Chapter 4. QUANTIZATION IN FIVE DIMENSIONS

Transcription

1 Chat QUANTIZATION IN FIVE DIMENSIONS Th cding dvlomnt ovids a tmndous walth o mathmatical abstactions Howv th sms within it no adily aant mthod o intting th nw ilds I th aas to b no hysical ntity which may b associatd with th nw ild quantitis thn th dvlomnt will hav gon o naught On th oth hand with th notion o nucla ilds in mind it sms that i th nw ild quantitis a includd in a quantizd ictu thn has th lation to nucla ilds may b mad In th ollowing th quimnt o quantization is ovidd by aoiat stictions uon a systm whos dscition is tan om th Dynamic Thoy Howv th us o th ivdimnsional Diac quation has not yt bn shown to sult om th Dynamic Thoy Schoding's quantum mchanics may b obtaind using London's wo but I am not awa o a ocdu to aiv logically at Diac's quation vn though I l that th mthod xists As it now stands th us o th gnalizd Diac quation must b acctd as an indndnt undamntal assumtion Quantization Th systm und considation now is a ivdimnsional systm with ac lmnt ( dq ) (dσ ) Now sinc ou systm is an Econsvativ de systm th incil o incasing ntoy quis that (dq ) > so that (dσ) Intoducing th quantization conditions sults in φ dx πin wh φ l n ± x / and x ct x q x q x x a I w stict ouslvs to a (dσ) sac which is th local Euclidan sac thn (dσ) is th ivdimnsional Minowsity maniold; using London's wo w would oduc a ivdimnsional quantum dynamical systm

2 FivDimnsional Hamiltonian W viously showd that th incil o incasing ntoy sultd in δ ( dq ) as th vaiational incil o a local Euclidan maniold Sinc multilication by a constant dos not chang th oblm w may ta ou vaiational oblm to b δ c ( dq ) Dining th vlocity vcto as u dx /dq and th momntum as L/u g u wh w hav usd th act that g u u thn w may om th contavaiant momntum as g g g l u l so that ( g l u )( g g ( g i c l u l u ) δ l u u ) l g u () sinc c g u u Equation () is th ivdimnsional "momntumngy" quation W may now ollow London's ocdu to obtain ou wav unction o th ivdimnsional systm Howv a quic way to invstigat th ct o th Dynamic Thoy uon quantum mchanics would sm to b that o adoting Diac's quation in a ivdimnsional om and ollowing a dvlomnt analogous to standad oudimnsional lativistic quantum mchanics With this in mind thn w shall adot th om ~ ~ ~ ~ + ~ i + + β x x x x h () to b th ivdimnsional sciic Hamiltonian oato Th atial divativ oatos a sciic oatos and hnc a dimnsionlss in natual units In Eqn () th 's and β do not involv divativs and must b Hmitian in od that h b Hmitian By taing th ou atial divativs in Eqn () as th ouvcto sciic momntum oato w may wit h ( ~ ~ + β ) ()

3 FivDimnsional Diac Equation I w ta > h > and qui that th 's and β a chosn such that solutions o this quation a also solutions o Eqn () w ind th stictions imosd uon th choic o th 's and β to b: ( ~ ) ~ β and ~ ~ ~ β + β ~ () wh natual units c a usd A st o 8 x 8 matics satisying th quimnts o Eqn () is ~ β i β ~ i i ~ i β A A (5) wh I β O I i σ i σ i i and A I I O and i and σ σ σ i I Thn th ivdimnsional Diac quation may b tan to b i Ψ (x)(i ~ ~ β ) Ψ (x) (6) t wh th 7is a oudimnsional oato By dining ~ µ ~ β ; β ~ µ ( µ ) thn Eqn (6) may b wittn as (i +) Ψ (x) By vitu o th otis o th ~ s and ~ β lus th act that o g { o o

4 th anticommutato o th matics must satisy i { } g i "Lontz" Covaianc Und a ivdimnsional Lontz tansomation w shall suos ach comonnt o th wav unction Ψ(x) tansoms into a lina combination o all ou comonnts: wh S is a Diac sino satisying x L Ψ (x) LT Ψ (x ) S Ψ (x) x S S L (7) By using an ininitsimal Lontz tansomation givn by L g + dθ ε (8) wh ε 8 a a st o 6 numbs thn S (θ) may b shown to b givn by wh th matix T is givn by θ S ( θ ) x (T dθ ) (9) T ε Equations (7) (8) and (9) suic to guaant th Lontz covaianc o th ivdimnsional Diac quation Following standad quantum mchanical ocdu w shall adot th obability cunt dnsity to b with th quimnts that: (x) ψ (x) ψ (x) i tansoms as a contavaiant vcto and must b al

5 5 Sin In th thdimnsional sac th angula momntum is givn by th vcto L as th coss oduct o th coodinats and momnta W shall thn din th angula oumomntum to b th oudimnsional coss oduct wh x is th mass dnsity and L ε i x ε i { i any two indics a ali o vn mutation to align indics in ascnding od o odd mutation to align indics in ascnding od Thn th commutato o th comonnts o th angula oumomntum with th sciic Hamiltonian is not zo; o instanc [ L h] i i + i i + i i Now suos th xists a ousin vcto S such that th sum o th angula oumomntum and th ousin vcto commuts with th sciic Hamiltonian; thn i w din a nw thsin vcto u givn by th comonnts u i i u i / and u i / and ta th usual sin vcto s givn by s i / s i / and s i / th comonnts o th ousin vcto may b shown to b S s u u S s +u u S s +u+u and s s + s S In analogy with standad lativistic quantum mchanics th ignvalus o th ousin comonnts can b shown to b + _ 6 It may also b shown that th st o obsvabls P h and S wh P is th oumomntum and S is th ousin om a comlt st o commuting obsvabls

