy xds {VC`Y~UE`[email protected]`[email protected]`@`WDCUBUSV`QUWXD C E B B c c i B WE Sl S WfE W Q Y QEC WE W c Qg Q BC S V x ' u %pV " B r DIt " VpV " ' )" u u r ( u %Vp " B ' r r D%Idt " VpV " ' n " u u WXVC`YbEXkVUCbS`Y`cUBwuegbWDY~TXSeYDcUBwwthgX`XEGafGXwUadQXGda`QsSX|~CDYB E B Yq v S Q v v BE aE Q v v W Q B i Y Yq gq S c Qq E S Y HlE B itUE`[email protected] dWVogVfUQ`@W WX8xxs CDY~dEeQdW`[email protected]`QuC`YGdEwvXa rHRXDkX`@W c c f S c fE Q Yi S Y y C E B B c cfE W Q Y iC B Q B E xt x u %VV " %xt " r u aE u X VpV " UWDQDeB}X`qdQX`@W B W Q Bl S eBdcsqbfAWX`B `Cc|VpV " DQDeB{egDDcoBX`q&V`CcXE`YeBDB ex 0hX B eWVDC`dcsVC Q Y Y v u B W c Y c E v i c r aE c B [email protected] egDbBXGTX`~q da`QUE`QUW`[email protected]`Q TeQ X B UldWbQdfS`YeBX)rHk9 r UQ`@hBdcsqfX`drEy c YE Q B g B c c c aE [email protected] C B C S W Q Y Q B xxs XVeCVSeYDceBwuegW @ ci Qvv Q [email protected] ci E Q E W c X`XVeCV{bYdfsq evVC u B UBeQGDqdlXfx r ! B r XX`QevdcX`eBX" W Q B W y W QECl cC Q c V x E WC WE c VbCDYbBeYVs`@X`8y r 2 rgoBDY~dEXray r 2 B B Q 2WBSQ q 3`UUe7 VC C BC E B B [email protected] Q B Q WE S W Qw x c Q B @ ci Yg Q B Y f c Q B B f c RrHCDYGkevdcsX`RdcGa`YGVdCfGfUBegVUCXDTGV`CVC UHQuXDTbE`WbBXedW`QXDDQXkevVC Q H c p C W W W E W W Y [email protected] Q B C f c Q B f Q Qi B Q cC [email protected] UvDQeBuhg)DCB VbCDYeBdS`QXVCueg`QXGDk`WbBXedWeQXDdQ$ Y`dcXUEVDCdcXeaV2rEy wz wdz!F 2 ) Uz z 15Fwd zU 1 k ! 2 { (I s A 2 VV xs r s s r qA s r t gpA t s dEX`XleBdSeQ7 [email protected]`{`bB||UBUSeQ7 VDBGD{VC`YG`vdcs}XD87DoB|UWDQGdEXX! y Q B qC B9 Q B Y H qC Q BC E B B [email protected] Q B Y H BC Qa QEC x s s r 2 s s r 2 ss r B 2 W E CY B B v c Q 3GV`G`[email protected] bSYUftegq daUgQG[email protected] dEUgff UxDW~dEwdvbQdfh f`bB|H XVeCVjGDB S S g Q Y S Qq E S WE B B Q Q Q B Q Q r 0s Y @ ci Q UcXaeYXVdC}QGGxW i )XVeCVXXeUWX X|XVC`Ytk`vUcUEVUgbfs6UB`QXy Q W E S Q H f @ v Q x C @ c i q W Q E Q a W E B B Q @ Q Q f Ct 5 W Q y Vo x XuY fbY r 2 3 I aaC W u UEVegbW|oY r 2 43I Q Q Y u UWXY GGV{X EDGVDC{egD`WUWdQUSXXVbCDYB QE Qa QEC aE l Y BqC B c QE WE i `YVs`@GD`cB [email protected]`QecVdCXuA)[email protected] UxXVbCDYB WC WE C Qq vE Q B B i YaEC c S aaC cC Q Q f S Y E B B Q WE i `YVs`@GD`cbgeY`WdWUQdSXe)~)UaXEDGVksdS CDYGkevdcsF~DxHX`dcX`eBX)dkXE WC WE B Q S W gq Q Y BqC Qq E E B B [email protected] gE B B C W Q c E S Q V x B r 2 B r B B B Y Q v Y W E S Q `wogdf`QXUawv}UQ`Q}XE xwxw V( x )p xxw P $r p p xxww v sv s r 2 2 ww 2 p ww G ( qd )' E # E q0 D7 I XAYP T X p C P'@ P a E p x 2 p7 P $r 7xwxwxwxwww ss r 2 G blGi sUlDWeBdSeQ7 VwRrHTbgGV}DQGkevdcsjbSX|Gl qd A(' E ' G ( 3Iqd # FCE `Q~hBVsbvbgbfUYdSs`@z x xQ qC C B fEC B [email protected] Y H G c BE [email protected] Y f Q W 1 xx ww 1 1 1 v ~ x xxww t s p ww DWUBUSeQ7 VC p XVbCDYDGk`vUc0 !`WeBbW`YXVUCS i X`CVXy q C WE B B [email protected] p C WE @ ci Q x ( Ya [email protected] y fC ' $ l Q x x k)[email protected] l`y!xVel"&%!"#"! eY`WUWjx olXniQ ldQ`QUlWfUBGVUEVogVeQk`WbBXedWeQbgXE i XV`CwbS`Y`cUBwuegXD8dlDWeBdSeQ7 C |dXVeCVi W Y Qa cC" x Q Yi c f c f x @ ci Q v v W Q B q p C @ c E B B [email protected] Q B C WE B B c c f S Y Q B BCq E B c QE Q H W W CDYGkevdcsX`XVbCDYDUE`QUW`[email protected]`QwQXbqdYXXUa`QeccX`G6VbCDY`vCGEY daUQXRzdlwdvDQhBuegW [email protected] FDCVbCDYd$ YDdcGdEVXD}VxVbC`YXUEbvXrapiXVTeQXXVeCVYUfq XVC`Y~UE`QW c p B E B f Q Qi Q B cC" fE W Q Y QEC c [email protected] ci E Q C WE B B pi`[email protected]`QbQXqUYXXUa`Q`cbc}`XE`YhBjUv`QVUCXDB DdcGdEVX`weQS UYdSs`@`QUcXUcDQ~XVUCXdE c c f S Y B B i B Wl c Q f Q Qi Q B C W f Q W Q BE S Q Q VI 0 6 A 6 @ 6 7 6 2 4 2 0 F eP RP R y a ) v ` R y 5CB898531!UiIq ge 5 =1 4 =2 ' ay V a %# V F X ! u @ u ' (TYP &$IWe " sAa 9 2WQc W C HCc 3de`7x`{DB Q B C c aE W Q B C WE S c c Q y W c B Q Wq Y Q c X`jxwxww txH`C)GVdCUS`Q}XDpl A`WeBWeYXVUC7xH`CAeBeW Gdx|xHeC}UEXdeQXX`W bEFdaeQdcXeaVC lUdW`Q`W ~TXzoBfXGzWTbSX|Vt# va )#gh qd 1( w~UaDQ~dEeQdW`[email protected]`Q|sVbCDYbB`YeBXT c p gq @ Yq Y Y Hl 0 C ' a p gq B c c Qq E S E [email protected] S x B [email protected] v S Qq vE Q B c W QECl W Yg x p p `WUEXVsdC8C dcskX}XDfC|}XVVog`fXCu~q 6!C d A(' E ' FEC # I dabQfdWY f S V xs w w w xx s `bB|H Y xs X xxw v r p w w Qq vEf c BE B p Qq vE Q B C E [email protected] v Qa Q Yaa X`WdcskX`XDDC~EY dcskX}X`8CDYoB`YVswVdCUSXVogDYbBGXjE dxGVC`YbBX Xa WE YE Q aE Q Q W Q c W QE QEC l C WE B B c c f S Y S WE S B Q cC XUWUvX`SdcX`eBXkGdadQGXV!|s XVbCDYDUE`QUW`[email protected]`QwQXbqdYXXUa`QecceBX``ceBXVUC)DCdcXeaV2rEy dxGVC`YG`vdcsQGq`YeWVsfsqeYdc`SUWXa WE B B [email protected] f [email protected] f Q Q `CB 2 u 2 u B eWVDC`dcsVGDAeQX`Cf~dEbQdYw`WoBXdlwdv`QeBuegGusrqeiUQ`QX`XVbCDYXdbSwEGdlXy c B [email protected] Q B W B B S W Y Y W W W gaC c BC WE B f f S Y W v ~ xs 2 2 u u B r s5 Q QH W Yg VgRTbgefXVbCt~q