XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos. Fo a gve pemutato ths umbe s always eve o always odd the same pemutato ca be acheved by dffeet sequeces of taspostos. Theefoe we ca speak about eve ad odd pemutatos. The sg of a pemutato P: sg P P eve P odd
Ieducble epesetatos of the symmetc goup IRs of coespod to cougacy classes of ad ae labelled by a patto of the tege postve umbe. { } { } We ode these tege umbe as. Ths patto ca be epeseted gaphcally by so-called Youg dagams. A Youg dagam cotas λ boxes ts th ow. Few examples: {} {} {33} 0.
Cougate dagams: ows colums o equvaletly eflecto wth espect to the dagoal. {λ}={33} ~ {λ}={53} Cougato s deoted by ~ Youg tableau: a Youg dagam boxes flled wth tege umbes.... tadad Youg tableau: umbes each ow ad each colum ae placed the ceasg ode. Example: all stadad Youg tableaux fo {λ}={3}. 3 4 5 3 4 3 5 4 5 3 5 3 4 5 4
The dmeso d {λ} of a IR of chaactezed by the Youg dagam {λ} s equal to the umbe of coespodg stadad Youg tableaux. The dmesos of IRs chaactezed by cougate Youg dagams ae the same Note that { } ~ { } Hook legth h of the th box of a Youg dagam: d h = umbe of boxes to the ght the ow + + umbe of boxes tbelow the colum + Hook legth fo some {λ} d d! { } { } 3 3 3 4 3 5 4 3 A ease way to calculate d {λ} s gve by a theoem: d { }! h
XII.. ystems of detcal patcles. Cosde detcal patcles wth the sp s. multaeous pemutato of both co-odates ad sp vaables σ of each pa of patcles multples the -patcle wave fucto by + f s s tege bosos B ad by f s s half-tege femos F. A wave fucto satsfyg ths symmety equemet ca be costucted may ways coespodg to vaous Youg dagams Hee sum s take ove all d {λ} stadad Youg tableaux whee we use sp vaables σ stead of umbes =... to fll the boxes whe we costuct the sp pat X. The co-odate pat s obtaed by eplacg the sp vaables wth the co-odates ad the case of femos oly by cougatg the Youg tableau. We focus o the costucto of the sp pat X of the wave fucto the coodate pat beg bult a smla way. ; ; } { { ~ } ~ } { } { F B
. Choose a ceta stadad Youg tableau lke that: σ σ 3 σ 6 σ σ 4 σ 5 σ 7. Costuct a poduct of sgle sp wave fuctos whee m s = s s +... s s s the poecto of the sp of the th patcle to the quatzato axs. 3. Apply to ths poduct a Youg symmetze.... Fo a gve Youg tableau we select fom all! pemutatos oly those opeatos cludg the detty opeato! whch do ot pemute sp vaables belogg to dffeet ows. We deote these pemutatos by P. How may ae thee P s? Also we select pemutatos cludg detty! whch do ot pemute sp vaables belogg to dffeet colums deotg them by P c. How may ae thee P c s? The summg by all possble P ad P c we defe the Youg symmetze: Y { } sgpc P Pc P P c s m s
p wave fucto X... Y{ }... { } The maxmum umbe of ow {λ} s s + sce atsymmetzato ove vaables colums cotag moe tha s + wll eque atsymmetzato ove agumets σ ad σ of two fuctos < σ s m s > ad < σ s m s > wth the same m s. The esult of the atsymmetzato wll the be detcally zeo. I patcula fo sp- patcles such as electos the sp wave fucto ca be chaactezed by Youg dagams wth ow fully symmetc o ows oly. It s possble to calculate whch values of the total sp bult fom detcal sps s coespod to a gve Youg dagam. If the total sp appeas W {λ} tmes fo the {λ}-type wave fucto tha obvously W { } s { }
Fo s = ½ ulke othe o-zeto values of patcle sp thee s a oe-to-oe coespodece betwee {λ} ad. The ule: If we add to a state of sp-½ patcles wth the total sp ad {λ} = {λ λ } λ + λ = a ew patcle of the same kd the we obta the states: {λ + λ } + ½ {λ λ +} ½. The vaat s possble fo λ > λ oly. Recall that W shows how may tmes the total sp appeas fo a gve {λ}.
Addto of sps s =. = W= =3 W= = W= = W=+ =3...ad so o. = W= = W=+ = =0 W= =0 W= Youg dagams ot appeag fo s = Fo s > fo a gve {λ} W may be o-zeo fo dffeet
Explct costucto of wave fuctos of may sps s = ½ Fo each patcle two states: sp up ad sp-dow. p-loweg opeato fo th patcle: Collectve sp-loweg opeato: The hghest possble = coespods to the fully symmetzed state Youg dagam {} cossts of a sgle ow. The hghest possble collectve sp poecto M = + = +: all atoms the sp-up state. Applyg the collectve sp-loweg opeato we decease M by : fo 0 0 0 M } { M M! } {! } {
But thee ae lealy depedet states wth M = : oe sp dow all othe sps up:. Fom them we ca costuct lealy depedet fuctos othogoal to These fuctos ae to be detfed as dffeet ealzatos of = fo the Youg dagam { }. We apply σ oce aga ad obta fuctos fo the states wth = ad = wth the total sp poecto M =. But thee ae l.depedet states wth sps dow ad sps up. By othogoalzg them to the aleady defed states we obta = 3 states fo = ad {λ} = { }. {λ} = { m m} whee m s tege ad m < coespods to = m. Calculate W {λ} fo {λ} = { m m} = m. { } M