6 6 Diac Equation with Filds In analogy with lativistic quantum mchanics w ta th ivdimnsional Diac quation to b wh ϕ is ivvcto otntial By oating on th lt with i [(i φ ) ] 8 and saating 9 into symmtic and antisymmtic ats as [(i φ ) +] Ψ () i i i { }+ [ ] g + σ thn Eqn () bcoms Saating φ [(i into symmtic and antisymmtic ats as and dining th ild tnso as φ )(i φ )+( + φ φ iφ i φ ) σ ] Ψ () φ ( φ )+ ( φ φ ) F φ φ Eqn () bcoms [(i φ )(i φ ) if σ ] Ψ (8) Now sinc x& x& x& x& x& i s i s n σ x& i s i s n x& i s i s n x& n n n wh σ n and 5

7 F E E E V V V V lus calling th svn Maxwllty quations om Eqn (5) E B B V E + + a c t E B B E B B V V + t V V V V π J c V E π J V B a c t c B xv + a x B B x E + c t V E πρ a () thn Eqn () may b wittn as and thus bcoms th Diac quation with ilds E B V and V Suos w consid a systm without an lctic chag so that J thn by Eqn () w still hav and tho th will still b a magntic momnt 7 Allowd Fundamntal Sin Stats In th ivdimnsional quantization o th sactimmass maniold th sin vctos aa On o ths is th amilia thcomonnt sin vcto o lativistic quantum mchanics Th scond o th th is a nw thcomonnt sin vcto whil th maining on is a oucomonnt sin vcto Using th thom: [(i φ )(i φ ) + B s ie x iv x& in V ] Ψ V E V a and x B a c t E I satisis a wh a is a numb thn th ignvalus o a ±a Thn it is not diicult to show that th comonnt ignvalus a s + _ u + _ S and 6

8 I in analogy with th ignvalus o th total angula momntum w wit S S ( S +) thn th ossibl ignvalus bcoms s + _ u + _ S Howv th ollowing lations which w shown to b quid o S to commut with th sciic Hamiltonian stict th numb o ossibl combinations o ths ignvalus S s u u S s +u u S s +u+u and s s + s S Th qustion to b asd now is how many combinations o th abov ignvalus a allowd? Fo S / th combination s / and u / is imossibl Fo S / th combination s / and u / is imossibl Fo S / th combination s / and u / is imossibl Fo S / th combination s / and s / is imossibl Fo S / only on combination is ossibl: s / u / and u / Fo S / only on combination is ossibl: s / u / and u / Fo S / only on combination is ossibl: s / u / and u / Fo S / only on combination is ossibl: s / s / and s / Now bcaus S is a combination o th ist tms o ach o th comonnts S S S not all o th abov listd 6 combinations a ossibl Fo S / th ollowing combinations o (s s s u u u ) a ossibl 7

9 () ( ; o S S S () ( ; ) o S S S () ( ; ) o S S S () ( ; ) o S S S Th maining combinations a: (5) ( ; ) o S S β; S S (6) ( ; ) o S S β; S S (7) ( ; ) o S S β; S S (8) ( ; ) o S S β; S S Thus th is an octt o ossibl combinations Th a also som obvious symmtis in ths combinations An aid in sing ths symmtis is th vcto dind as t wh Thn o ach o th ight combinations abov w ind (t t t ) givn by () t () (5) t ( ) () t() (6) t ( ) () t () (7) t ( ) () t () (8) t ( ) Thus th ight combinations cosond to ou distinct t vctos which cay a ± sign O t ( ) ;t ( ) ; t ( ) ; t ( ) Fo + t w hav: t (s;u) ( ; ) t (s;u) ( ; ) t (s;u) ( ; ) t (s;u) ( ; ) Fo t w hav: t (s;u) (su) ( ; ) t (s;u) (su) ( ; ) t (s;u) (su) ( ; ) t (s;a) (su) ( ; ) Now by dining th vctos: a ( ) ; b ( ) c ( ) ; d ( ) t ( u +u ) ; t u u ; t u +u 8

10 W may wit t (s;u) (a;b) t (s;u) (c;d) t (s;u) (d;c) t (s;u) (b;a) t (s;u) (a;b) t (s;u) (c;d) t (s;u) (d;c) t (s;u) (b;a) Th octt is thn mad u o th combinations: (a;±b); (c;±d); (b;±a); (d;±c) Th aaanc o octts o basic quantum numbs is miniscnt o lmntay aticl thoy Thus th Dynamic Thoy sms to giv omis to th ho o tying lmntay aticls to undamntal incils in a nw way B Quantizd Filds Much diiculty was ncountd in tying to ind a solution to th wav quations This stimulatd a tun to thoughts o undamntal aticls Th motivation o this chang was imaily th ling that it would b mo oductiv to gt away om th wav solutions o a whil but also th was th haunting ling taind o som iv yas that th nw ilds layd a ol in aticl stuctu This ling was basd imaily on th ol th nw ilds aa to lay in th ivdimnsional quantization and thi ol in th slngy o chagd aticls 8 Quantum Condition Alid to Paticls Th quantum condition φ dx πin n () was quid whn gnalizd isntoic stats w considd Givn th thmodynamic basis o th th undamntal laws it sms natual to thin that i th Dynamic Thoy w to say anything about undamntal aticls thn it should obably com om considing gnalizd isntoic stats Thus th quantum condition Eqn () should lay a cucial ol This also was th condition om which London bgan his wo which showd that this condition oducs quantum mchanics but quantum mchanics dscibs intactions btwn aticls such as lctons and nucli It dos not sciy what tys o aticls a allowd 9