s VV xs q r s s tA q r 2 r t t t ~ xs r s tA qpA r 2 r EdXyxA r lbGi l t VbCDYbBeYVs`@XsW `Dc X 2 ~Ftda`QDUcXdE|`WUEUvQUfGtXy Q x xQ E WC E B aE gq B Q Qi Qq E S B Q Q aaC Q s A xG xs 2 r s s r t g t s r t g t s 3 =3 2 =4 1 =6 aE X y o I UxdWuXa QC vnx XbiY " GXFbQXeaXDeBT0xHX`XdWuXs x XbiY " dxUcX`aV{XbEXXUEdQ`SEkeQwXdCS x Bq fq B caE W C W BCE QCa x Q cC i Ya W Y c E v f a H c S Q Y Qq vE Q B WC B y x %p @B # Q B c BE [email protected] Y f Q W XE IeCdbE`WUcXAX`AeQVXDwrE!zit# 7 gQ# E h C h d # UEG GD|`Q~DeBVsbvbgbfUYdSs`@z 0 6 A 6 6 6 2 4 20 y [email protected] 97 53 ge Xa cb h VFX %UvWa fIWe ! u @ u ' "sAa 9 x p @ B a p f S Y H Y B Q Y BqC Qq v C E B Y t# 7 &# 4 d 1( dabQfdW{gffbWXDUaXEDGV|`WUcXE CDYtXqeYDceBbWXa DXXGV!dWeQ`GD`WVeCdcghgD`cbBXeqEpeuxwxwwtV}`WdcskXE p S x v ci Ya iEC C c W Q B W S H c Y c E Y u u Qq v Q B B Ya QEC c B S W Y E B B c c Q B i S WE S c c S [email protected] Q X`UWDQGGq`Y`ceBbWXTXeBeWADB XebW2VC`Y~UE`[email protected]`FXbEDYUBG`DchBXVUCVCw`QXGdadQu`CGXy dxwDVXXX~UE`Qdc' Gw`CB W v ci Ya iEC B Q Ya aEC c S WE B B c c B f Y iEC x C E B B c c f S Y E XVs`@UW`QecVdCXCDY~dEeQdW`[email protected]`Q~dEQogXdQ tVAAs VC`Y~UE`[email protected]`Q`cbcDWeBbWY i Q Q B c pCBC Yq v ci Ya iEC S Qc x Yl pC E [email protected] S QcC Vx`QUcXDdlUW`QeW GGVoBfXXt`VGAXGVdVCxblQ xGVbCDYoBDYeBdV" 2 W H Cff C W a Q B f v c C Q q E 3|xbV{d`Gk`VsjS @ ci E B B [email protected] Q B C WE B B c c C c Q B Q v f c BE [email protected] Y WC v Q X`CVjVC`YG`vdcskX`zXVbCDYD~dE`QUW`[email protected]`QjegdCX`XDB VeCcbBG`dWeQ~DeBVswvhBVwkXy 11 12 10 8 7 5 13 9 6 3 1 4 14 15 2 16 I 2 W v c i Y a i E C Q B a E W E CY BY B c @ Qf qY W W C @ Q 3w`VGwGXV XD{XfXV`b`eXb``Xy s &p 7 @ I Qi r 0 # % x v ci Ya iEC Y aE j`VXXXV `WbBX EVC`YbBDYhB!bWXDADC8XVCUfbSX|VVbCDYDUE`QUW`[email protected]`QQXbqdYXXUa`QeccfX`kVbC`YXUEbvXfX`B dWbgV~6sC [email protected] Y B BWiE Qq Y HlE B B c c f S Y Q BC E W Q Ya Q Q Yil p E [email protected] c aE Qq E S Y Hl fq B caE W fq Yi Ya f C Qq vE Q VC`YbBDYhBVCGXVC}jSX|Xx QG`aGhBQX`bWXVXbEDYeBbWXbfzdcskXXy 0 6 A 6 6 6 4 0 y [email protected] 97 2 2 ge Xa c b ya Fa c PQQ %UvIa h H WH TR !vW!bCTYe ) u @ u ' sAa 9 9 6 2 4 1 13 12 14 8 3 11 15 5 7 11 16 VFy QXbqdYXXUa`QeccX`dW`Q)`hrExv u X!s u VCDXXVbgXWkeQdcX`dEUQ`QkshW f S Y Q B W S BCq y aE v c v ci Ya QEC fEC Y Q Bl W Qq E S t @ 4 y g Xa cb e f !vWa h sAa 9 u @ u ' Standard - Tableaux for N = 3 Partition 3=2+1 3=1+1+1 3=3 [2,1] [3] [1,1,1] abbrev. notation 3 1 1 3 2 1 [ 1 , 1 , 1] = [ 1 ] [2,1] [3] 2 2 3 3 1 2 3 Young diagram GFy eBdcsqbfAkUBXXu`CGbQdfS`[email protected]{XD6E2VbCDYDUE`QUW`[email protected]`Q [email protected] CDYeBX``ceBXdCbBdYbSYXtxXy Q Y Sa [email protected] c i p Q B Y E B B c c S Q W E S WE S [email protected] Q Q x v ci Ya iEC Q v fq Y BqC fq B B Q Ya t `VGXGV XDB V`CcbQXbEtVx QG|~dEeQdc' X8C UckGuXDDCUAbW)VC`Y~UE`[email protected]}QXbqdYXXUa`QeccGDB jVbC`YXUEbvXGDDwbgUfDQXdawwvY Qq vE Q B Bf Q Y E B B c c f S Y Q C E W Q Ya Q B B B B Y Q v W H f Yl W W i Qq f Y B C Q [email protected] v S Q B C W Q W c [email protected] aE f Q B |IbCfVCbBjUvDQhBthgTXbEYUfWGD}`WUWXdE`QUBQGVUCXDeQXUS dxG`a~XhX}B eQX`DCB `Q`QUcdSXhBXkbQXDXGV RdWeQ`|X`bE`WdcskX|X`D!dQUfzWbBCDYeBX``ceBXdCS W Qa W v fq B iEC C c W Q B Y Qq vE Q B B B c S Y YlE S WE C QEE v Q B g xEC W aE !UcXXX`R V`CX~xl ! !v DVXXGV XD8dWVogVdaxgUv`QdEX|~bSX|f`Q`W ci Ya iEC Q B Q Yil Q C c Q Hl Y H c B B Y [email protected] Y Qq vE Q y DEGXzbW!I dcskXAG!px DVXXXV X`UWVogVG VeCcdaxgUv`QTUEX|H v ci Ya iEC Q B Q Yil v Q C c Q l SX|keQ`Dw DC~E dcskXkGDtXDCWeWdcskXkX`TXbE`Ydc`Q~dEC`QXGdadQu`CGkXy Y H c W B B B Y Qq vE Q B [email protected] B Y Qq vE Q B i B Q c c S [email protected] Q 7 54 y g Xa cb e f !vWa h sAa 9 u @ u ' mxXVCeYXdEwvXa RrHfWFCDY~dEeQdW`[email protected]`QQXbqdYXXUa`QeccY fE W Q Y iC B Y E B B c c f S Q B W Bl fq B caE W C B c Q B u cC" fE W Q Y QEC c WE B B c X`XDux bQXD`aGhBrHeQeQdcX`ul3vt u qgmxVCeYXdEwvGraeiGVeQ|XVbCDYDUE`QUW`[email protected]`Qc Standard - Tableaux for N = 4 [4] [31] [22] [1111] [211] 1 3 1 4 1 1 3 1 4 3 2 4 2 2 2 2 3 3 4 2 2 1 4 1 3 1 2 4 3 3 4 1 1 2 2 3 3 4 4 4 ~Fy 2 xdsdVoV}`f~ xs `V`DUGD{X{`bX{`XDeGUDVXa H Cf Q q E Q gY i Q c C W c C B c Q @ C Q B a E x Qf q B a c a E B W l W v c i Y iEC WE [email protected] Q y s i Y Q Yi f c Q B Q c B Q Q XGV dlXCDYoBDYhBGexV x RIt x EjdEVogVDWbBG`dWeQGDAUcVxgdCUS`QDCgRH r V" c C xs r p aE r p W S [email protected] W WC v C B Q B Q WE S Q H [email protected] v Q t#1jGw #&UldW eQbQXwv`Y|hBVwTRrH}GDdcXba`YXdC}XUldWbQXXx|W xE [email protected] Q WE S Q BC x fq B caE W B Y VC`YbB`YeBjdaeQdcXba`YXVUC}XD`bQX{`aXDeB~dE`QUcQ ' Xa UcXkXDj~dEVogVbWVbCDYDUE`QUW`[email protected]`QkX`z)VbC`YXUEbvXkX`[email protected] C Qq vE Q B gq Q Yi YlE B B c c Q B C E W Q Ya Q B c Qg B Qa Q