PHY 309: QUANTUM MECHANICS I (3 UNITS) COURSE GUIDE

Transcription

1 PHY 9: QUANTUM MECHANICS I UNITS COURSE GUIDE I Qutu Mchics I PHY 9, you t bout th iqucis of Cssic Mchics th ffots by Physicists to ss th shotcoigs o th ptfo of Qutu Mchics. You hv t oo t th thtic foutio cssy to ust tht cous. Oc tht ws o, you t th ppictio of qutu chics to th ifiit s w s th fiit potti w. Th qutu ttt of th hoic oscito foow, cuitig i Hit s poyois th opto ttt of th hoic oscito. PHY 49 is th sco i th sis of qutu chic couss. I th pst cous, you wi stt with futh thtics, yig foutio of tic spcs vtuy, Hibt spcs o which th i pouct is fi. You wi gt to ow tht vy Hibt spc hs wht w c copt othoo bsis, big us to h syst which c b i isct st of stts. Lt i th cous, you wi bout th hyog-i to, th sphicy sytic potti, ig to sphic hoics vtuy, spcific i fuctios, coptig th ttt of th sipst to, tht of hyog. I th st ou, futh s of qutu chics iscuss: ptubtio scttig. I ptubtio thoy, you wi to h s ptubtios by oifyig th o fii t pob. Put oth wy, you wi b b to uc th gy vs ig fuctios of th ptub syst i ts of th igstts ig fuctios of th o fii potti. You y fii with scttig i cssic chics. Is th y iffc i th cs of qutu scttig, i which cs, you hv pst ptic by wv? You c s tht you hv ot of itstig topics witig you i this cous. Qutu chics is phps th ost itstig pt of Physics, fis Ipott ppictios of qutu thoy icu qutu chisty, qutu optics, qutu coputig, supcouctig gts, ight-ittig ios, th s, th tsisto sicouctos such s th icopocsso, ic sch igig such s gtic soc igig cto icoscopy. Eptios fo y bioogic physic pho oot i th tu of th chic bo, ost otby th co-ocu DNA. COURSE AIMS Th i of th cous is to big th stuts to th owg of fut spcts ivtios i th ifft s of qutu chics spciy i thtic foutio, tic spc, out pouct, th poctio opto th coptss tio, gu otu spi of toic uc ptics, th-isio sphicy sytic pottis th po g qutio, g poyois.

2 COURSE OBJECTIVES *To itouc th stut to th cocpt of Covgc of Squc, Cuchy Squc Bch Spc, Hibt Spc Li Mp Li Fuctio. *To fi giv ps of i p, i fuctio, pi fi th u vcto s w s to ccut th tsitio pobbiity btw o igstt oth *Ls wou b b to fi th ti ts of i opto, giv th vt bsis vctos. *To how to wo with th spios, fi th Pui spi tics ccut thi igvctos igvus. I itio *Fi th soutios to th g Lg qutio, th ssocit Lg fuctios *To vut itgs t to th hyog to pcttio vu of so physic obsvbs i giv igstt of th hyog to *To show tht i wvfuctio fo th-isio ptic f to ov isi sph stisfis th Bss qutio *To stt th ssuptios of th Bo ppoitio fi th iffti coss sctio fo giv scttig potti i th Bo ppoitio WORKING THROUGH THIS COURSE Th cous is stuctu ito fou os. A th Mous cosist of th uit ch cpt.it is cssy tht fo th stut to stuy ust th cott of th uits i th spctiv ous. COURSE MATERIALS You wi b povi with th foowig tis:. Cous Gui. Stuy Uits Mou : Th Mthtic Foutio Uit : Mtic Spc, Hibt Spc Uit : Li Mp, Li Fuctio, Du Vcto Spc Uit : Coutto Agb, Mti Ets Of A Li Opto Uit 4: Th Out Pouct, Th Poctio Opto A Th Coptss Rtio Mou : Agu Motu A Spi of Atoic A Nuc Ptics

3 Uit : Uit : Agu otu Uit : Eigfuctios Of Agu Motu I Uit : Eigfuctios Of Agu Motu II Uit 4: Ecto Spi I Uit 5: Ecto Spi II Mou Uit : Th-Disio Sphicy Sytic Pottis Uit : Th Po Ag Equtio, Lg Poyois Uit : Associt Lg Fuctios, Agu Eigfuctios Uit 4: Th Ri Equtio Uit 5: Th Hyog Ato Mou 4: Ptubtio Thoy Qutu Scttig Uit : Ptubtio Thoy Uit : Sco O Ptubtio Uit : Qutu Scttig I Uit 4: Scttig Ii - Th Bo Appoitio Uit 5: Scttig Iii - Pti Wv Aysis

4 MAIN COURSE MODULE : THE MATHEMATICAL FOUNDATION UNIT : METRIC SPACE, HILBERT SPACE CONTENTS. Itouctio. Obctivs. Mi Cott. Mtic Spc.. Dfiitio.. Covgc of Squc, Cuchy Squc. Bch Spc, Hibt Spc. Li Mp, Li Fuctio 4. Cocusio 5. Suy 6. Tuto-M Assigt TMA 7. Rfcs/Futh Rig. INTRODUCTION A vcto spc is o i which w c cy out th itio of two vctos th sc utipictio of vcto without woyig bout whth th outco of ou optio fs outsi th owb possibiitis, which i this cs is th vcto spc itsf. I th gug of thtics, this wou tht vcto spc is cos u th two optios of vcto itio sc utipictio. W hv s i PHY 9 tht th i of vcto hs b gis to icu y thtic stuctu o which w c fi ths two optios hv show tht th sutig spc is cos u th two optios. W so gv so ps, icuig th spc of th usu Euci vctos, th spc of tics th spc of squ itgb fuctios of. W shou b b to su fo of istc o vcto spc. Fo this, w fi o, fuctio tht ssigs positiv gth to ch vcto, pt fo th zo vcto, which hs gth zo, th istc of poit fo itsf. Th o c b visuis s th istc of vcto fo th oigi. I this Uit, you wi gt to ow o bout vcto spcs th itio quit vcto spc shou possss to b usfu i th fi of Qutu Mchics. Hvig fi th istc of vcto fo th oigi, w cou go o to fi th istc btw th poits of two vctos. Fo this, th t tic is fi. Oc th tic is fi o vcto spc, it bcos tic spc, w i positio to stuy th covgc of squcs i such spc. This tuy s to th cocpt of coptss of tic spc. A copt o vcto spc is Bch spc. If th o is fi by th i pouct, th copt o spc is Hibt spc. Qutu Mchics is o i Hibt spc, o put oth wy, th Hibt spc is th spc i which vctos i Qutu Mchics iv.. OBJECTIVES At th of this stuy sssio you wi b b to: 4

5 fi tic spc ist pi th poptis of tic spc pov whth o ot giv spc is tic spc giv ps of tic spc fi pi covgc of squc i tic spc pi wht is t by Cuchy squc pi wht is t by copt tic spc copt giv tic spc fi Bch spc Hibt spc iscuss th tioship btw Bch spc Hibt spc pi why Qutu Mchics is o i Hibt spc. MAIN CONTENT. Mtic Spc.. Dfiitio A tic spc is o-pty st V such tht fo vy y i V, th is o-gtiv ub, c th tic, i of istc, which stisfis cti st of ios. Ntuy, wh such cocpt of istc ivovs th poits, it shou stisfy th w of tig iquity, which stiputs tht o sig si of tig shou b og th th su of th iig two. Th iu possib vu of th gth of o si is qu to th su of th iig two. This is th cs of gt tig i which th vtics coi og th s i. W wou so pct th istc fo poit A to poit B to b th s s th istc fo poit B to poit A syty. Fiy, sic tic is i of istc, it shou b such tht th tic of poit fo itsf b zo, zo tic ipy th istc fo th s poit to itsf fiitss. Ths outi bow i th gug of Mthtics. Mthticy, tic is fuctio : V V R, stisfyig fo f, g, h V : i f, g f, h h, g Tig iquity. ii f, g g, f Syty. iii f, g if oy if f g Dfiitss. A i pouct iucs o, o iucs tic., b, b Rc tht w si tht th o is th istc of vcto fo th oigi. You cou s tic s th istc btw th s of two vctos, o quivty, th istc btw th two vctos. W fi tic spc s vcto spc V quipp with tic, witt s V,. W t so ps: i O of th sipst ps of tic spc is th ub i. A p of tic o th vcto spc is th bsout vu tic,, y y.4 5