11 That th th adotd laws must aly to individual undamntal aticls is tantamount to th notion that ths th laws must sciy what aticls a allowd and thby must sciy thi allowd ilds I w again loo at th quantum condition Eqn () w s that it is givn as a lin intgal that must hav a quantizd valu Ou usual ist ncount with a lin intgal involvs th valuation o a givn lin intgal whn th ath is sciid Bcaus th quantum condition snts a lin intgal that may only hav ctain valus London asd a lgitimat qustion whn h asd what aths would b allowd givn th lctostatic otntial in advanc 8 This oints out that th a th ats to any lin intgal th intgand th ath and th intgal valu Anoth qustion that may b asd o th quantum condition is givn that th intgal valu may only b πin what a th ossibl ϕ allowd o a aticl that must tain its idntity along any ath? This is quivalnt to asing what ilds a undamntal aticls allowd to hav i w a to mov thm anywh in th maniold? To b mo sciic w a asing what θ a allowd by th quantum condition i th dx a to b indndnt? I th dx a to b indndnt thn w may choos all dx to b zo xct dx Thn th quantum condition quis φ dx πin (no sum on ) (5) Equation (5) must b tu o all and bcaus w a to st th ath thn th ϕ must lct th quantization sntd by th intg N Tho ~ φ N φ (no sum) wh th may not b quantizd Thus Eqn (6) snts th ist sons o th quantum condition to th qustion concning what ϕ a allowd o undamntal aticls; th gaug otntials must b quantizd Th dinition o th gaug otntials is wh was th gaug unction Th ild tnso was dind by wh covaiant dintiation is quid Th a stictions lacd uon ths ilds o thy must oby th st o ight dintial quations givn by Eqn (5) (6) ln φ (7) x F φ φ (8)

12 9 Radial Fild Dndnc ~ Any otntial φ N φ 6 allowd by th quantum condition must also satisy Eqns (5) Evn so Eqns (5) (7) and (8) snt th stags o dintiation stating with th gaug unction In looing o th stictions Eqns (5) lac uon th quantizd otntials w may mloy a tchniqu o mathmatics in th solution o dintial quations W ty to ind a solution in th om o th oduct o unctions o th saat vaiabls Howv ou tial solution must oduc otntials o th om in Eqn (6) Tho suos w ty to ind a solution o th om ln FG (9) wh F t θ φ with t unction o tim only unction o shical adius only tc and th unction G is dind by th systm o atial dintial quations G ( N G) F x F x (no sum) Th dinition o th gaug otntial using th tial solution o Eqn (9) now oducs φ ln x (FG) F G G + F x x x but by th dining lations o G this bcoms F F F φ G +( N G) N x x x (no sum) I w din φ ~ F x thn w hav th o om ~ φ N φ (no sum) () W may now us ou tial solution to wit th otntials in shical coodinats:

13 F N a N a ln a F sin N sin N ln sin F N N ln F N N ln t t t t φ θ φ θ φ θ φ θ φ φ θ θ φ θ φ θ φ φ wh th notation t 7 dnots d t /dt and F dnotd d df F d df F d df F dt df F φ θ and d df F Substituting th otntials givn by Eqn () into th dinition o th ild tnso and using th quid covaiant dintiation w obtain th ild comonnts F ic N ic N (ilc) ln t φ θ φ ()

14 ( N N ) E F c ( N N ) Eθ F c ( N N ) Eφ F csinθ a ( N N ) F V c ( N N ) F B N cotθ F sinθ () ( N N ) N F Bθ F sinθ ( N N ) N F Bφ F V ( N N ) a F V θ ( N N )a F and V φ ( N N ) a F Ths ild comonnts lct th quantization o th otntials Howv th quantization o th ilds is not a siml quantization bcaus ach comonnt dnds uon th dinc o quantum numbs Th ild comonnts givn by Eqn () must satisy th dintial quations o Eqn (5) Tho i w substitut Eqn () into Eqn (5) w will obtain th stictions uon th quantum numbs N and th unctions t θ ϕ and quid o ths ilds to b th ilds o a undamntal aticl W bgin with th quation This quation bcoms B (5a) B (sinθ Bθ ) + + Bφ sinθ θ sinθ in shical coodinats Substituting om Eqn () and simliying w inally aiv at sin Θ This quis that {( cotθ( + [( +( ]} N N ) N )+ F N N ) N N N ) N ( N N ) N cot ( F + F F [( N N )+( N N ) N ] θ Substituting om th dinition o th F w ind

15 I 78 this may b wittn as t φ [ F φ + ] [( N N ) N +( N N ) N ] ( N N ) N cot θ t θ t θ φ t θ φ Howv w may divid by cot and saat th quation into [ + wh ( ) [( +( ] N N ) N cot θ + N ( θ θ N ) N N N ) N ] θ () cotθ θ ( N K N ) N +( N N ) N ( N N ) N Th lthand sid o Eqn () is a unction o only whil th ighthand sid is a unction o θ only Tho Eqn () must b a constant W can thn wit [ + ] θ n () cotθ θ wh th constant n dnds uon th st o quantum numbs N N and N o th aticl Thus n dnds on th aticl und considation Th adial quation in Eqn () may b intgatd immdiatly with th sult R n (5) Th aaanc o this xonntial unctional om o th adial dndnc is suising and at ist lasing Th suis is that this unctional om coms only om th gaug unction laying th guiding ol in th gaug ilds and th ild quation B which is a uly classical quation Thus th xonntial nocoulombic adial unction dos not aa at ist to dnd uon th ith dimnsionality but only uon th quantum condition so that vn a oudimnsional aoach would hav oducd this sam adial unction