Qa Y c [email protected] UaXEGV`QhB2egUcVegXRdldcUgRxHA f C Qq vE Q y B Qa Q Qa Y c Q f Y H Y Q Y BqC B W Q Q BCE Q Q C v x )exxtV " puwwwu u r " U2 " VCDYSbYXX~dEXd eQXXewXs |DW~dE wdvbQdfQ [email protected]# dcIXXeA`QUcX|H E C E B f @ @ B Q W q W a E C B Q @ Q g C W E c Q xXv x u E B Yq v S c Q f Q B i Y Yq g VbCDYbEXkVUCfdXEbYGDkXbEXabfXX~q UaXEDGVRkS bW`YXR`QUBQGVUCS x VC`[email protected]`cbcXEGXVs`@dWeQ`cVUCX`DCVXCdfsq Q Y BqC Qq E Wq [email protected] v E B B c c f S Y i YaEC c S Q B BWiE Q V~v x puwwwu u %FpxxtpV " r f " 2 DQDeBXy B W Q xs r r 2 VVX`eV}`{x dXB f E C i C B c C Q c Qf q `aXDeB~dEeQdc' Xa VeCcUaX X}X eWVDC`dcsVXDIG`dXV}VbCDYbBGXrEy caE W B Q Y v QE Qa aE c B [email protected] B B HC W E S QEC lE Yaa Vy xs u r blQ xXdlV`[email protected] x Y c B [email protected] E 7 c Y B B E C W Qq E S y x S W Q Y Sa [email protected] Q B Y c B [email protected] W aC QaE Qq |bWPXDj|xHXe}{dAB [email protected]`CX}X`{bE|`WV`CDUcV|jut`CeB`WUcXXX}sDCB ga f c WC WE B B [email protected]f Qg aEf BE cC f Q C Q B v W Q B Q Gd`QceQ`QXyfdxGVC`YG`vdcsbSDYeBUcVX~VUC$ eYVXfdcxg)dE# |WY XeX``QUcX|H 3 3 ~ xs u I r I r 2 c B @C Q B QE Qa QECl fq B S Q W Q B cC" x DC`dcQsV}XD|dWXXY X}XVGbQXD}X [email protected]}XDfVi C @ ciq c f C Q B aE f W Q y B [email protected] c E v f S Y q XVeCVXXeW`vC bf!AeB`QW}X`{G bf8CUBeQGx daDQtk`vUceQ{GVUCbE`WUBUSeQ7 VC bgX2SX|[email protected] daDQtk`vUceQ5xH`C [email protected]`WUBUSeQ7 VC fEC Y Y l WE B B Q f Q f E B [email protected] c c S Q W Y q bgfXSG|bEYRwXVC`YBtk`vUcfD~VdC$ `YVCXf`WUcXaYeXVdCXVRdQXDWX`qdQXB EC Y H l WE B [email protected] BE c f Q WE S QEC l fq B Y B cC" x fq `aaXDeBX [email protected]`@WX`vV`Ccz%)pDUuxwxwwupV " `QhBW `dcXUEVz~s x dQX`@W c E W S Q W aE W Y B u u B f Q Qi W C S s VFy V vx u su VpG " %xtpGs " %~DVt " u u suu uu %pVGs " %Itt " %~DtpV " r u su su u %~DstpV " ( r 2 ( " ' u u aE X v x u su %pGV " Itpts " ~Vdt " u u su u uu %pVGs " IGu " ~tV " r u su su u %~supV n r 2 n " ' u u `QfVV xs DCXEG`aVUC dSl ( X n `bB|H c t B i Y c S a E Y UaDQXdbS )" X n " dW`QhBX8x C `W~dEwdvQUf~f dcQ IXeDCRXVVUldWeQDskrEy B f f S ( aE B W Q y B Q Q f gC v W B W QEC W S BCq t v ~Vv x I r 2 ( Q v v BE c B Q C E B Q v v BE Q B v aE ~ dcUQ$ eYDcUBwwthgrWi DY~~ VDC`[email protected] VbCDYd$ eYDcUBwwthgW`YGDGl3v u `Cc{X 3 s V~v x r2 n Q v v W c B [email protected] UcdQ$ eYDceBwuegV VDC`dcsVC E B Q v v W Q B