VII. Central Potentials

Transcription

1 VII. Cnta Potntias Bfo going any futh with angua ontu, it is bst to bgin using th ations w aady hav so that w can gt so ida what thy a good fo. Phaps th bst appication of th angua ontu ignfunctions w dat with in th pvious sction cos whn on das with a sphicay sytic (o cnta) potntia. In this cas, th potntia ngy is ony a function of th distanc to th oigin, and th knowdg w aady hav wi t us a gat da about th ignstats of th syst ispctiv of th paticua potntia. a. Sphica Poa Coodinats Sinc w a daing with a potntia that is a function ony of th distanc to th oigin, it is by fa pfab to wok in a st of coodinats wh is on of th basic vaiabs, ath than so function of x, y,.to this nd, w nd to convt ou quations to sphica poa coodinats - x, y,,θ, φ. In ost ath txtbooks, φ is dfind to b th ang ativ to th axis whi th vast ajoity of quantu chanics txts us θ in this capacity. W wi us th att dfinition, but b θ cafu that any quations takn fo oth soucs us this sa convntion! H a so usfu ations in sphica poa coodinats: y x cosφ y sinφ cosθ x φ

2 θ φ cosφ i + sinφ j + cosθ k cosφ cosθ i + sinφ cosθ j k sinφ i + cosφ j + φ + + θ + sin θ b. Cnta Potntias Fo an abitay potntia V (), w can wit Hˆ + V ( ˆ ) At this point, w convt th quations to natua units by choosing ou unit of ngth and unit of ass so that. Not that this avs us on f standad unit (ti, o, quivanty, ngy). It is convnint to fix this dinsion basd on th pob at hand; fo xap, in a haonic osciato, it is usfu to choos th ngy so that ω, whi fo th Couob intaction it is usfu to choos th unit of cton chag to b that of on cton. Ths units a y out of convninc and in th nd, onc w hav cacuatd an obsvab (such as th position) w wi nd to convt th sut to a st of standad units (such as ts). Th ain bnfit at th ont is that it ovs th ativy unipotant factos of and fo ou quation, so that in natua units: Hˆ + V ( ˆ ) ( + + ) + V ( ˆ ) sin θ wh th scond quaity just infocs th goy dtais wappd up in th Lapacian opato.

3 c. Obita Angua Montu Opatos In od to s what angua ontu has to do with this, w nd to xpss th angua ontu opatos in sphica poa coodinats, as w. In this cas, th vant typ of angua ontu is that of th patic obiting aound th oigin: ˆ pˆ i iφ θ Pugging in ou xpssions fo { θ, φ } fo abov: i { sinφ i + cosφ j} { cosφ cosθ i + sinφ cosθ j k} i sinφ cosφ cotθ i icosφ sinφ cotθ j i k x y Futh, aft so agba, on can show ˆ ˆ + ˆ + ˆ L Lx Ly L +. sin θ At this point w notic that pays a conspicuous o in th Haitonian: ˆ Hˆ L + + V ( ˆ ) Hnc, a of th angua dpndnc of Ĥ is containd in and w idiaty concud that: Hˆ, L ˆ 0 Hˆ, [ ] [ ] 0 which ans that th ignfunctions of Ĥ a aso angua ontu ignfunctions! That is, fo any fixd, ψ Eignfunction of Haitonian i, Quantu Nub fo L Quantu Nub fo L

4 Of cous, w ay want to know what th ignfunctions ook ik in a spac ath than witing th as abstact vctos. Fist of a, notic that non of th angua ontu opatos dpnd on, and so th ignfunctions dpnd ony on th angs θ and φ. W wi dnot ths functions by Y ( θ, φ) θ, φ, wh indxs th ignvau of and indxs th ignvau. Thn, w can wit, θ, φ ψ i R ( ) Y ( θ, φ) Th adia function wi dpnd on th fo of V (), but th angua pats a univsa thy a just th spatia psntation of th obita angua ontu ignfunctions. Thy a cad sphica haonics and w pocd to dfin thi pcis fo d. Sphica Haonics Th ignvau quations w divd pviousy fo angua ontu now bco patia diffntia quations that a not aways asiy sovd. Th quation is tivia to sov: Y θ, φ Y θ, φ i Y ( ) ( ) ( θ, φ) Y ( θ, φ) iφ Y ( θ, φ) P ( θ ) Unfotunaty, th quations fo ( θ ) th P a o difficut. To sov fo P s, w foow two stps: ) Rca that ˆ L ( θ, ) 0. O, in diffntia anguag: + i + Y φ iφ iφ φ cot θ P i i ( θ ) 0 ( cotθ ) P P ( θ ) sin θ ( θ ) 0