16 It is lasing to s th aaanc o th xonntial nocoulombic adial unctional o Eqn (5) bcaus th lcton catastoh has hauntd thoticians sinc th invs adial dndnc o th columbic otntial was ist sn Th adial unction in Eqn (5) is wll bhavd vywh; as and as A quic glanc at th unction might caus on to thin it is th Yuawa otntial but a clos loo will show that th xonnt is th invs o th xonnt in th Yuawa otntial Th valu o θ may b obtaind by intgating th maining otion o Eqn () This intgation oducs θ ( sin θ ) I th xonntial nocoulombic unction cosonds to ality thn n must b small lss than 7 m o lctons and o th od o magnitud o 5 m o otons Tho must b vy small which in tun imlis θ is vy clos to a constant Th quations sulting om substituting th quantizd otntials into th maining nonsouc quations [5b] [5] and [5] oduc th ollowing stictions ( N N ) N φ [5b] ( N N ) N φ ( N N ) N θ ( N N ) N t φ [5] ( N N ) N t φ ( N N ) N t θ [5] (satisid idntically) Whn th otntials a substitutd into th quations with souc tms Eqns [5c] and [5h] th sulting quations a vy comlx To duc th comlxity o th quations th assumtion was mad that all souc tms w zo; that is ρ ; J ; J This assumtion ducd th comlxity somwhat but still lt a systm o quations that thus a is unsolvd Howv an intsting asct o this assumtion is th ossibl xistnc o a adial lctic ild without th snc o any lctic chag within o uon th aticl This ossibility sts uon th ssu o th tm V / 86 in th Eqn [5d] Much was land about th intaction o chagd aticls by considing only th adial dndnc o th lctic ild whil tmoaily nglcting th magntic ild o any otntial vaiation o th lctic ild with azimuthal angls This latt is th shically symmtic ild assumtion Having not yt obtaind a comlt solution to th systm o quations that is th sult o substituting th quantizd 5

17 ilds into th ight ild quations it ovd bnicial to ma th assumtion o shically symmtic ilds in which th only vaiation o th ilds is th adial dndnc sciid by th nocoulombic adial unction Thn i w want to xlo th adial dndnc o static ocs btwn th undamntal aticls allowd by th quantum condition w must consid th oc law K F c J whos satial comonnts may b wittn J K E + ( J x B )+ ρ V c c By sticting ou concn to static ocs w can concntat on th oc dnsity J K ρ E + V (6) c Thus th adial dndnc o th lctic ild E and th V ild a all that nd to b considd at th momnt Substituting th adial unction Eqn (5) into th ild xssions Eqn () w ind Z n E n (7) and Wg n V n wh Z (N N ) and W (N N ) so that th quantum numb Z aas in th adial lctic ild th sam as it dos classically Fom Eqn (7) th lctic ild o undamntal aticls allowd by th Dynamic Thoy is quantizd by th quantum condition and th quantum sts may only b intg sts This would ncssaily clud any aticls with actional chag sts Substituting th adial ilds o Eqn (7) into th oc law Eqn (6) and intgating th chag dnsity ov th hysical xtnt o th aticl w ind th adial oc btwn two aticls is q Z n n g Wg n n F + (8) wh J q ρ d(vol) g d(vol) c I w consid th lctic oc in Eqn (8) and stict ou attntion to such that >> n thn th lctic oc bcoms th columbic oc 6

18 q Z F E Futh th oth oc tm basd uon th V ild may b sn to also hav th sam longang om g Wg FV Thus this oc is also an invs squa long ang oc Tho th nonlctic oc in Eqn (8) cannot b a nucla oc What thn must b th inttation o this oc o must its aaanc b inttd to man that natu cannot hav a ivdimnsional chaact? Th only oc nown in natu that has a longang chaact in addition to th lctostatic oc is th gavitational oc But how can w intt th V ild as th gavitational ild whn Einstin showd that th gavitational ild could b xlaind by a vcto cuvatu in a oudimnsional maniold and th V ild is a gaug ild in a ivdimnsional maniold? I th V ild w to b considd as gavitational thn th bnding o light aound th sun dictd by Einstin's Gnal Thoy o Rlativity must hav anoth xlanation Is this ossibl? I th V ild is gavitational thn is th oom in th Dynamic Thoy o an xlanation o nucla hnomna o must it also ollow th cunt aoach to nucla hysics thby quiing simila additional assumtions? Ths and oth qustions occu whn w s th longang chaact o th V ild oc It is this thotical quanday sntd by th V ild that soils th lasant suis o sing a nonsingula lctostatic ild mg om th quantum condition A numb o ossibilitis aa dndnt uon th answs obtaind to th vious qustions and/o oths Pimaily th ossibilitis tain to th validity o th ivdimnsional viw o it is om this ivdimnsionality that th V ild coms On asonabl aoach in attmting to ind a ossibl way out o this quanday sms to b to suos th gavitational inttation is a ossibility and thn s how th Dynamic Thoy comas with masuabl ximntal vidnc SlEngy o Chagd Paticls On o th diicultis in Maxwllian lctomagntism is th ininit slngy that is dictd o a chagd aticl This "lcton catastoh" o singulaity dos not xist with th nonsingula nocoulombic ild and th slngy o a chagd aticl may b ound In classical lctomagntic thoy th slngy o a chagd aticl is discussd but its valu has not bn stablishd This is 7