6 ii Giv th vcto spc V, w c fi th tic, if y, y, othwis Thus, o pi of poits cos. Evy pi of poits is pt. This is thus c isct tic. You c siy s tht if w pc i th fiitio by, th istc btw two istict pi of poits is. iii I th cs of th two-isio Euci vcto spc V, if th positio vcto of is i y tht of b is ib by, w c fi th tic, b b by y.5 This is c th usu tic o th two-isio Euci spc. It is th istc btw th two poits b o th -y p. It is th usu tic bcus tht is th cocpt of istc o p you hv wys b fii with. iv Cosi th -y p s vcto spc, togth with th ticb tic,, y,, y y y.6 As you c s, with th ticb tic, th istc btw two poits is th su of th bsout iffcs of thi Ctsi cooits, ui th usu tic tht is th squ oot of th su of th squs of th iffc btw thi Ctsi cooits. iv Th tic iuc by th supu t tis shot sup o o,, y,, y y y.7 Th sup o is, wh, fo. v Fo th spc of cotiuous -vu fuctios of ov th itv, b, b f, g f g.8 Ep. Giv th tic spc X,, show tht, z z, y, y. Soutio., z, y y, z tig iquity.9 fo which,, y, z y, z. Siiy, fo tig iquity, y, z y,, z. 6

7 o y, z, z, y. Mutipyig qutio. by vss th iquity: y, z, z, y Cobiig qutios..,, y, z y, z, y. Hc,, z z, y, y.4 Sf-Asssst Ecis List pi th poptis of tic spc... Covgc of Squc, Cuchy Squc Lt }, wh N, b squc i tic spc V, {. W sy tht covgs to, th ists N such tht tht is, covgs to V o, if fo, fo N. Mo oosy, w sy covgs to if i,. I o i spc vcto o i spc o which o is fi, covgs to ipis i. A squc is si to b Cuchy squc wh, wh,. O o pcisy, fo Cuchy squc, fo, th ists N such tht, fo, N. I o pi ts, Cuchy squc is o whos ts bco bitiy cos s th squc pogsss. A covgt squc is Cuchy, bcus, owig to th tig iquity,,,, / /.5 sic fo giv, th ists N tu ub N, such tht, /, /, bcus th squc is covgt. Rc th tig iquity which stiputs tht o sig si of tig cou b og th th su of th iig two sis. Th ogst it c b is th su of th oth two sis. Thus, covgt squcs Cuchy, but ot Cuchy squcs covg. As p, i th tic spc, with th bsout vu tic, y y, th squc / os ot covg i,. It os covg i [, ] howv. Not tht / covgs to, which is ot icu i,. Sf Asssst Ecis 7

8 Why is th fiitio of Cuchy covgc ivovig s ub cosi o pcis th tht fo which th tic,?. Bch Spc, Hibt Spc W sy tic spc V, is copt if vy Cuchy squc covgs to poit i th tic spc. Not Cuchy squcs covg. Fo istc, th st of tio ubs is ot copt, bcus, fo p, is issig fo it. Yt, w c costuct Cuchy squc tht covgs to. Th squc,.4,.4,.44,.44,.44,... which c b show to b Cuchy, covgs to, which is itio ub. Hc, th st of tio ubs is ot copt. Howv, w c fi th hos. Tht is, w c g copt tic spc to coti th iit of y of its Cuchy squcs. If this tic is o o i spc, this givs us Bch spc. If w g i pouct spc to icu th iit of y of its Cuchy squcs, w gt Hibt spc. Put oth wy, th Cuchy coptio of o i spc is Bch spc, th Cuchy coptio of i pouct spc is c Hibt spc. A o i spc is spc o which o is fi. A i pouct spc is o i spc i which th o is iuc by th i pouct. I th p i.., o th ub i,, is op st ttivy op itv. As such, ot ts of th st. [, is hf-op, cotiig, but ot., ] is so hf-op, cotiig, but ot zo. [, ] is cos st, is th cosu of th op st, o th hf op sts [,, ]. Th cosu of st is witt by witig th sybo fo th st puttig b o top. So, if T,, T [,]. This is chiv by ig th iits of th st, i this cs, th sigtos st with oy o t {} {}. Thus, [, ] =, {} {}. You c ow s tht th squc i qustio covgs withi th cos itv. Thus, w sy tht th cos itv [, ] is copt with th bsout vu tic. Fo y tic spc M, w c quy costuct copt tic spc M, which cotis M s s subspc. Tht is, fo M, y ighbouhoo of cotis t st o poit fo M. Wht os this? Lt us t M M, th is othig to pov, sic y ighbouhoo of cotis t st o poit i M. W shou thfo both bout poit o th bouy M. I this cs, w ot tht th sst spc tht cotis M s pop subst is M. Thus, fo M, y ighbouhoo of cotis t st o poit i M. A subspc ihits th poptis of th spc of which it is subspc. Fig..shows op b Fig.. its cosu Fig..b. Poit o th bouy of th cos b hs ighbous both withi outsi th op b. 8 Nighbouhoo of poit o th bouy of M

9 b Fig..: Showig tht th op b is s i th cos b b Cosquty, Bch spc is copt o spc, tht is, o spc tht is copt u th tic iuc by th o. A Hibt spc is Bch spc i which th o is iuc by th i pouct. O Hibt spc is Cuchy copt i pouct spc. A Hibt spc is Bch spc, but ot vy Bch spc is Hibt spc. Hibt Bch. Sf Asssst Ecis Stt which of th foowig is op o cos st. If op st, stt th st to it cos st. i -, ii [-, ] iii [, 9 iv, 7] Why Hibt Spcs. Fo o, Hibt spc is copt, ig tht w hv o pob of covgc. Coptss i this cs s tht if squc of vctos is Cuchy, th it covgs to iit i th spc. I ffct, igvctos o igfuctios i Qutu chics iv isi Hibt spc. Moov, Zo s vi th Hh-Bch tho i Fuctio Aysis sus tht vy o-pty Hibt spc hs othoo bsis. Thus, w c p y wvfuctio s i cobitio of ts of th ppopit othoo bsis. I i pouct spc, th i pouct fis th o of vcto 9 i th vcto spc s, so bs us to wit, A o A, th pcttio vu of th physic obsvb pst by th opto A i stt. Th o is pcuso to th oistio of vcto o wvfuctio, which i tu s to fiitio of pobbiitis i Qutu chics. Not: A t of Hibt spc c b uiquy spcifi by its cooits with spct to othoo bsis, i ogy with Ctsi cooits. Wh th bsis is coutby ifiit, this s tht th Hibt spc c so b thought of i ts of ifiit squcs tht squ sub. You wou c fo PHS 7 PHS tht if is ois, { i} i is othoo st, th,

10 i i c, i i c c * i i c, i c * c c.6 i i i i Eps of fiit-isio Hibt spcs icu th ubs with th ot pouct T u, v u v th cop ubs with th vcto ot pouct u, v u v. Eps of ifiit-isio Hibt spc, th st of squ itgb fuctios with th i pouct fi s f, g f g th st of squ itgb cop vu fuctios with th i pouct fi s f, g f * g. Squ itgb i ths css is t i th ss, spctivy, f, f [ f ] f, f f. 4. CONCLUSION I this Uit, you hv t th bcgou thtic cocpts tht fi th viot i which Qutu Mchics opts. Th wo qutu itsf ipis st of stts i which giv physic syst c ist. Ech of th possib stts is c vcto. You so t how to fi istc o vcto spc, so tht you c fi th istc btw y two poits o th vcto spc of itst. Th vctos i Qutu Mchics iv i Hibt spc, suig tht w c p y giv stt of th syst s i cobitio of th possib stts i which it c b fou. This is possib bcus Hibt spc is copt. 5. SUMMARY I this Uit, you hv t tht: tic spc is copt if vy Cuchy squc covgs to poit i th tic spc o i vcto spc is spc o which o is fi copt o i spc is Bch spc copt o i spc with th o fi by th i pouct is Hibt spc vy Hibt spc is Bch spc Asw to Sf Asssst Ecis A tic is fuctio : V V R, stisfyig fo f, g, h V : i f, g f, h h, g Tig iquity Th w of tig iquity stiputs tht o sig si of tig c b og th th su of th iig two sis. Th poits of th vctos f, g h fo tig. As such th is oiig th poits ust fo tig, obyig th tig iquity. ii f, g g, f Syty