Yg W Y BqC QEC v B i YaEC c fq B Q B v c" x VC`Y$ Y`DceBwugeX`}VV xs uXEDGVGV!ls `CFXbEXXVs`@UW`Q`cdCS dQXTX``Cft!x cC x Y Q Yi g f fq B caE aE W v ci Ya iEC f Y BqC Q Xi8" EY dEbgVGad`Qc x d QX2`aXXhBW Xku w`VGXXV f bEGH r V" xE W Q B WC v B QiE S c B [email protected] WC B aEl c i`Y}X`eBVwVXXeWVDCDdcsV}`QXDXXw r fVC B 0 dQX`bBddXbE`WdeVDBedTiXE`eDekGDz|[email protected]`bBX)EDGVbgGVseIGv x DXEXeVdU r c YQ QS Y lWc C SQg YBcBW Q B C W f Y f v YB q C f E C Q H l CB i Y ac CSS WeVDCDdcsVC `vVC)R} Ux ( " DQDeBFbS`Y`cUBwuegbWDY~GDbEG3v VC XUl n " c B [email protected] c QH y B W Q v v BE Q B Y BqC Q H c aE `QhBbS`Y`cUBwvuegX`RdWQbgV!~Vv x `CGEXeaVdCUS{da`QUBXeDceBGVdCRVDCeBdSV|X`uly u u CDY oBDYhBXDzV " B W Q v W Q B Yi B i Y c S S WE S c Qg Q B v E [email protected] Q B cC 54 A y Xa c b ya FaR H Fa ' yV P H V F X Y ge !vWa R H WH TCQ IrQ $a !a8fIWe 1 2 ! u @ u ' sAa 9 [2] [11] 1 2 E = 1+P E = 1-P v Fy v ~x " r v ~Vu " U%VtpV " U%IdGt " %xtpGs " u r su u su u u su u u u su VpG " U%pVGs " U%VtpV " %VVt " A " 2 u u s u u s u u r u t aE X s Vx " U " r v VpVdGs " UVpGDV " UItpts " I GDt " u r u u u su u u u su v %Itt " UItGs " U~Vu " ~tV " A " 2 u su u u s u u s u u r u t XXXVG C `Qc evdcX`eBX" WaE QE l C Q c Vx " r " W QEC XGVl B r eQXUS C W Q x S W fE W Q Y iC B E W aE B Qa Q Qa Y c Q f c [email protected]`YXUEbvXrarH{[email protected]@XXE}dXEbYeQ a E W Q B B W Q " X " dDeXy xtx %pGsV " U%pVGs " U%~DstpV " %VsVt " r u u u u u u u u " r 2 " 2 C H B Q W Q B C E C Y B S Y f @ @ Q f @ Y B f v g q a Q E Y B C W Y c C B c Q @ QRr}`dGDVbDdGXbX`bX{{UXDGVq bVD`dsVC UcXD6hgdcV! xxA 5r 2 VbCDYDGk`vUcbSYdfuegwGDGA 5r VbCDYbBeYVs`@X`DcXD`QGw Q BC Qg E B B [email protected] S S Q B aE E WC WE B Q B W Q xeIXnvx `WDC`dcstVkX`UBXeDceBGVdC`CFhg{ogdfVogDYdfeQWAbBel~Vs x XA~ xs pl x XEeYA9 C c B [email protected] Q B S WE S B W Q Q B c Y Y s aE i W Vx %xGst " U%xtpGs " U%~DsVt " %VstpV " r B dXVDGl G ' `WE ( i5 t# AP' gI XDRdWdQGXteCDc~EGV !dxUWbvDYqfCX``cRW|~UEUvDQDeBW bWXDdEX`XAs FVbCDYDUE`QUW`[email protected]`QfXdbSDYhBFbEbvDYeg`DcbBG`qwV`CWY UvDQeBuhg! " Y B Q Bl C E B B c c c f [email protected] Y Q B c Y c Q v W B W W C `QhBX`B bY `QCU`QUcXjUvDQeBuhgX`!XVDCoBfwXD|[email protected]`DcDCbWX y B W Q l c Q y x W W Q B C E YE Y v Q B c f Q Q Q B Qq B W Y x )v FpxxtpV & u r puwwwu u u u 2 s XV`tk`UjDo|DGkwVUw`XonQmY C W E CY B B v c Q @ ff BY H Q B v v C S