19 bcaus th xssion o th slngy is a unction o th adius associatd with th hysical xtnt o th chag distibution Thus th adius o th chagd aticl must b nown bo th slngy can b dtmind Cuntly th slngy o a chagd aticl is quatd with th ngy associatd with its intial mass by E mc Thn th adius associatd with its ngy is tan as th "adius" o th aticl Th is no intntion that this adius b th hysical adius o th aticl though it comas avoably with ximntal valus Th qustion aiss h o whth o not th Dynamic Thoy with th ivdimnsional viwoint can thotically dict th slngy and/o th adius o th hysical xtnt o th mass o chag distibution o th aticl On o th bnicial ascts o th gnalization o hysical thoy as don in th Dynamic Thoy is th ossibility o using conctualizations and ocdus dvlod in on banch o hysics in anoth banch This asct o th thoy aas alicabl h Th slngy o a chagd aticl is th notion that a ctain amount o ngy b associatd with th xistnc o th aticl and its chag This notion may b associatd with th notion o ngy usd in thmodynamics o i th slngy o th chagd aticl is its ngy thn it snts th ngy which may b "d" uon convting th aticl into ngy Convsly this would snt th ngy quid to assmbl th chagd aticl With th conctualization o ngy th scond law ovids th condition o a stabl quilibium stat namly that a chagd aticl in an quilibium stat must xist at a minimum o its ngy Thus i th slngy o ngy o a chagd aticl is sought thn minimizing its ngy will yild th dsid sult Th ngy was dind in analogy with th thmodynamic cas as wh dnds uon th alicabl wo tms which h will b tan as th th satial dimnsions so that Th ist law is givn by whil th scond law yilds G U φs x F (9) de du o a quasistatic vsibl ocss Tho th dintial chang in th systm ngy is 8 F φ ds du F dx dx du φ ds + F dx ()

20 Dintiating Eqn (9) givs th dintial chang in th ngy as Substituting Eqn () into () yilds dg du φ ds Sdφ F dx x df () dg Sdφ x df () Th oc in Eqn () is considd to b th Lontz oc so that Eqn (8) bcoms I w wish to consid th chang in ngy with sct to a chang in th chag at a constant vlocity w ind that sinc ρ is a unction o vlocity only dρ Th sciication o constant vlocity stms om th dsi to obtain th slngy o a chagd aticl; tho th aticl should b considd as sitting still so that it will hav no intic ngy Th dintial chang o ngy o a stationay aticl is thn so that o ρ constant but is indndnt o th chag q and tho I th chag is not in motion thn F q[ E +( vxb ) ] dg Sdφ x d{ q[ E +( vxb ) ] } dg Sdφ x d{q[ E +( vxb ) ] } Sdφ x {dq[ E +( vxb ) ] + qd[ E + [( vxb ) ] } G q φ x [ E +( vxb ) ] G q φ E +( vxb ) [ E +( vxb ) ] x q q x [ E +( vxb ) ] φ G q φ x E () 9

21 sinc v I G is th slngy o a chagd aticl thn by Eqn () G q E th chang in th slngy is givn with sct to a chang in th chag q I w assum a shically symmtic chag dnsity ρ thn dq ρ dv π ρ d W may thn ind th ngy by th intgation G dg R π E ρd wh R snts th adius within which th chag dnsity ρ is containd Th ild Eqn [5d] is q G q E dq ( ε V ) ( ε E ) πρ a Th nti ighthand sid o this ild quation bhavs as a chag dnsity; tho w may qually wit ( ε E ) πρ () wh it is undstood that ith ( ε V )/ 9 is zo o ρ is considd to b a total ctiv chag dnsity In ith vnt Eqn () givs us ε ( E ) πρ whn w consid only a adially symmtic ild E Thus Substituting Eqn (5) into Eqn () th slngy is thn ound to b I w now us th nocoulombic lctic ild givn by ε d d( E ) ρ (5) π ε ( E )d( E )+ G (6) G

22 E πε in th intgal o Eqn (6) w ind R πε d πε +G G which may b intgatd so that + (π ) ε R R R G R +G (7) To ind th sciic valu o th slngy w must ind th R that minimizs G Tho st G R At caying out th quid dintiation and simliying this is satisid i Equation (8) only has on ositiv oot which is R + R (8) 5 5 R Substituting this sult into Eqn (7) th slngy bcoms (679)+G (π ) ε G o 765 x MV mi +G (mi) G (9) whn is givn in units o mi An xaml may b a oton o which is aoximatly mi i th otonoton scatting data is considd Thn i ~mi so that G 765 x MV + G o G o 986 MV ()

23 is th at o th oton st ngy indndnt o its chag Th chag ngy o th oton would thn b G c 765 V which is ngligibly small comad to th nonchag ngy G o What is th natu o th ngy G o? It is not ngy causd by th snc o lctic chag on th otons Also th slngy G was ound o a sting aticl I w associat th sting slngy G with th st mass as G m o c G c + G o and G c is th otion o th oton's st ngy that is du to its chag thn G o must b that otion o th st ngy that is du to th oton mass abov In this cas th oton mass ngy G o is givn by Eqn () Suos w consid an lcton and assum that ~ mi Thn G 5 MV 765 MV + G o th mass ngy o th lcton would thn b whas its chag ngy is G o 777 MV G c 765 MV Nucla Phnomna Th lctostatic oc aaing in Eqn (7) dis signiicantly om th columbic oc only whn bcoms small nough to b o th od o magnitud o th n Th ist ximntal vidnc that th scatting o chagd aticls by oth chagd aticls was not always columbic was th Ruthod scatting data Th aaanc o th xonntial multili in th nocoulombic oc o Eqn (7) omts us to as whth o not th dinc btwn this oc and th columbic oc suics to xlain nucla hnomna without soting to th ostulation o a nw shotang oc such as th nucla oc An obvious stating oint to xlo th ossibility that th nocoulombic oc might aly to nucla hnomna would obably b th