11 Th istc fo o poit to oth ust b th s, ispctiv of th stt poit th poit. iii f, g if oy if f g Dfiitss Th istc btw poit ust b zo, if th istc btw th poits of th vctos f g is zo, thy ust b th s poit. Asw to Sf Asssst Ecis A squc is si to b Cuchy squc wh, wh,. O o pcisy, fo Cuchy squc, fo, th ists N such tht, fo, N. Th cs ivovig is o stict bcus w b to fi fiit tu ub N such tht th coitio of Cuchy squc is stisfi. Asw to Sf Asssst Ecis i -, op st, {-} {} to to gt th cos st [-, ] ii [-, ] cos st iii [, 9 hf-op st, {9} to b to gt th cos st [, ] iv, 7] hf-op st, {} to b to gt th cos st [, 7] 6. TUTOR-MARKED ASSIGNMENT. Justify whth o ot ch of th foowig is tic spc. with,, y, y y. b Th ub i with, c with, b b y by. Wh is squc si to b Cuchy? b Giv p of Cuchy squc tht os ot covg. c Wht is th tioship btw Hibt spc Bch spc?. Cosi Hibt spc with sc pouct., Pov th w of tig iquity, f g f g. Hit: f, g f, g Cuchy-Schwz iquity, f, g f g 4. Pov th Cuchy-Schwz iquity, u, v u v, fo two vctos u v i th -isio cop Hibt spc. Assu th cop ub vu, v th fct tht uv. 7. REFERENCES/FURTHER READING

12 Byo, F. W. J. & Fu, R. W. 99. Mthtics of Cssic Qutu Physics. NY: Dov Pubictios. DG, T. 7 Qutu Mchics, Rtiv fo Giffiths, D. J. 5 Itouctio to Qutu Mchics. Upp S Riv, NJ: Pso Ptic H. Kipf, J. A Bif Itouctio to Hibt Spc Qutu Logic, Rtiv fo Ls, N.P. 6 Lctu Nots o Hibt Spcs Qutu Mchics, Rtiv fo Pio, M. Qutu Mchics, Rtiv fo

13 UNIT : LINEAR MAP, LINEAR FUNCTIONAL, DUAL VECTOR SPACE CONTENTS. Itouctio. Obctivs. Mi Cott. Li Mp.. Eps of Li Mps. Li Fuctio. Du Vcto Spc.4 B Kt Vctos.5 Tsitio Pobbiity 4. Cocusio 5. Suy 6. Tuto-M Assigt TMA 7. Rfcs/Futh Rig. INTRODUCTION I PHY 9, you otic tht th cocpt of vcto spc is cssy i th sciptio of qutu-chic syst s it c ist i st of possib igstts. W oft hv to go fo o vcto spc to oth. A i p is goo fuctio tht bs us to t vctos fo o vcto spc to oth. If i fuctio cts o th su of two vctos i th fist vcto spc, th sut is th s s th i fuctio tig ch vcto ito th oth vcto spc ig th two sutig fuctios i th w spc. This s th fuctio is itiv. I itio, i fuctio tig i utip of vcto to oth vcto spc is quivt to tht i utip ot pow of it ifft fo utipi by th output of th fuctio i th w vcto spc. A i p is btw two vcto spcs. A i fuctio is i p btw vcto spc its uyig fi. Rc tht whi fiig vcto spc, w f to uyig fi, i which th sc i th fiitio of th vcto spc sis. I PHY 9, you otic tht w too th Hiti cougt of th fist of two vctos ivov i i pouct. I this uit, you wi gt to ow tht such vcto sis i wht w c u vcto spc.. OBJECTIVES At th of this Uit, you shou b b to: fi giv ps of i p fi giv ps of i fuctio pi th t u vcto spc fi th u of giv vcto wo with th Dic b t ottio ccut th tsitio pobbiity btw o igstt oth

14 . MAIN CONTENT. Li Mp A i p f : X Y btw vcto spcs X Y is fuctio which psvs vcto itio sc utipictio, i.., f = f f. f = f fo K, costt,, X. W cou up ths ito sig quit: f b f bf... Eps of Li Mps. Th zo p f : X Y, tig vctos i vcto spc X to th zo vcto i vcto spc Y.. Th itity p f : X Y, tig ch vcto i vcto spc X to itsf i th s vcto spc X.. Th p f : X Y, such tht f. f b b b f bf. Li Fuctio W cou cosi th uyig fi, K, of th vcto spc s vcto spc. I ost pobs i Physics, th uyig fi is ith th ub i o th cop p. Ech of ths c b cosi i spc su of y two vctos is so i th st, s w s sc utipictio. A i fuctio o i fo o o-fo o covcto g : X K is i p o i fuctio fo th vcto spc X to K, th uyig fi of X, with K cosi s vcto spc, i.., g = g g.4 g = g fo K, costt,, X.5 Equivty, g b g bg.6 Thus, i fuctio is fuctio fo vcto spc to its uyig fi tht is itiv hoogous. Rb ths th two o chctistics of vcto spc. I, th poptis of vcto spc so th s fo i fuctios.. Du Vcto Spc Th st of i fuctios fo X to K is vcto spc ov K, c th u vcto spc of X. Fo p, if vctos i R pst s cou vctos, 4

15 ..7. th y i fuctio c b witt i ths cooits s su of th fo f....8 which is th ti pouct of th ow vcto.. ] : f [.. ].. [ [.. ] is ti of costt ts th ight h si of.9 givs ub i th uyig fi. As you c s i this pssio, you u vcto i fuctio o covcto fo th u vcto spc of th vcto spc o th ft vcto fo th vcto spc o th ight to gt i pouct. Thus, i fuctio [.. ] hs t vcto.. fo th vcto spc to th uyig fi. Wht you gt out of this is b of th uyig fi. Fo vcto spc, you wi gt ub; fo cop vcto spc, you wi gt cop ub i g. If you put th vcto o th ft th u vcto o th ight, you ot goig to gt sc. Tht i wou b th out pouct. Th t ots th vcto i Hibt spc whi th b ots i fuctio o covcto i th u vcto spc. This is why th u vcto spc is so c th spc of i fuctios ov th vcto spc. I, th u of Bch spc is so Bch spc. Th cooy is y ipi: th u of Hibt spc is so Hibt spc. i Lt us t ict p: Th vctos i th vcto spc of cou vctos ov th fi of cop ubs. Notic tht th sco vcto ight ot hv tis ivovig i. Nothss, it is sti fo th vcto spc of itst. It ust so hpps tht th igiy pt of ch copot is zo. You ight s th two vctos o gy i this i i wy. To t th i pouct, w wit fi th u quivt of th i i fist vcto, o i ou pst gug, th quivt i fuctio:.9 5

16 * T i * * * i i i Th i pouct is th i i i i i is i th cop p, th fi uyig ou vcto spc. Thus, th i fuctio i fo th u vcto spc ov th cop ubs hs t vcto fo th vcto spc ov th fi of cop ubs to th uyig fi of cop ubs viw s vcto spc. Now, cst you i bc to PHY 9. How i you cov th cofficits of th psio of vcto witt s i cobitio of th vctos i th othoo bsis? Tht pocss is th s s th o scib h. You p th vcto s cii, wh { } is othoo st. To gt c, you too th i pouct i i,, c c, c c i i i i i i i i i You too this stp futh, fiig th pobbiity tht th syst u cositio is i y pticu stt. povi is oiz, * *, cii, c ci c i, ci c i ci ci ci i i i i i I this cs, th pssio o th ft is th o of, th output of which is wys ub. Nvthss, th fist b of th i pouct is fo th u vcto spc. You c ow gis wht w hv o so f to ti. Lt A B b vctos i th vcto spc of tics ov th cop p. Th i pouct, T A B ts vcto B fo th vcto spc to th fi of cop ubs viw s vcto spc. Notic tht A is ti. Fo p, i i i A, B i i i Th u of A is, i* * i A * i * i i* * i i i 4 i i 4i i T A B T i T i i i i i i i i i 6