5 ) Using this sip sut fo, w can gnat th sphica haonics fo oth vaus of by patd appication of th owing opato: P θ P θ ( ) ( ) ( ) Th scond stp is ath tdious, and so w sipy stat that th sut + ( + )! d P ( θ ) ( ) sin θ sin θ + π!! d cosθ ( ) ( ) iφ ( θ, φ) ( ) P ( θ ) Y Ths a th sphica haonics and thy a th ignfunctions of and. Th noaiation constant has bn chosn so that Y * ' ( θ, φ) Y ' ( θ, φ) dθ dφ δ, ' δ, ' that is, w hav chosn it so that th sphica haonics a othonoa. Now, on ipotant constaint on ths soutions is that ust b an intg. To s this, not that haf-intg woud ipy hafintg. In this cas w hav a pob, bcaus as w sp aound an ang φ π th wavfunction nds to tun to its oigina vau; that is, it nds to b piodic. Howv, if is hafintg, this is not tu. Fo xap, if, ( φ+ π )/ iφ / iπ iφ / iφ / i Bcaus th haf intg soutions do not oby th pop bounday conditions, thy ust b discadd. Hnc, vn though ou divation abov sd to indicat that angua ontu coud b haf-intg, fo th spcia cas of obita angua ontu, this is not possib. W wi s shoty that haf-intg angua onta a cucia fo th dsciption of patic spins. In any cas, this shows how th gna quantiation 3 conditions 0,,,... and, +,... can b vn futh stictd whn on is daing with paticua typs of angua ontu. W wi nv hav an that is not an intg o haf

6 intg, but oftn ony ctain intg o haf intg vaus wi b pissib.. Th Radia Equation Cobining ou xpssion fo th sphica haonics with th pvious suts, w find that th ignfunctions fo any cnta potntia can b wittn, θ, φ ψ R i iφ ( ) P ( θ ) that is, th th dinsiona wavfunction is spaab into a poduct of th on dinsiona wavfunctions. This is not gnay th cas, and is on of th paticuay nic poptis of sphicay sytic potntias. Th adia function wi gnay dpnd on th fo of th potntia, but it wi oby th quation: ( + ) + + V ( ) R ( ) EiR ( ) This quation can b sovd xacty fo ony a fw cass (th haonic osciato and th Couob potntia a th ost notab). Notic that th ignvau quation dpnds on th vau of, th quantu nub fo, but not, which indicats th pojction of th angua ontu aong th axis. Hnc th R s do not actuay dpnd on. Futh, w anticipat th appaanc of anoth quantu nub (ca it n ) that indxs th soutions to this adia quation. Hnc, w pac R ( ) Rn ( ) in what foows. Th adia quation can b sipifid futh if w ook at th quation ρ R : satisfid by th functions n ( ) n ( ) ( + ) + + V ( ) ρ ( ) E ρ ( ) n n n wh, on th ight, w hav notd that th ngis aso dpnd on n and. Notic that sbanc of this quation to th D Schöding quation. Indd, if w dfin th ffctiv potntia by ( + ) V ff ( ) + V ( )

7 thn this is a D Schöding quation, with th ffctiv potntia abov. Not, howv, that th bounday conditions a diffnt than th typica D cas: ρ ( 0 ) 0 ρ ( ) 0 n n Bcaus th additiona t in Vff aiss fo th angua otion of th patic aound th nucus, it is usuay cad th cntifuga potntia. f. Hydogn-ik Atos W a now ady to spciai to th paticua cas of th hydogn-ik atos that is, atoic ions with ony on cton (H, H +, Li + ). Fist, w wi ak th infinit ass appoxiation fo th nucus - w pac th it at th oigin and assu it nv ovs bcaus it is uch o assiv than th cton. This is a faiy good appoxiation, sinc / 800, but if on wishs to b p o pcis, on y nds to pac th cton ass with th ducd ass µ ( + ) in what foows. Hnc, th nucus ony N psnts a potntia in which th cton ovs: Z V ( ) wh is th chag on th cton and + Zis th chag of th nucus. At this point, w ov fo natua units ( )to atoic units ( ). W can now xpicity stat ou fundanta units of ass, ngth and ngy: 8 unit of ass 9. 0 g 9 unit of ngth Boh a unit of ngyhat E h J a0 Th att two units giv th typica distanc an cton is fo th nucus and a typica ngy fo an cton in a Couob potntia. Hnc, w want to sov th quation

8 ( + ) Z + ρ n ( ) Enρn ( ). Lik th quation fo P ( θ ), this is faiy tdious to sov, and w y outin th stps ) Notic that fo ag, th potntia ts vanish and w just hav ± E ρ n ( ) On ust tak th - soution, sinc othwis th wavfunction wi not go to o at infinity. ) Wit E ρ n ( ) f ( ) and thn xpand f () in a pow sis about th oigin. 3) Inst ρ n into th Schöding quation abov and quat ach t in th pow sis xpansion fo f () to o. Aft a significant aount of agba, on finds that ( ξ Z / n ) 3 ξ + n+ ( n )!4 Z d d ξ ξ n+ Rn ( ) ( ξ ) ξ n+ (( n + )! ) n dξ dξ which a known as th associatd Lagu poynoias. Exaps fo ow vaus of n and a poducd in any txtbooks. Th a two oth vy intsting things that co out of th agba that ads up to th Lgnd poynoias: ) On finds that soutions ony xist if < n. Hnc, whi a Hydognic ato can ony hav any intg angua ontu, ths vaus a futh stictd fo fixd n. W typicay dnot th stats as s, p, d, f, g, h obitas, fo 0,,,3,4,5. Hnc, w hav s, s, p, 3s, 3p, 3d, tc obitas, but not d obitas o 3f obitas. Z ) Th ngis of th Hydognic ato a E n, which n w known xpintay ong bfo Schöding v