24 Ruthod scatting omula This may b don; howv th aaanc o th xonntial tm mas an analytical xssion diicult i not imossibl to obtain W may aiv at a solution o limitd usulnss i w assum that >> Futh in considing aticl scatting w shall stict ou considation to scatting o li aticls only so w a guaantd that only on is involvd Th bst way to invstigat th scatting coss sctions is to stat with th solutions o th quations o motion o lantay obits in which th oc is givn by th nocoulombic oc instad o th siml invs oc om Nwton's gavitational oc W ind th adial quation bcoms wh u / is th gavitational constant and L is th obital angula momntum Th xonntial unction may b xssd in tms o a ow sis and ou adial quation bcoms by assuming >> so that / u << thn w may nglct th tms with n in ou adial quation Th sult o this assumtion is with This quation may b comad with th classical quation o with th gnal lativistic quation 6 which has th idntical om o ou quation Thus th sam mthod o tubations may b usd to obtain a solution as was usd o th lativistic cas Th sult o this calculation is th solution u d u M +u ( u) d θ L d u M (u ) (u ) +u ( u) +( u)+ ++ d θ L! n! M (n+) n + (n) ( u ) L n! u M M + u + u d θ L L d M + L d u M +u d θ L u M +u + Mu d θ L d n +

25 M [ + cos ( θ )] θ δ θ () L u wh M θ δ θ L is th incas in th ihlion Notic w hav shown that th nocoulombic oc will dict an advanc in th ihlion o lantay obits with th solutions to ou lantay obits quation W will discuss this uth at a lat tim W ndd th solution givn by Eqn () in od to obtain an xssion o th scatting coss sction o li aticls I w now consid th solution Eqn () obtaind with th assumtion that >> thn th scatting coss sction may b xssd as q q π θdθ dσ sin mv θ sin δ () wh E θ ( π θ ) θ +6 + sin tan δ E θ + sin sinθ( π θ ) Th aaanc o th acto δ xsss th istod dviation o th scatting coss sction o th nocoulombic oc om that o th columbic oc Howv th assumtion that >> imlis a limit on th minimum imact aamt o which this coss sction tains validity Tho a comut solution is obably ncssay to ally invstigat th scatting o chagd aticls using th nocoulombic oc Figu 6 Comaison o coulomb and nocoulomb ocs at shot ang Anoth way o visualizing th nocoulombic oc is to ma a lot o it and coma it with a lot o th columbic oc Figu 6 comas ths two ocs lottd with th saation vaiabl in mions and nomalizd so that th columbic oc at onmion saation is unity Not that this lot comas th ocs o li aticls to nsu that is th sam o both aticls Figu 6 shows that th nocoulombic oc

26 is vitually indistinguishabl om th coulomb oc o saations gat than aoximatly Howv at a saation o xactly th oc is idntically zo In tms o th classical notion o nucla ocs w would say that at saations gat than th nucla oc is ngligibl whas at a saation o th magnitud o th nucla oc was qual to th magnitud o th coulomb oc Th nocoulombic oc bcoms an attactiv oc o saations lss than This is xactly th bhavio to b xctd o a nonsingula otntial Fo a otntial to b nonsingula it must tnd to zo as gos to zo Such a otntial which tnds to zo o tnding to zo and o tnding to must hav a maximum absolut valu in btwn At that maximum th oc bing dtmind by th slo o th otntial will go to zo and will b o th oosit signs on ach sid o th zo Now lt us loo at th oc btwn unli aticls say a oton and an lcton Consid th lcton and oton to b lacd on a hoizontal suac saatd by a distanc with th oton to th ight o th lcton Thus th longang attactiv ocs btwn ths two aticls will caus th oton to xinc a oc to th lt whil th lcton will xinc a oc to th ight W may than wit th oc on th oton that is du to th ositiv chag o th oton bing in th lcton ild as F q E () ( uˆ x ) wh th lcton ild involving th lcton lambda has bn accountd o Th lcton oc owing to th lcton chag bing in th oton ild is givn by F () ( uˆ x ) Figu 7 lots both ths ocs as a unction o th saation wh 5 m o mi has bn assumd Th lctonlcton scatting data show that th lctonlcton intaction bhavs in a coulombic mann vn whn saations a aoximatly m To b consistnt with this data w hav assumd mi Fom this lot o th oc on th oton and th oc on th lcton w s that o saations lss than about mis th ocs bcom xtmly unsymmtical This immdiatly and visually dmonstats that th nocoulombic xonntial oc violats Nwton's 5 q E