17 i 4 i 5i T i i i 4 i i 4 i i i i i Th fuctio bov is fi with th pouct of two tics. As oth p of i fuctio, t V b th vcto spc of tics ov th fi K, which cou b th ub i o th cop p. Th scs ts of th ub i o th cop p s th cs y b. Th tc p, T : V R is i p. T AB A B A B... A B A A... A B B... B T A T B T A A A... A A A... A T A Equivty, T A B A B A B... A B A A... A B B... B A A... A B B... B T A T B T is t i th u spc V * of vcto spc V. Th p T is th ow cotiig th tc of th tics i th vcto spc. Ay oth i fuctio fi o this spc is so t of V *, th u spc of V. Th tc i this cs cts oy o ti, ui tht i th pvious p i which it cts o th pouct of two tics. Fo p, th p T : V R tig th fist t Aof ti to th uyig fi is so i fuctio o V *. T A A. Sf Asssst Ecis Epi th ts i p, i fuctio, vcto spc u spc i sig pgph..4 Th Dic Nottio B Kt Vctos To vy vcto i th vcto spc, th is cty o u vcto i th u vcto spc. Th u of th t vcto is th b covcto Th u of th b covcto is th iiti t vcto Put oth wy, th u of th u vcto is gi th vcto itsf. 7

18 Fo opto A ctig o th vcto igvcto This is quivt sic A A A to Moov, sic AB B A, AB is quivt to B A B, this bcos BA. i th vcto spc, w wit A. A i th u spc. A. Fo Hiti optos Sf Asssst Ecis I ts of vcto spcs, u vcto spcs tics, wht b t vctos? How c you fo i pouct fo th two is of vctos?.5 Tsitio Pobbiity I PHS 7, you p th wvfuctio of giv qutu-chic syst s i cobitio of th possib othoo igstts. Th, you w b to with th hp of th i pouct cov th cofficit of th psio th uc th pobbiity of fiig th syst i y pticu igstt. I ust th s wy, th i pouct so fis th tsitio pitu fo stt to stt, i..,. Thfo, i i with th Bo itpttio of th wvfuctio, th tsitio pobbiity is, P *. This is th tsitio pobbiity of th ptic u cositio fo stt to stt. You c s tht th pobbiity tht th tsitio to th s stt is uity ch igstt is ois: P. I th ti-pt cs, t ie, t,. t ie ', t',. th tsitio pobbiity fo stt to stt is, P ie t iet', t', t,,.4 ie te t'.5 Sf Asssst Ecis I ts of b t vctos, wht o you ust by tsitio pobbiity? 8

19 Ep. A ptic i bo of gth L with ifiit ws is i its gou stt. Wht is th pobbiity tsitio pobbiity tht th ptic is i th gou stt if o w is suy ov outw, ig th w bo of gth L? Soutio. L Lt us ot th gou stt of th L gth w th gou stt of th L gth w L. Th owb wvfuctios fo th L gth w, L si L L Th igfuctios fo th L w ust b tsfotio L L : L si L L Thfo, L L L L si si L L L L si si L L L You c s this s th cs wh th ti t is zo bcus th chg occu suy. Not tht th itg os ot to t to L, bcus th wvfuctio fo th L w is zo outsi th L g. L L L si si L L L L cos cos L L L L L L 4 cos cos L L L L 4 cos cos L L L 4 si si L L L 4 L L L 9

20 L L L L L L 4 4 si 4 si 4 L L Hc, th pobbiity tht th ptic is i th gou stt of th w bo of gth L is P 4. CONCLUSION I this Uit, you t bout i ps, which fuctios btw two vcto spcs, i fuctios which fuctios fo vcto spc to th uyig fi. You hv so s tht th i pouct of two vctos ivovs vcto fo th spc of u vctos o th ft vcto fo th vcto spc o th ight. This sus tht th output of such optio is i th fi uyig th vcto spc. You so t how to ccut th tsitio pobbiity btw t b. 5. SUMMARY I this Uit, you hv t th foowig: i p is fuctio btw two vcto spcs which psvs vcto itio sc utipictio i fuctio is fuctio fo vcto spc to th uyig fi which psvs vcto itio sc utipictio th st of i fuctios of ov vcto spc is th u spc of th vcto spc th i pouct of two vctos is btw vcto i th u spc of th vcto spc s b o th ft vcto fo th vcto spc o th ight how to ccut th tsitio pobbiity fo t stt to b stt. Asw to Sf Asssst Ecis A i p is fuctio btw vcto spcs tht spcts iity hoogity. Th cooi of i p is vcto spc. A i p thfo ts vcto spc to

21 tgt spc tht is so vcto spc. Th output is vcto. A i fuctio is i tsfotio whos tgt spc cooi is th sc fi, which is o-isio vcto spc o th cop p if fo cop vcto spc. Th output is sc. A u spc of vcto spc X is th st of i fuctios fo X to th uyig sc fi of X, fo which th scs fo X w. Th u spc is so vcto spc if w fi itio sc utipictio copotwis. Asw to Sf Asssst Ecis Kt vctos vctos fo th vcto spc cou vctos. B vctos fo th u vcto spc cou vctos. A i pouct is fo by hvig ow ti o th ft cou ti o th ight. Th sutig pouct givs sc. Asw to Sf Asssst Ecis Th t vcto c b s s th iiti stt i tsitio. Th b psts th fi stt. Th tsitio pobbiity is th pobbiity tht syst iitiy i th t stt is vtuy fou i th b stt. It is ot sufficit to ust t th i pouct with th b o th ft th t o th ight s th output ight b cop. W t th squ of th gitu which ust of cssity b ub:. 6. TUTOR-MARKED ASSIGNMENT. Dfi th foowig ts: Li p b Li fuctio. Show tht th oiy iffti qutio opto P L is i opto.. Pov tht th ifiit itg opto f f is i opto. 4. Cosi th spc of fuctios. Show tht th fiit itg f b f is i fuctio fo th vcto spc C[, b] of cotiuous fuctios o th itv [, b] to th spc of ubs. 5. A ptic is cofi i o-isio bo with isios is ow to b i th fist cit stt =. If suy, th with of th bo is oub without istubig th stt of th ptic, sut of th gy is, fi th pobbiity tht th syst is i th gou stt th fist cit stt of th w w. 6. Giv othoo st { }, Show tht psttio of th itity opto. I, th spct

22 7. Th stt of syst is giv by A[ i i /6 i ] Wht is th ssocit b vcto? b Nois th vcto. c Ccut th pobbiity pitu th pobbiity of fiig th syst i ch of th possib i ch of th stts,. Wht is th ost pobb stt th syst c b fou? Aoth stt of th syst is giv by i /6 [ i ] Ccut th pobbiity of obsvig th syst i stt if it ws iitiy i stt. 8. A ocu is copos of th tos. A cto c ttch to th ocu by ttchig to y of th tos. Th cto c b i o of th possib igstts, o, wh th stt is th stt i which th cto is ttch to th th to. Wht is th isio of th igspc of th syst? b Th igstts, o fo copt othoo bsis fo th syst. Epi th t copt bsis. c Wht wou hv b th ipictio if th th igstts i ot fo copt bsis fo th syst? If th syst is i stt [ 4 i ], o which to you ost iy to fi th cto? 9. Show tht th tsfotio i tsfotio. T : R R, giv by, y y,, y is ot 7. REFERENCES/FURTHER READING Css, J. D. Qutu Physics ots, Rtiv fo Gsit, J. Lctu Nots o Qutu Mchics, Rtiv Giffiths, D. J. 5 Itouctio to Qutu Mchics. Upp S Riv, NJ: Pso Ptic H. Pio, M. Qutu Mchics, Rtiv fo