9 ca aong. Th intsting thing h is that th ngis do not dpnd on! This is a fatu pcuia to Hydognic potntias and is atd to an additiona syty possssd by th Couob potntia. This is td an accidnta dgnacy of th vs. Finay, bfo oving on, w not that ths a ony th bound stats of th hydogn ato. Th a aso positiv ngy stats that a osciatoy instad of dcaying. W wi not concn ousvs with ths stats, xcpt to say that th bound ignfunctions by thsvs a not a copt basis ony if th unbound soutions a incudd is coptnss achd. g. Ecton Spin Up to this point, w hav bn tating th cton as a stuctuss patic that has a ass and an ctic chag. Howv, th cton actuay has intinsic angua ontu, as w now show. It tuns out that th cton has a agntic ont. This can b asud xpintay in a Stn-Gach xpint. H, on taks a ba of atos that hav on xcss cton byond a fid sh (ost oftn Siv, but on coud us Sodiu, as w). Th ba is passd though an inhoognous agntic fid. If th cton has a agntic ont, th cassica foc on th cton is F B, wh is th agntic ont and B is th agnitud of th agntic fid. Thus, patics with diffnt onts wi b dfctd diffing aounts by th agntic fid. Whn on pfos th xpint, on obsvs: Ba of Atos N Magntic ont up S Magntic ont down

10 Thus, th agnitud agntic ont of th cton is fixd, and th diction it points is quantid and can tak on on of two vaus. Now, how dos this ad us to concud that th cton has an intinsic angua ontu? Th a two agunts that ad to this concusion: ) Cassicay, agntic onts a aways du to cicuating cunts this is known as Ap s hypothsis. Thus th intinsic agntic ont of th cton ads us to postuat an associatd angua ontu, cad spin. Th fact that th a ony two possib ointations fo th spin ipis that th cton is spin-/, fo thn th two ointations cospond to s ±. Cassicay, on associats th agntic ont with th angua ontu via S. Howv, this tuns out to b c wong fo th cton; a fu ativistic cacuation shows that a ag nub of sa coctions to this foua xist and th agggat ffct of ths ts noais th g ffctiv agntic ont of th cton so that S, c g, o, fo a pactica puposs, g. wh ) Again, cassicay, a agntic ont oving in a potntia xpincs a foc p V ( ). Fo th cnta potntias w a daing with, th gadint of th potntia wi aways point in th diction. Thus th foc is popotiona to p L. Now th diffnt coponnts of do not cout, and so it is ca that if w add th appopiat quantu coction fo th intaction of th agntic ont with th potntia ( ˆ ) angua ontu wi no ong b consvd! This can b aioatd if w assu th cton cais an intinsic angua ontu and that it is th su of th spin and obita angua onta that is consvd.

11 Fo ths asons, w concud that th cton has an intinsic angua ontu of agnitud /. W can thus wok out th coutation ations and ignvau ations fo spin by spciaiing ou gna suts fo angua ontu. Fist, th a th ignvau quations: Sˆ s, s s( s + ) s, S s, s s s, s ±, ± it is convntiona to ak th dfinitions α, +, β. Thn, in th α, β -basis, th spin opatos tak th fo of sip x atics: ˆ 0 ˆ 0 i 0 S ˆ x S y S 0 i 0 0 On can asiy vify that ths atics satisfy th coct coutation ations. Th a a ot of things on can an about quantu chanics vn fo a syst as sip as this. But fo th ti bing, w wi b contnt with ths ationships. How dos a this affct ou pvious cacuations that ngctd spin ntiy? Thankfuy, th ffcts a ath id. To a good appoxiation, w can sipy think of th spin as an additiona dg of fdo. Opatos that act in coodinat spac wi cout with th spin dg of fdo, and vic vsa. Sinc non of ou Haitonian opatos, to this point, hav invovd spin, s and s hav bn good quantu nubs. Hnc, w can sipy think of ach wavfunction as actuay psnting on of two dgnat coponnts that a idntica in coodinat spac and diff ony in spin spac on has spin α, th oth spin β. Howv, ou pvious agunts indicat that th Haitonian fo a cnta potntia shoud contain a t popotiona to g ˆ Sˆ c thus, th is an intaction btwn spin and obita angua ontu. In od to da with this, it is advantagous to fist consid how on das with utip angua onta, in gna. s 3 4, s