27 thid law quiing that th oc on th oton b qual in magnitud and oosit indiction to th oc on th lcton Th qustion aiss whth o not a violation o Nwton's thid law has v bn sn as th sult o an intaction btwn an lcton and a oton? Th answ basd on a nuton disintgation om which a oton and lcton mg is dinitly ys; Nwton's thid law was sn to b violatd To instat Nwton's thid law in nuton disintgation and all oth bta dcay Pauli ostulatd th xistnc o th nutino Fmi lat dvlod his thoy o wa intactions om which aad th ncssity to tal o a outh oc in natu Can it b that th nocoulombic oc which quis distinct o distinct undamntal aticls accounts o th action o th wa ocs also? Th ossibility that it might ons th thotical lood gats and a vitual tidal wav o qustions sugs oth Dos this man th nutino dos not xist? What about th ximntal vidnc submittd in suot o th catu o a nutino? Could this man that th nuton might b bound stats o an lcton and oton? This qustion should b ollowd by what about consvation o angula momntum in nuton dcay (i sin) consvation o lina momntum and Hisnbg's unctainty incil? Figu 7 Nocoulombic ocs btwn unli aticls at shot ang Th cding qustions do not bgin to scatch th suac o th thotical qustions that nd to b answd as th sult o considing th ossibility that th oc law o Eqn (7) with only a gavitational oc lus th nocoulombic oc might xlain th hnomna now thought to qui ou distinct ocs in thi xlanation Howv th aaanc o a nonsingula oc with th aant ang o th nocoulombic om cannot b thown out ohand Tho it sms that th only asonabl choic is to systmatically and thooughly xlo th ossibilitis I w again consid th lots o th oton and lcton ocs in Fig 7 w s that at atomic saations and gat distancs th ocs oby Nwton's thid law and th dinc btwn th nocoulombic and columbic ocs is so small that it could not b dtctd in atomic o macoscoic hnomna But as th saation bcoms small th ictu bgins to chang Whn th aoachs th lcton is no long attactd to th oton as stongly as th oton is attactd to th lcton I th saation is xactly thn th lcton is indint to th oton's snc Th oton on th oth hand is still vy much attactd to th lcton I o th momnt w igno th inttation o Hisnbg's unctainty incil that would say it cannot b thn w could asily imagin a cicula oton obit aound a stationay lcton duing which th oton stays at a adius o om th lcton Th 6

28 lcton should b stationay duing such motion bcaus it would xinc no oc W now consid a saation btwn th lcton and oton which is som siml action o H w ind th lcton ulsd by th oton but th oton is still attactd to th lcton Notic that th oc on both aticls om ou initial ositioning o th oton on th ight is to th lt I both aticls w givn an angula momntum such that thy w lacd into synchonizd cicula obits thn bcaus thi synchonous motion always sults in th oc on both aticls bing dictd along th lin saating thm and om th oton towad th lcton o om th lcton away om th oton thn again ignoing agumnts om th unctainty incil cicula obits in which th lcton is in a small obit about a sac oint could b imagind wh th oton is in a much lag obit about th sam sac oint Lt us ollow this ictu a littl ath and wit siml Nwtonianli oc laws o this situation Th situation nvisiond is sntd in Fig Th lcton osition is givn by om th oigin and th osition o th oton is givn by Th saation btwn thm is (5) Bcaus th oc is always dictd along th lin saating th two aticls w may wit th adial quation o motion o th oton as m v (6) wh th assumd cicula motion has bn tan into account and v is th tangntial oton vlocity Th lcton quation o motion is givn by v m (7) Figu 8 Elcton and oton obits In both quations < Th ighthand sid o Eqn (6) and (7) a both unctions o th saation whas th two lthand sids a individually unctions o and A solution is ossibl only whn th th quations Eqns (5)(7) a solvd simultanously An altnativ aoach is to add Eqn (6) and (7) to obtain th quation o motion o th cnt o mass m v v + m 7 (8)

29 o MV R wh R (m + m )/(m + m ) and M m + m Fom Eqn (8) w s that bound stats wh th cnt o mass is in motion as th sult o th asymmtical oc may only occu whn th saation is lss than All o ths quations o motion xhibit a atu not usually ound in quations o motion That is bcaus th oc dnds on th saation btwn th aticls and not stictly on th osition o th lction thn th usual intgation o th oc ov a chang o osition which oducs th otntial ngy cannot b adily don bcaus V( ) F d K (9) d Howv Eqn (5) and Eqn (7) may b usd to obtain as a unction o o vic vsa so th intgation o Eqn (9) may b comltd Th tanscndntal unction in th oc law ohibits an analytical solution o as a unction o Tho only numical o gahic solutions o ths quations a ossibl Hisnbg's Unctainty Pincil and Gomty Th suggstion that bound stats o lctons and otons might xist wh th obits a o th aoximat od o magnitud o nucla dimnsions is ssntially a tun to th notion that a nuton might b such a stat This ida gav way und agumnts o consvation o momntum and Hisnbg's Unctainty Pincil to th viw that lctons a obiddn to b ound within th nuclus Tho lt us ta anoth loo at thos undamntal tnts o quantum mchanics th Poisson bacts Th classical Poisson bact is dind by F q { FG} wh F and G a any two unctions o th canonically conugat vaiabls q and Th scial lations that occu whn F and G a q and sctivly a scially imotant in quantum mchanics; ths a classically: 8 G F G q

30 {q q } { } (5) {q } wh δ is th Konc dlta Th classical Poisson bacts o Eqn (5) a obtaind whn Euclidan sacs a assumd Howv th dinition o Poisson bacts mains valid o gnal mtic sacs whn th notion o covaiant dintiation is usd I w now consid th momnta xssd in a gnal coodinat systm th covaiant comonnts mg x& i and (5) g l m x& l a th contavaiant comonnts Covaiant dintiation must b caid out with sct to contavaiant vcto comonnts Th in a gnal sac th canonically conugat vaiabls to b considd a x and and th Poisson bact o th osition and momnta bcoms { x } x x n + x + l l l l x s l x n l s δ l + x δ l s l (5) o s { x } δ + x Quantum mchanics adots th oato s _ i x o th momntum This in gnal cas bcoms th covaiant oato _ i ( ) (5) Th oato o th contavaiant momntum comonnts is thn 9