23 Uit : COMMUTATOR ALGEBRA, MATRIX ELEMENTS OF A LINEAR OPERATOR. Itouctio. Obctivs. Mi Cott. Th Coutto. Coutto Agb. Mti Ets of Li Opto 4. Cocusio 5. Suy 6. Tuto-M Assigt TMA 7. Rfcs/Futh Rig. INTRODUCTION I Qutu Mchics, vy physicy obsvb is ssocit with Hiti opto. Th physicy obsvb poptis icu positio, ti, gy, gu otu, tc. Quit ui th cs with Cssic Mchics, ot physicy obsvb poptis of qutuchic syst c b su pcisy siutousy. This is u to th Hisbg uctity picip. If y two obsvb physic poptis c b su siutousy with ifiit ccucy, th thi optos ust cout. I, you wi gt to ow tht two such obsvbs c hv th s igvctos. I this uit, you wi so how to fi th ti ts of opto i giv qutu-chic stt. Thus, you wi b b to ccut th pcttio vu of th physicy obsvb popty i such stt. I itio, you wi bout th out pouct of two vctos s w s th poctio opto.. OBJECTIVES At th of this stuy sssio, you wi b b to: o coutto gb wit th ti psttio of giv i opto show tht two optos tht cout c hv th s igvctos fi th out pouct of two vctos wo with poctio optos. MAIN CONTENT. Th Coutto Lt two optos, cospoig to two physic obsvbs b A B, t th hv th s igvcto, with igvus b spctivy. Th, w c wit A. B. b

24 Mutipyig qutio. o th ft by B, BA B B. b Mutipyig qutio. o th ft by B, AB A A.4 b b b Subtctig qutio. fo qutio.4, AB BA.5 b b AB BA.6 sic th igvus ust ubs. Gy,. Hc, AB BA. This is witt s [ AB, ], wh [ AB, ] is th coutto of A B. Th coutto is ticouttiv, tht is, Sf Asssst Ecis Pov th tio.7. [ B, A] [ A, B].7 Lt th uctity i th sut of physic obsvb with cospoig opto A b th uctity i th sut of physic obsvb b with its cospoig opto B b b. Th, th uctity tio is i b [A,B] If th coutto [ AB, ], th th pouct of th uctitis i th suts of th cospoig physic obsvbs is zo. It s th two physic obsvbs c b su siutousy with ifiit ccucy, b. W th sy A B cout. Msuig o of th physic obsvbs vs th syst uistub tiv to sut of th oth obsvb. W sy th two obsvbs coptib. 4

25 If th coutto [ AB, ], A B o ot cout. Th th pouct of th uctitis i th suts of th cospoig physic obsvbs is ot zo. It s th two physic obsvbs cot b su siutousy with ifiit ccucy, wou ipy ifiit o i th sut of b. Msuig o of th obsvbs istubs th syst, cusig o i th sut of th oth obsvb. W sy th two obsvbs icoptib. Th coutto ss to s, Wht is th ffct of suig th physic qutity cospoig to opto A fist, th th physic qutity cospoig to opto B, suig th i vs o? If th optos cout, it shou ot tt which physic qutity cospoig to th opto is su fist. Sf Asssst Ecis Wht o you ust by coutig obsvbs.. Coutto Agb H so poptis of th coutto:. [ A, A] AA AA.8. [ A, B] AB BA BA AB [ B, A].9. [ AB, C] ABC CAB ACB ACB CAB ABC ACB ACB CAB A BC CB AC CA B A [ B, C] [ A, C] B. 4. [ A, BC] ABC BCA ABC BAC BAC BCA ABC BAC BAC BCA AB BA C B AC CA [ A, B] C B[ A, C]. 5. [ A B, C] A B C C A B AC BC CA CB AC CA BC CB [ A, C] [ B, C]. 6. [ A, B C] A B C B C A AB AC BA CA AB BA AC CA [ A, B] [ A, C]. 7. [ A,[ B, C]] A[ B, C] [ B, C] A.4 Sf Asssst Ecis Pov th Jcobi itity [ A,[ B, C]] [ B,[ C, A]] [ C,[ A, B]] 5

26 6. Mti Ets of Li Opto Suppos opto A ctig o t suts i th t, w wit A W c so p i othoo bsis i i } {, i v i i O th oth h, w c p icty i th s bsis, usig ifft uy i s th cout, v ' so tht w c cov ' v. i i i v A A v ' i i i A v i i v i A Ty to wit out this su, wht you gt is, A v A v A v v... ' Thus, w c wit this s th th ow of ti A utipyig cou vcto v. v v v A A A A A A A Th, v v A A A A v v ' '.5 So, w c pst y i opto A by squ ti i i A A.

27 7 Mo spcificy, w c wit, A A A A A A A A A T oo t ch cou o ch ow you wi s th is ptt: Th ti is up of coctio of cou vctos o w cou s it s ow vctos:...,...,... o wh A, A, tc., A, tc. Ep. Fi th ti of psttio of th itity opto. Soutio. Fo th itity opto I, i i I i I I i = i i Hc,

28 I Sf Asssst Ecis 4 Fi th ti of psttio of th i tsfotio i th usu bsis i L : R R, such tht L,,,,. R, 4. CONCLUSION I this Uit, you t tht two physic obsvbs c b su siutousy with ifiit ccucy if thi cospoig optos cout. If th optos o ot cout, suig o of th obsvbs iucs o i th sut of th oth physic obsvb. Moov, you hv b b to fi th ti ts of i opto, giv th vt bsis vctos. 5. SUMMARY I this Uit, you hv t th foowig: coutto ts us if w c o cot su two physic obsvbs siutousy with ifiit ccucy if th coutto is zo, th two qutitis cospoig to th optos c b su siutousy with ifiit ccucy if th coutto is ot zo, th two qutitis cospoig to th optos cot b su siutousy with ifiit ccucy how to fi th ti ts of i opto Asw to Sf Asssst Ecis [ B, A] BA AB AB BA [ A, B] Asw to Sf Asssst Ecis If th coutto [ AB, ], th th pouct of th uctitis i th suts of th cospoig physic obsvbs is zo. It s th two physic obsvbs c b su siutousy with ifiit ccucy, b. W th sy A B cout. Msuig o of th physic obsvbs vs th syst uistub tiv to sut of th oth obsvb. W sy th two obsvbs coptib. If th coutto [ AB, ], A B o ot cout. Th th pouct of th uctitis i th suts of th cospoig physic obsvbs is ot zo. It s th two physic obsvbs cot b su siutousy with ifiit ccucy, wou 8

29 ipy ifiit o i th sut of b. Msuig o of th obsvbs istubs th syst, cusig o i th sut of th oth obsvb. W sy th two obsvbs icoptib. Asw to Sf Asssst Ecis [ A,[ B, C]] [ B,[ C, A]] [ C,[ A, B]] A [ B, C] [ B, C] A B[ C, A] [ C, A] B C[ A, B] [ A, B] C ABC ACB BCA CBA BCA BAC ACB CAB CAB CBA BAC ABC Soutio to Sf Asssst Ecis 4 L L,, L L,, L L,, Hc, th ti pstig th i tsfotio is, L L L L L L L L L L Notic tht w ight s w hv g th ti by witig out th th cous L L L L Attivy, L L,, L L,, L L,, Hc, L L L L L L L L L L Aoth ppoch: 9

30 L L L L L L L L L Agig ths i ti givs yt gi, L 6. TUTOR-MARKED ASSIGNMENT. Show tht [ A, B] A [ A, B], povi [[ A, B], A], fo, th st of tu ubs.. Th ctio o stt of th owig th isig optos fo th hoic oscito, spctivy,,.. Wit out th ti ts of tht of. b. Fi th two poucts of th tics,. c. Wit th ti fo of th Hitoi opto fo th hoic oscito, H.. i Fi th igvctos of th tics 4 ii Show tht th tics cout. iii Cot o you fiigs i i ii. 4. Fi pssio fo th coutto [, ] AB CD i ts of th couttos of pis of th optos.

