hw-.nb: //::9:5: omwok: Du -- Ths st (#7) s du on Wdnsday, //. Th soluton fom Poblm fom th xam s found n th mdtm solutons. ü Sakua Chap : 7,,,, 5. Mbach.. BJ 6. ü Mbach. Th bass stats of angula momntum a typcally labld by S jm, wh J S jm = j j + S jm and J S jm = m S jm. Th a j + m-stats fo ach valu of j, wth - j < m < j. As J, J =, th valu of j s unchangd by opaton by J. Accodngly, th full angula momntum vcto spac can b bokn nto subspacs labld by j. Each subspac s spannd by a st of j + bass stats S jm. Wthn ach subspac th acton of J x, J y, o J can b numatd as a matx wth th convnton that th ow and column ndcs of th matx cospond to stats of dcasng m. Ths matcs a th gnatos fo a j + dmnsonal psntaton of SO. Fo J and J th matcs a spcfd by j' m' J jm = md j' j d m' m j' m' J jm = j j + d j' j d m' m Fo J x, J y t s usful to us th asng and lowng opatos J = J x J y whch lads to J x J + + J - and J y = - J + - J -. Usng th latons j' m' J + jm = c + jm d j' j d m' m+ = j - m j + m + d j' j d m' m+ j' m' J - jm = c - jm d j' j d m' m- = j + m j - m + d j' j d m' m- Puttng t togth fo
hw-.nb: //::9:5: J J x J y J - - - - - - - - 5 ÅÅ 5 Å 5 Å 5 ÅÅ - - - - - ü BJ 6. A otaton by f aound th axs ǹ s gvn by Df, ǹ = -fǹÿj = -fj ǹ, wh Jǹ = ǹ ÿ J and J cosponds to th angula momntum gnatos n th psntaton of ntst. In patcula, ths s tu fo th cas of obtal angula momntum, wh J = L. Th gnato cospondng to an nfntsmal otaton aound th ǹ-axs s thfo L ǹ = ǹ ÿ L = snq cosf L x + snq snf L y + cosq L Usng L = L x L y, o L x L + + L - and L y = - L + - L - gvs L ǹ snq cosf L + + L - - snq L + - L - + cosq L snq -f L + + f L - + cosq L a) Th s nothng spcal about th x, y, codnat systm. Takng ǹ to play th ol of `, wll lad to th sam st of quantaton uls as fo L, L. On could thfo choos th bass sts of th systm to b Slm ǹ, wh -l < m ǹ < l and m ǹ s th gnvalu of L ǹ Slm ǹ = m ǹ Slm ǹ. If on masus L ǹ, thn th masumnt pocss wll tun m ǹ and lav th systm n th stat Slm ǹ an gnstat of L ǹ, wth pobablty P m m ǹ = Slm lm ǹ W. Typcally all P m m ǹ a non-o, but und ctan spcal cass, fo xampl ǹ = `, th st of possbl valus of m ǹ wll b stctd. b) On mthod s to substtut th laton L ǹ = snq -f L + + f L - + cosq L nto th xpctaton valu lm L n lm = lmw snq -f L + + f L - + cosq L Slm. snq -f lmw c+ lm Sl, m + + f lmw c- lm Sl, m - + cosq lmw m Slm = mcosq
hw-.nb: //::9:5: Fo L n th s a bt mo wok lm L n L n lm = lmw L n snq -f L + + f L - + cosq L Slm snq -f c+ lm lmw L n Sl, m + + f c- lm lmw L n Sl, m - + mcosq lmw L n Slm snq -f c+ lm lmw snqf L - Sl, m + + f c- lm lmw snqf L + Sl, m - + mcosq lmw cosq L Slm = sn q c+ lm lmw c- lm+ Slm + c- lm lmw c+ lm- Slm + m cos q sn q c+ lm c- lm+ + c- lm c+ lm- + m cos q sn q l - m l + m + l + m + l - m + + + l + m l - m + l - m - l + m - + + m cos q sn q l - m l + m + + l + m l - m + + m cos q sn q l - m + l + m cos q. ü Sakua.7 Although t s unstatd, th matx U s a otaton matx actng on som spac of stats. Fo xampl Ua, b, g mght cospond to a D j matx fo ul angls a, b, g actng on th stats of th j-psntaton. Th matx R s th cospondng otaton matx actng on a vcto. If th componnts of A k oby th laton U A k U = R kl A l thn A s known as a "vcto" opato. On xampl of a vcto opato s th poston X. Consd th acton of X on th poston-kts x X x = x dx - x Thn x U X U x = x ÿ R T X R ÿ x = x' X x' = x' dx' - x' = R ÿ x dx - x Rtun to th mo gnal cas of abtay stats Sm, whch bhav as Sm Ø U Sm und a otaton. Thn, und a otaton th matx lmnts of A chang as mw A k Sn Ø mw U A k U Sn = mw R kl A l Sn = R kl mw A l Sn.. th ndvdual matx lmnts of A tansfom lk catsan vctos. ü Sakua. Th smplst way to do ths poblm s to compa