C B Q g lx x y F y C E YE Qa Q B B i Y c VbCDYbBGY XX`DCXbEX`aVUCdSS x Q x W Y c Ya Y C S Q W Q dat& XbEdv`Q#s euxwxwwxu puq UWdQSGXERTUQXdEGuU`Q`QUcX|H pl euxwxwwu pu XVbCDYDGk`vUcfUcVxgC Xe|DCzGE UxVDCeBCDY$ bY`vVXbWY `QUcX|H CWE B B [email protected] f Q v W BW Q c S E B f cCE Q v ~!x zv R %&xwxww p g Iw e & I 0 p ' r S Uuxwxwwdu pu )" u & 2 ( u W Y BqC QEC aE v XbEtV}XV{XplVnvx ( dcdQ$ `Y`cUBwuegW oi`YAGD`QX}`C|XV~l qd A(' 0 Dh7R VCdlGu8exVV xs DCXEG`aVUCdSTdaDQeBX``ceBXdCS Q v v BE Q B W B W QEC G C c W y B i Y c S S WE `V`DUVGDFXbEubgXX2~q dGEDGVs)DVeVXhdUQ`edsVeXdhDewueW dec`UXa W c C B c Q @ C Q B i Y f @ @ g a Q Y B q C Q q C B Q g Q W C y x W Y B c Q @ C c @ g c B Q v v g a Q Y W Q Q B Q BC C f W WE SE Q Q Q B W S B Y Q HC WE S B Y Q X`6g{XE Xa bgbf`GXVC`YUBXGGdEVbidYhX`heQ{DB bEUldcQ Ug RI0UxXVC`YUB`dc`QbE# RH bgXXV`Cd`QVCeQXD|WFSX|XCDYTbvGxeCXXFtuVVbg`YUf`Qwff`YeB`Q~VUx XVbCDYeBXX fqE W c c W S Q B Y Y HlE B Y [email protected]@ aCCi Q B c W c s WE SE Q B aE u u X`jXpl)ppuxwxwwupV `vdc`QCDYeBDUcDQ~E~VUx `CXGGV!oaYbff`YeB|WbBG`dWeQAbWXy B E S B Y E B Waa QEC Y fg W Y f c Y xe~s x EkUEVbgP [email protected]`CXXDVC bW`YW dWbgVX1VC`YUBXGGdEVbidYUEXDWY Y Q Yi S W Q Y Sa [email protected] Q Bc q Q Yi aE C E SE Q Q Q B s VV!x & xwxww g Iw e & r 9 duxwxwwUu u g " & 2 CDYeBXXeBXXu`CXXy E SE Sa [email protected] Q ~!x 7 I7 # r SY7 u 2 & W 7 C WE SE Q Q [email protected] fi W Q B c H E S QEC Q fg Q Q f c WaE W Qa Y Q B GVC`YUBXXtdEVbidYQdfbSDYhBQVXbE`YkGD}DQbBeY|)XVdldWG~dEVbidYfVCGXhBd VxGXEkGDz y xG!x UxxdtpV9 e" 7 r 2 7 puwwwu u r u 7 u QE Qa aE aE WCC S Q x [email protected] B9 Q BC q cg Q B EC S fEC Q X X{X"7 X7 `QtX} bQdfbSDYeB{[email protected] `YXDV|`WUBbgXVA7 e`QUcX|H VV!x u 7 7 r [email protected] XD7 GXDCbBbfbQdfS`YeBXQVGE`Y}X`8bgUfVogDYbBGX c p Q BC WE YE Y v [email protected] fi W Q B C Q Yaa da`QUBXeDceBGVdC|sdPdvDQhBueg}Xe$XV`CbBbfAGy exT`vUcDQVC`YUBUS`Q`cdC`CzXbQkogXVC S WE S Qq E S W W S WC E YE Y v Q B E c S B Wa f fE WE Q Q B Q Qq E f c q v fC d Q B Q Hl v B E WE Q Q B c x xQl X`[email protected]`QkVCGVX``QUcX|IjV`CEY XV`C`cUBUSQdfRXDVColXnidaUQ$ Y oid`QfdfR}DQbBGuWVbCDYGwX`eWAWX`B dlwdv`QeBueg{~FrEy dxdWbQdfbSDYeBX`)dEUQrHeBXVbCDYUBDdc`Q~EY f c Q H Y Y E [email protected] v W Y W W W gE v [email protected] Q B Q Qq WE S B eBdSQdiXhdR`hCDYtXe`WwX`DbB|hBhBRdlbW XDq" Ua`Q`WGd`SbWX{`CwxgX G 7 ' FCE f QE E S Q H B B E [email protected] v W Q B Y H c W Q H Y B cC x W Ya Qq B Q TG E d7C ' 7 5 d 7 G P @ P G T 5 B E B c C @ vY B W C v Q B l E CY B W Q WY 7 0 0 U`WD7 qh WE R ( G 7 P `7 p Ugd ( Q# X' iI ewhVwwGDVbDhdXGDB dcReHG6 `QX`E{daUQ$ Y`Q`QAUWQeYDcUBwwthg}QXbq`Y`WXDB FbSX|7xHX|WjVC`YeBdWGuXy Q WE Cy c BE Y f c c Q v v W f [email protected] Q C Y H CE Y E Q Q exUWSeYt~z Wg bQdfbSDYeB AeQXDeBX``cB VC`YUBdSbQdfQ hiUcX X xVDI~xG~G dQGdE`QUcQGVd dQXUEVxg`Cz udErQxY c C c S E [email protected] Q y i [email protected]@ls S EC S Q c i i lXVAz [email protected]~EVgX`CVsrq`@ XbEX`@kUcVdi`Q~ErYioVgXVCw`vUcQ `XDB i x Y W Q B Y Q WE WC i c aE Y W Q B f Q WE Y i [email protected] B bgUfX!daxg`CXaXVUCS Uv`QVUCXYX GDf6VbCDYd$ YDdc GdEVi V DfdVeCVsrq`@ Q vE l Q [email protected] Qq f c Q y f Q B C E B f Q Q x #E c C E WC i c E WCq c Y c E aE l E c C E Y i [email protected] Y E Y c Y c E l W Yg `CVs2 egEXeaVC Xt# DCw`vdcQmDWVCw`vUcQ egEXeaVC fVbgefXVbCt~q y F UYQdfGuE GrgoBXaXX UcXe W S Y Y [email protected] dv`QeBuegbSXuC`[email protected] W W Ya Q DQ`XdEXGVdCS B W QaE UWdQXUEXdeQXVdCS S Q WE [email protected]`[email protected] oibYX EdY`QeBXbEz eieQV f S c f W Y WC dcX`eBX Q c f Q eBv f Qv eBEY Q e AxHw WE XV`C`cUBUSQdfQ W ! # Xz XV`[email protected] SE f WE BC c f CaYg `UXdQ gicQ E DoB|fdU`[email protected] ! 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[email protected] a 9 PyRv d PR Bu u ' !wTR t' ' Q # a 1URP p %Uv!PW URWX Py v d PR F Fa ` H c P c ! 2 W E CY B EY q v C S i EY H Cff C Q B W EY B q C Q E C l W E CY v c Q C HB v Q B W g W 3XDXVdX|xbVXDfGDGV}XVXUXVbedrC d`eue cC" q ex& %!$ & $ %!$ & # `UWevdcw bEdEVogVi # & l Q Q v x Y Q Y bWbQX`RWX`u`[email protected] SeYDcUBwwthgrWi DY~fbWVbCDYUBGXVgsDDCAXDRDkeC! ThgDceBwuegW Y f c Y BCC c x Q v v BE Y E SE Q Hf B B Q B B B W C Q v v E B B [email protected] Q B W QqC B B [email protected] cC Y W gq f f v Qq W v Y Q Bl v ci Ya Q CDYGkevdcskX`uUgsVXDhBVXbEX`@TF~{[email protected]`YbBXseBXoB)dEX`T DVXkGDB C Q v v E B B [email protected] Q B W QqC E SE Q H C [email protected] f B W Q B l [email protected] v Q cC eg`cUBwuhgW VbCDYDGk evdcsjX`TuUgsVCDYeBXXFgfh}eB|DYX`@FGDRQXwGVxwV" % % 0 Triplett States F!exw W gX vY rp R UvcYWUFQ WTr V I !FW Q PR F a V P v `R Qv P Rya P Q !wTR t' ' d Py v Bu u ' [email protected] a 9 ' CIa R y a c aP R Singlett States ...
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