31 5. Th usu bsis fo spi-hf syst is. W c wit th two vctos i i. Lt th opto Q b fi by Q ih Wit th vctos s t cou vctos. b Show tht th vctos othoo. c Wht is th ti psttio of th opto Q i th bsis {, }. Ccut Q Q. Ccut Q, Q. 6. Th opto of physic obsvb i th othoo stts, which sp th Hibt spc of syst is giv by Q i Q i i Q i i Wht is th ti psttio of Q i th {,, } bsis? b Fo th stt, ti Q Qi th 6 psttio of th vctos optos i th {,, } bsis. 7. I th isospi thoy, th ucos poto uto ssu to b ifft stts, ot spctivy by p. If th stt of th uco chgs s sut of coisio btw th uco oth ptic, chg fi by Q p p i Q i p Wit th ti psttio of Q i th bsis { p, }. b Assuig poto ugos such coisio, wht is th pobbiity tht th uco cou b obsv to b uto ft th coisio? c Nois th stt p i hc, ti th w stt of th syst ft th coisio hs occu. Wit th b vcto s ow vcto. Hc, fi th pobbiity tht th uco wi b fou i th stt ft th coisio. Suppos th uco is i stt c p c bfo coisio, fi its stt ft th coisio. f Fi, if possib, coisios i which th stt os ot chg.

32 Css, J. D. Qutu Physics ots. Rtiv fo Gsit, J. Lctu Nots o Qutu Mchics. Rtiv Giffiths, D. J. 5 Itouctio to Qutu Mchics. Upp S Riv, NJ: Pso Ptic H. Mzbch, E. 998 Qutu Mchics. Nw Yo: J. Wiy & Sos. Pio, M. Qutu Mchics. Rtiv fo

33 UNIT 4: THE OUTER PRODUCT, THE PROJECTION OPERATOR AND THE COMPLETENESS RELATION. Itouctio. Obctivs. Mi Cott. Th Out Pouct. Th Poctio Opto. Th Coptss Rtio 4. Cocusio 5. Suy 6. Tuto-M Assigt TMA 7. Rfcs/Futh Rig. INTRODUCTION Th i pouct of two vctos is obti by puttig b vcto, fo th u vcto spc o th ft t vcto, fo th vcto spc o th ight. Th outco of this pocu is sc. I cotst, out pouct hs t o th ft b o th ight. Th outco is ti, ot sc. A spci cs of th out pouct is th poctio opto, which.. Th coptss tio is sutio of th out poucts fo fo ch vcto i th bsis its u. This sutio givs th itity ti, ig tht th coptss tio psss th physic situtio wh suig istut pc i th pth of th syst u cositio, but ig o sut is th s s ot hvig th istut th i th fist pc. I this stuy sssio, you wi gt to ow th coctio btw coptss... OBJECTIVES At th of this stuy sssio, you wi b b to: 6. o coutto gb SAQ 6. SAQ wit th ti psttio of giv i opto SAQ show tht two optos tht cout c hv th s igvctos SAQ fi th out pouct of two vctos SAQ wo with poctio optos SAQ 6.6 SAQ 6.7, SAQ MAIN CONTENT. Th Out Pouct With th b-vctos, v, w t-vctos, v w, th i pouct of vctos v w is witt, v w th out pouct v w which, is opto A vw. A s v w s v w s v 4. vw wh w s.

34 4 Th oit of th out pouct w v is v w w v w v, cig tht AB B A otig tht th oit optio tus b ito t vic vs. Ep 4. Fi th i pouct, w v, th out pouct, w v, of th vctos v v v v w w w w i vcto spc. A vcto spc is such tht th tis of th cou o ow vctos. Soutio 4. w v w v w v w w w v v v T w v w v w v w v w v w v w v w v w v w v w v w w w v v v T vw w v Mo gy, b b, whi b b Tht is, th tspos is ot th vt optio, but th tspos th th cop cougtio T T * *, tht is, Hiti cougtio. Not tht th o c b vs: you c fist fi th cop cougt th tspos th cou vcto. Ep 4. Fi th i pouct th out pouct of th vctos b, th out pouct, b, giv tht, i i i i b Soutio 4. * * i i i i i i i i i i b I g, this shou b cop ub. I this cs, th igiy pt is zo.

35 i b i i 4i 4 i i i i i * i i i i i i i i i i i i i i Sf Asssst Ecis Giv tht A is ois, fi A. Hc, ccut P. 5. Th Poctio Opto Lt us ow t spci cs of th opto, wh i giv othoo bsis, wh y vcto i th spc c b witt, i c v 4. i i i c c i i 4. W c P 4.4 th poctio opto s its ffct o vcto, tht is i ictio. It so s ss, tht is to poct it i oy o ictio, tht of P P P 4.5 I, P P c c c P Thus, to pov tht giv opto is poctio opto, it is sufficit fo it to stisfy, P P 4.6 5

36 . Th Coptss Rtio ci i ci i c i i sic ci i c Rc fo Qutu Mchics I i Hc, w c s tht is itity opto. Thfo, w c wit Thus, I 4.7 I = 4.8 This is th coptss tio fo th othoo syst { }. Equtio 4.7 iicts tht th coptss tio s possib poctios of th syst hv b t ito cositio. But this so s tht th syst c b i y of ths stts i o oth stt outsi th os i th giv bsis vctos o stts. I th cs of cotiuous spctu, th coptss tio ts th fo of itg: b 4.9 Hc, b I = 4. Now, is th y i btw coptss s w iscuss u Bch Hibt spcs? Of cous, th w t bout fiig hos. With th hos if y fi, w c sfy y out ou vctos i such wy tht st of othoo bsis vctos c b utiis i witig o pstig y vcto i th spc. I Qutu Mchics, tht vcto cou b gy vcto, positio vcto o gu otu vcto. A vcto spc, you wi so c, is o tht ows us to vctos utipy th by ppopit scs o cop sti b cofit ou sut is sti i th vcto spc. I itio, th i of covgc ipis th is ifiit ub of vctos btw y two poits o copt vcto spc wht w f to s sss. Th is o ho. Thus, vy Cuchy squc covgs withi th vcto spc. Thus, fo p, th is ifiit ub of vctos btw,,,,, such s.,,,.,,,.,,, tc. I, btw y two of ths, th is ifiit ub of vctos. Dos tht ig b? Ys, th ub i itsf is Bch spc s th itios tht th hos hv b pugg i. Thus, btw.., th is o op spc th is ifiit ub of ubs btw th. Th o i this cs is th bsout vu o. Liwis, th cop p is Bch spc with th o fi by th Euci o 6 y y, wh vcto i th spc is pst by z iy. It shou so b c fo th fogoig, tht th -

37 isio Euci spc is so copt tic spc. H, covgc shou b s i viw of th tic fi o th vcto spc. W c tht tic spc is vcto spc quipp with tic. Sf Asssst Ecis Wht is th coptss tio? Outi th i, if y, btw coptss of copt o i spc th coptss tio. 4. CONCLUSION I this Uit, you t bout th out pouct, th poctio opto th coptss tio. You t to istiguish btw th i pouct, th output of which is sc, th out pouct, th output of which is opto. Th poctio is spci out pouct tht pocts vcto i oy o of th stts i which it c possiby ist. Epcty, ppyig th poctio opto oc o poucs th s sut, s th syst is y i tht stt i which it hs b poct. You t tht th coptss tio s th syst u cositio c oy b i th st of pscib igstts. This ws so i to coptss of o i spc s iscuss u Bch Hibt spc i Uit. 5. SUMMARY I this stuy sssio, you hv t: to istiguish btw i out poucts of vctos to fi th out pouct of two giv vctos to wo with poctio optos bout th coptss tio its tioship to th coptss of o i spc Soutio to Sf Asssst Ecis 8 ci A A 5 5 i 8 7 A, 5 5 A 7 5 Thfo, P c 5 Sf Asssst Ecis I th tt. 6. TUTOR-MARKED ASSIGNMENT. Fi th i pouct b th out pouct b of th vctos 7

38 , b. Giv othoo st { }, Show tht I.. Show tht th ti is pocto fo R to th y z p. 4. Show tht i opto P is poctio opto if oy if th ists oth i opto Q such tht, P Q I. 5. Wht o you ust by th coptss tio? 7. REFERENCES Css, J. D. Qutu Physics ots. Rtiv fo Giffiths, D. J. 5 Itouctio to Qutu Mchics. Upp S Riv, NJ: Pso Ptic H. Kipf, J. A Bif Itouctio to Hibt Spc Qutu Logic. Rtiv fo Pio, M. Qutu Mchics. Rtiv fo Sh, R. 994 Picips of Qutu Mchics. Docht, Th Nths: Kuw Acic Pubishs, 8