hw-.nb: //::9:5: Da, b, g = -a+gi cos b a+gi sn b - -a+gi sn b a+gi cos b wth Rf, ǹ = cos f - sn f - n sn f - sn f n - n n + n cos f + n sn f Isolatng th al pat of th componnt, f = Accos cos ÅÅ a+g cos b ü Sakua. Th ntal stat s S j, j, th hghst wght stat fo th angula momntum psntaton j. Ths stat s otatd by aound th y-axs. Th otaton matx s R, ỳ = - J y - J y - Å = - J +-J - - Å J y... Å Å J +-J - = - ÅÅÅ J + - J - + J + + J - - J + J - - J - J + wh th xpanson has bn don to od and J y has bn xplctly wttn n tms of asng and lowng opatos. It s now staghtfowad to valuat R S j, j = - ÅÅÅ J + - J - + J + + J - - J + J - - J - J + S j, j = + ÅÅÅ J - + J - - J + J - S j, j = S j, j + c - j j S j, j - + Å c - j j- c j j + c j j - S j, j - - c j j- - S j, j = - Å c + j j- c - j j S j, j + c - j j S j, j - + Å c - j j- c- j j S j, j - = - Å j j S j, j + j S j, j - + Å j - j S j, j - = - j S j, j + j S j, j - + Å j j - S j, j - wh at th nd us was mad of c jm ognal stat s P m= j = - j - j As a chck, not that P m= j- = j, so pobablty s consvd at od. = jsm j m +. Th pobablty fo th otatd stat to stll b n th ü Sakua. ü a) show J + J - = J - J + J Bgn wth J = J x + J y + J, and us J x J + + J -, J y = - J + - J -.
hw-.nb: //::9:5: 5 J J + + J - - J + - J - + J J + + J - + J + J - + J - J + - J + + J - - J + J - - J - J + + J J + J - + J - J + + J J + J - + J -, J + + J J + J - - J + J = J + J - - J + J o, oganng J + J - = J - J + J ü b) fnd th coffcnt c - n J - S j, m = c - S j, m -. Stat by consdng J - S j, m = j, mw J +, but also J - S j, m = c - S j, m - = j, m - W c - *. It follows that j, mw J + J - S j, m = j, m - W c - * c - S j, m - = Sc - W At th sam tm, usng th sult fom pat a, j, mw J + J - S j, m = j, mw J - J + J S j, m = j j + - m + m = j - m + j + m = j + m j - m + and so, wth a convntonal choc of phas, c - = j + m j - m + ü Sakua.5 W a gvn that V s sphcally symmtc and yx = x + y + f. a) Is y a an gnfuncton of L? Ys. Th componnts of th wav functon a all popotonal to l = sphcal hamoncs. Thus, = ÅÅÅ p Y x = ÅÅÅ p Y - - Y y = ÅÅÅ p Y + Y - Ths a all gnstats of L wth th sam gnvalu ll +. It follows that a lna combnaton of ths stats wll also b an gnstat of L wth th sam gnvalu.
hw-.nb: //::9:5: 6 Altnatvly, labl th stat by Sa, so that yx = x a. Spaat Sa nto adal and angula pcs, Sa = Sa Sa W, wth a = f and ǹ a W = x` + ỳ + `, wth x` x, tc. Nxt, xpss th angula pc as a sum ov angula momntum stats Sa W = S c lm Slm. Thn ǹ a W = c lm ǹ lm and cogn that x` = - ǹ, - ǹ, -, lm ỳ ǹ, + ǹ, -, and ` = ǹ,. In th psnt cas, th only non-o c lm a fo l =. Th opaton of L bcoms L è yx = x L a = S c m x L m = S c m ll + l= x m m m = S c m x m = x a = yx m b) Th non-o xpanson coffcnts n th angula momntum sum a c lm, wth c =, c, S. Suppssng th l = quantum numb, th pobablty to fnd a patcula m - stat s gvn by P m c m, wh S m c m = + 9 + =. So th pobabts fo th th m-stats a P Å, P ÅÅ 9, P - Å c) If y s an ngy gnfuncton, th gnvalu quaton s - Å m + V y = Ey Pfomng a spaaton of vaabls th Laplacan opato can b wttn as Å and so - m - Å L + V y = Ey Snc L y = ll + y = y, ths can b aangd as V y = E - m y + m y Dvdng though by y, V = E - y m + m y Å Å - S m c m Å L ª - Å L,