39 MODULE : ANGULAR MOMENTUM AND SPIN OF ATOMIC AND NUCLEAR PARTICLES Uit : Agu otu. Itouctio. Obctivs. Mi Cott. Agu Motu Optos. Th Agu Motu Cos. Eigfuctios of Agu Motu.4 Risig Lowig Optos of Agu Motu 4. Cocusio 5. Suy 6. Tuto-M Assigt 7. Rfcs/Futh Rig. INTRODUCTION Agu otu i qutu chics is quivt to th s cocpt i cssic chics. You wi c tht i cssic chics, th gu otu of isot syst is cosv. As usu, i qutu chics, th gu otu opto is opto. I qutu chics howv, ui th cs of th cssic ogy, th Ctsi copots of th obit gu otu optos which o ot cout. This ipis w cot su th ifft copots of th gu otu of boy siutousy with ifiit ccucy.. OBJECTIVES At th of this Uit, you wi b b to: fi th pssios fo th copots of gu otu fi th coutto tioship btw pis of gu otu optos ccut th igvus of th gitu of th gu otu ccut th igvus of th z-copot of th gu otu pi wht is t by gu otu cos. MAIN CONTENT. Agu Motu Optos Th gu otu of boy is th ot of its otu bout giv fc poit. I Cssic chics, th gu otu of boy c t o y vu. Is this th cs i Qutu chics? To stt with, s it is with physic obsvbs i Qutu chics, th gu otu of boy is pst by opto. W stt ou ysis by cosiig th gu otu opto L. L p. I th Ctsi syst, w c wit this s 9

40 L i y z p p p y z wh i y z p pi py pz spctivy th positio vcto th i otu of th boy. Thus, L i yp zp zp p p yp. z y z y Th couttio tios bow ho: [ L, L ] [ yp zp, zp p ] y z y z [ yp, zp ] [ y p, p ] [ zp, zp ] [ zp, p ] z z z y y z yp zp zp yp yp p p yp z z z z z z zp zp zp zp zp p p zp y y y z z y ypz zp zpypz zp ypz pzzp y zpzpy zpyzp pz ypz ypz pz this pt is zo Tht pt is zo bcus th optos cout it os ot tt i which o thy witt. Rc: [ q, p ] i i i Hc, Thus, [ Ly, Lz ] il z z y z y z p yp z p yzp p zp p p z p y p z zp p zp p z z z y z z p yz p p z p zp p z z z y z z p y[ z, p ] p [ z, p ] z y z ihp y ihp y ih p y p ih yp p ihl y y z [ L, Ly] ih Lz. Siiy, [ L, L ] il.4 y z z [ L, L ] il.5 y A ttiv wy of povig ths couttio tios is i Appi. 4

41 Equtios.-.5 c b su up by Fig... Cocwis ottio is co positiv, whi coutcocwis ottio is co gtiv. Fo istc, tio. givs positiv i, whi coutcocwis [ Lz, Ly ] il L z Fig..: Figu osttig th couttiv tioship og th, y z copots of th gu otu Th vt Hitoi s Appi, is h L H V h.6 Usig th tio [ q, p ] i, i, =,,, w c wit i i [ L, H] [ p, H] [ L, p ] Show.7 z z Sf Asssst Ecis Show ch of th pssios i qutio.6. z z Sic [ AB, C] A[ B, C] [ A, C] B, I, [ L, Lz ] L[ L, Lz ] L[L Ly Lz, Lz ] [ L, Lz ] [ L, Lz ] [ L, Lz ]. L couts with ch of th copots of L, i.., [ L, L ] [ L, Ly ] [ L, Lz ] L, L y L z. Tht is,.8 Sic th th optos H, L z p z cout with ch oth, it is possib to fi stts tht siutousy igstts fo th optos. Th igvu qutio fo L z c b witt s L z.9 wh th igvus of L th stts hv b b by. z W cocu tht th igvus of L gt: to ch igvu L, th iy ipt igstts, fo vus of i th g. I oth wos, ch L igvu is -fo gt. 4

42 Th i is c th gtic qutu ub, whi is c th obit gu otu o ziuth qutu ub.. Th Agu Motu Cos Th igvus of th gu otu L, ig tht sut of th gitu of gu otu wi oy fi o of th isct st of vus, L, =,,,,. Aso, sut of th copot of gu otu og cti is,.g., th z - is, wou oy fi o of th possib vus, o of th igvus, L z, {,,,...,,, }. W ow ot th iffcs btw gu otu i cssic i qutu chics: Th gu otu i y oth ictio, th z-ictio sy, is wys s th th gitu of th tot o-zo gu otu. Sic L z is th gst possib vu of L, ;, z. W cocu tht th gu otu c v b copty ig i y pticu ictio. I, if th gu otu i poit i y o fiit ictio, th th copots wou b fiit. Howv, w ow cospoig optos o ot cout; s such, ths copots cot b su ccuty siutousy. W ow i cssic physics, tht L is vcto i fiit ictio. I qutu chics, i viw of th Hisbg uctity o-couttivity of th optos ivov, this cot b th cs. Hc, it is btt to visuis th gu otu ssocit with giv igstt s co, th gu otu co. Nvthss, i igstt w ow tht th gitu of th gu otu L th z -copot L z fi, L, L z. Whi th vus of L L y ifiit, th squ pcttio vus ust stisfy L Ly Lz L.4 Thus, u to o-couttivity of th copots th ttt uctity, ifiit ub of owb vctos fo co tht c hv th s pitu z-copot. 4

43 Sf Asssst Ecis Wht qutu-chic cocpt uis gu otu cos? 4. CONCLUSION I Uit of this Mou, you t tht th cssic gu otu hs qutu ogu. Howv, ui th cssic cs, th copots of th qutu-chic gu otu o ot cout. Th ipictio of this is tht it is ot possib to su th copots t th s ti with ifiit ccucy. It so s tht thy cot hv th s igfuctio. Howv, th z-copot of th gu otu couts with th squ of th gitu of th gu otu s w s with th Hitoi. A th optos c thfo hv th s igfuctio. I itio, with th o-coutig tu of th copots of th gu otu, it is ipossib to su y pi of copots siutousy with ifiit ccucy u to th Hisbg uctity picip. As such, th gu otu ssocit with giv igvcto is bst visuis s co, th gu otu co. 5. SUMMARY I this Uit, you hv t: th qutu-chic gu otu is th qutu quivt of th cssic gu otu Ui th cssic gu otu, th copots of th qutu-chic gu otu o ot cout No-couttivity ipis th copots of qutu-chic gu otu cot b su siutousy with ifiit ccucy th Hitoi, s w s th squ of th gitu of th gu otu cout with th z-copot of th gu otu th c hv th s igfuctio th couttio tios fo th Ctsi copots of th gu otu opto. th gitu of th obit gu otu its z-copot fi; s such, u to o-couttivity of th copots, ifiit ub of vctos fo co gu otu co tht c hv th s pitu z-copot. Asw to Sf Asssst Ecis [ Lz, H] This is bcus H qutio.6 ivovs t i, which couts with t i, [ L, L ] z Lz which hs o h [ pz, H ], H i z bcus H hs o picit pc o z. 4

44 [ Lz, pz] sic Lz py yp [ py yp, pz ] [ py yp, pz ] sic copots of th i otu cout o with oth. Asw to Sf Asssst Ecis Hizbg s uctity picip. 6. TUTOR-MARKED ASSIGNMENT. Epi th t gu otu co.. Pov th foowig ititis: i [ L, L ] il L il L y z z ii [ L, ]. Fi th possib vus spctu L i th stt L y. 6 Gsit, J. Lctu Nots o Qutu Mchics. Rtiv Giffiths, D. J. 5 Itouctio to Qutu Mchics. Upp S Riv, NJ: Pso Ptic H. 44

45 UNIT : EIGENFUNCTIONS OF ANGULAR MOMENTUM I. Itouctio. Obctivs. Mi Cott. Agu Motu i Sphicy Cooits. Th Aziuth Equtio. Th Po Equtio 4. Cocusio 5. Suy 6. Tuto-M Assigt TMA 7. Rfcs/Futh Rig. INTRODUCTION I Uit, w itouc th cocpt of qutu-chic gu otu w b to itify so coutig optos, two of which th squ of th gitu of th gu otu th z-copot of th gu otu. As sut, th two optos c hv th s igfuctio. I this Uit, w po this possibiity. As i th cs of th hoic oscito, w itouc th optos, sttig with th highst igfuctio, w c th us th owig opto to obti oth igfuctios.. OBJECTIVES At th of this uit, you wou hv: ow th fo of gu otu igfuctios, th sphic hoics itifi th wvfuctios fo th squ of giv obit gu otu th z-copot of th gu otu ti th igvus of th squ of th gitu of th gu otu s w thos of th z-copot of gu otu. MAIN CONTENT. Agu Motu i Sphic Cooits I th cs of th hoic oscito s tt i PHY 9, th gou stt wvfuctio ws fou by sovig th fist o iffti qutio,, wh is th owig opto fo th hoic oscito, oth igstts by ppyig th isig opto I th cs of gu otu, w us sii ppoch. Lt b giv. Th, w sov th fist o iffti qutios, L, y, z. z L, y, z.. th obti oth owig opto L. wvfuctios i th utipt by ppyig succssivy th 45

46 I viw of th sphic syty of th pob, it is uch si to sov ths iffti qutios i sphic cooits: i, po, ziuth, z cos si cos y si si. This is bcus i sphic cooits, th vib ops out of th gu otu optos. Th sutig gu otu optos bco s Appi fo poof: L i si cot cos.4 L y i cos cot si.5 L z i.6 L si.7 si si Sf Asssst Ecis Fo th pssio fo L i qutio.7, show tht [ L, L ] [ L, H] [ L, p ] W sh cout qutio.7 whi sovig th gu pt of th Schoig qutio fo sphicy sytic potti. Th igfuctios of L of th fo, y, z f Y,.8 wh Y, th sphic hoics. W c thfo wit th igvu qutio, L Y Thus th igvus of h. Sic, Y,.9 L h. A sut of z z L c oy giv st of vus L L z cout, thy c hv th s igfuctios. So, w c so wit L Y, Y,. z I i, th igvus of h. Lz h. A sut of f is y fuctio such tht th oistio coitio = y *, y, z, y, z Lz c oy giv st of vus 46

47 * = si f * f Y, Y, is stisfi. Covtioy, w ois th Y such tht th itg ov gs is so qu to uity, * si Y, Y, = With this oistio, th Y, ow s sphic hoics. I sphic cooits, th isig owig optos L L il y. i i cot. L L il y. Not tht L L. i i cot.4 W c th sov th fist-o iffti qutios L Y, LzY Y.5 Th fist qutio i qutio.5 is cosquc of ou ibiity to is stt high th th highst possib, which i this cs is =. Sf Asssst Ecis Usig th isig th owig optos, spctivy, L L il y, pov th foowig: i L L L ; Ly L L ii L Lz L L LL iii [ L, L ] L z 47 L L il y,. Th Aziuth Equtio By th tho of sptio of vibs Y, A B.6 Th L z igvu qutio fo y is, B i B Hc, B i B i.7

48 o B i c Tig th Npii ogith of both sis, i B D c wh D. W c st D qu to uity sic w yt to ois th fuctio Y, A B. Th, B i.8 Sic th g is th s s th g, so th wvfuctios ust stisfy th pioicity coitio Y, Y,.9 Sic th pioicity coitio ust b stisfi, if icss by, B ust i th s, i.., B B i i. i i i i Th, i cos i si i Equtig th pts, cos This is possib oy fo itg positiv, gtiv zo vus of :,,,,. is th gtic qutu ub As sut, sic, th possib vus fo =,,, So w hv i Y, A. i pticu, sttig qu to, th gst possib vu of, i Y, A.. Th Po Equtio Appyig th isig opto to th stt giv i qutio., w ust hv isig th highst stt givs you zo, sic o stt is high th th highst stt: L Y = i i = i cot A i i i = i cot A A 48

49 = i cot A = i cot A. Th fo of th iffti chgs fo pti iffti to oiy iffti, s A is fuctio oy of o A cot A.4 Hc, A cos A si f ' Th itg f + c. f A si c si c, sic b b Tig th poti of both sis, D A si c c si si c wh D, costt. W c thfo wit big i i tht A D si.5 Th, Y, i A B N si.6 wh N is oistio costt, which is ti fo th oistio coitio = si * Y Y = N si /! = N Th -fuctio is spci fuctio with th popty Equtig qutio.7 to, w gt,.7 49

50 / N /! Y, so ps o i such wy tht w choos obti fo th oistio. N.8 tis th pssio i.8 Fo qutio 7.4, N /! Thfo, / / i Y, si. /! Asw to Sf Asssst Ecis L is picit oy i th gs ifftis with spct to ths gs. Th opto couts with y opto tht is ot picit i ths gs ifftis hvig to o with th gs. A th oth optos ot picit i th gs o thi ifftis with spct to th. Th pioicity coitio o th ziuth pt of th sphic hoics sus tht th igvus of th z-copot of th gu otu oy, positiv gtiv itgs. This so iposs coitio o th Asw to Sf Asssst Ecis.9 L L L il y L il y Aig subtctig iviig by, givs th si pssios. L L L y L z But, L L L L L ily L ily L ily L ily L il L y il y L L y 5 L il y L il L y L y

51 L Ly L L y L L L L Hc, L Lz L L L L [ L, L ] L L LL L ily L ily L il y L ily L 4. CONCLUSION il y L y il y y L L y y L I this Uit, you t tht th sphic hoics th igfuctios of both th squ of th gitu of th gu otu opto th z-copot of th gu otu. Th pioicity coitio o th ziuth pt of th wvfuctio sus tht th gtic qutu ub c oy t itg vus, ig to qutistio of th gtic qutu ubs. This so ipis tht th ziuth qutu ub c so oy b zo o st of positiv itgs. By ppyig th isig th owig optos of gu otu, you w b to obti th igfuctios sphic hoics fo giv vu of th ziuth o obit gu otu ub. 5. SUMMARY I this Uit, you hv t th foowig: th squ of th gitu of th gu otu opto th z-copot of th gu otu cout th sphic hoics th igfuctios of both th squ of th gitu of th gu otu th z-copot of th gu otu with th hp of th optos, w obti th igfuctios fo pticu vu of th gtic qutu ub is qutis oy, positiv gtiv itgs th obit gu otu qutu ub is qutis, positiv itgs oy pssios fo th copots of th gu otu i sphic cooits. pssios fo th igvus th igfuctios of th squ of th gitu of th gu otu th z-copot of th gu otu. il L z y il i L L L L ] i[ L, L ] i il L y L z L y 6. TUTOR-MARKED ASSIGNMENT 5

52 . Fi th ti of psttio fo th isig th owig optos of gu otu.. Fi th pcttio vus show tht L = L. y L, L, L, y z L i th stt,. Fi th ti psttio fo th isig th owig optos of gu otu. z Byo, F. W. J. & Fu, R. W. 99. Mthtics of Cssic Qutu Physics. NY: Dov Pubictios Gsit, J. Lctu Nots o Qutu Mchics. Rtiv Giffiths, D. J. 5 Itouctio to Qutu Mchics. Upp S Riv, NJ: Pso Ptic H. 5

53 UNIT : EIGENFUNCTIONS OF ANGULAR MOMENTUM II. Itouctio. Obctivs. Mi Cott. Risig Lowig Optos of Obit Agu Motu 4. Cocusio 5. Suy 6. Tuto-M Assigt TMA 7. Rfcs/Futh Rig. INTRODUCTION I Uit, w itouc th isig th owig optos of obit gu otu. I this Uit, w sh cotiu with ou ysis of th igfuctios of gu otu. W sh b witig fw of th sphic hoics.. OBJECTIVES At th of this stuy sssio, you wi b b to: is ow th gtic qutu ub of sphic hoics with th optos ccut th utipts of giv obit gu otu qutu ub. MAIN CONTENT. Risig Lowig Optos of Obit Agu Motu I th st Uit, w got th wvfuctio sphic hoic i Y, si /! W c gt th oth Y by ppyig th owig opto, L, / L Y, C Y,.,, giv th costt C,,. Cospoig costts C fi fo L Y, C Y,. To gt th L. C costts, w o th foowig: L L L L L z L z L L z L z.4 5