31 Now i w loo at th quantum Poisson bact wh th oatos a oating on a scala Ψ thn This may b wittn in tms o th classical Poisson bact Eqn (5) as I th sac is Euclidan thn th g l bcom th Konc dlta and th Chistol symbols vanish and th quantum Poisson bact o Eqn (55) bcoms bcaus g l l δ l l Howv om Eqns (5) and (5) w s that th mtic dos lay a ol in th quantum oatos This should also b sn in th us o th oatos in th Schoding Hamiltonian oato bcaus bcoms th oato to b usd in a gnal sac and o cous is th oato usd in alying Schoding's quation to th hydogn atom ( ) g i _ l l (5) x s l + g i x x g i _ x s l + x x i _ g x g i _ x )l x ( g i _ x g i _ x ] x [ s l l l l s l l l l l l l ψ δ ψ ψ ψ ψ φ ψ h (55) [ ] { } x _ g i x l l ψ ψ [ ] i_ x ψ δ ψ ( ) x g g x g _ x g i _ G x g i _ g x g i _ l l l (56)

32 Th gomtical ct may b sn also in Diac's quation by considing that th stictions ( ) β and β + β must b mt in od o solutions o (57) o > H > wh is also a solution o m in natual units Th ist stiction may b wittn as by th dinition o th momnta Thn i w xand th lthand sid and quat coicints o th w ind that and H ( ) i ( + βm) m x& x& g () g () g () g (77) Fom Eqn (77) o a Euclidian mtic wh g δ ths stictions duc to th usual stictions Any mtic otis will act ths stictions and will tho d into th solutions Now o what bnit it this discussion o gomtical ct uon quantum mchanics in considing th nocoulombic oc? Rcall that th nocoulombic oc cam om a gaug unction in a Wyl sac A gaug unction has a gomtical ct that could b thought o as ctivly changing th unit o action in quantum mchanics To s th basis o this statmnt lt us call th quantum Poisson bact oations on a scala g + + { } g { } g { } g + +

33 l s [ x ] ψ i _ g δ l + x ψ s l (7) and lt us din l s _ δ _ g δ l + x (78) s l thn w can wit [ x ] i _ δ ψ (79) which has th sam om now usd but th ctiv unit o action _ 58 dnds on th gomty as sn by Eqn (78) W may loo at th ctiv unit o action in yt anoth way Rcall om th incil o maximum ntoy that th gnalizd ntoy is th action Thus quantization o th action is a quantization o th gnalizd ntoy But bcaus th ntoy sac is tid to th sigma sac w hav ( dq ) (dσ ) Th gaug unction is a unction o th sac oint; tho th gaug unction vais continuously om oint to oint in th sac Thus i th gnalizd ntoy is quantizd so must b σ W may wit q n_ σ n_ wh th dinc btwn _ 6 and _ 6 contains th gomtical dinc btwn q and σ But how can w dtmin th lationshi btwn _ 6 and _ 6? Fom th incil o maximum ntoy th otntial ngy unction was dind as th ngativ intgal o th oc though a distanc ust as it is in classical mchanics bcaus dv F dx but th otntial ngy lays th ol o th gaug unction Th quations o motion o th asymmtical ocs btwn unli ocs showd that an analytic om o th otntial ngy may b unobtainabl owing to th tanscndntal natu o th ocs and bcaus th ocs dnd uon th saation btwn th aticls not thi ositions Thus th aas no way at th momnt to obtain an analytical xssion o _ 66 and w must sot to a numical solution o th unli aticl cas

34 Th absnc o an analytical xssion o th ctiv unit o action _ dos not comltly sto us om considing th ossibility that a nuton may b a oton in a lag obit about an lcton in a small obit W may o th momnt acnowldg th diiculty o obtaining an analytical xssion o _ by allowing th _ 5 o unit o action o th oton and lcton to b a unction o thi obit and w may dsignat _ 6 to b th ctiv unit o action o th lcton obit in a nuton and _ 7 to b th unit o action o th nuton's oton obit I th ctiv unit o action dnds uon th obit as it aas h that it must thn th inttation that Hisnbg's Unctainty Pincil uls out th ossibility o an lcton bing containd within nucla discussions is inalicabl Anoth agumnt against th nuton bing an lcton and oton in nuclasizd obits is basd on an agumnt that th incil o angula momntum cannot b consvd Th nocoulombic ocs which qui that th oc btwn th lcton and oton b dictd on a lin btwn thm quis that angula momntum b consvd Howv th ctiv unit o action o th lcton obit quis that in th nuton th obital angula momntum would b givn by _ 8 and its intinsic sin angula momntum would b _ 9 Similaly o th oton th obital angula momntum would b _ and th sin ( )_ At th nuton dcays th angula momntum is th sum o th two aticls' intinsic sin angula momnta which is givn by _ bcaus both aticls a and tho ach has an intinsic sin angula momntum o ( )_ Tho th consvation o angula momntum is xssd as ( + _ + ) + _ + _ (8) Eximntal vidnc o obital and/o sin angula momntum is containd in th ximntal magntic momnts I w quat th intinsic and obital magntic momnts o th lcton and oton whil thy a in th obital coniguation to th ximntal valu o th nuton's magntic momnt w hav wh µ β 6 is a Boh magnton and µ n 7 is a nucla magnton Equations (8) and (8) snt two quations in th two unnowns n 8 and _ 9 which may b solvd to obtain th ctiv units o action o th lcton and oton obits maing u a nuton such that angula momntum in consvd duing nutons' dcay and that th coct magntic momnt o th nuton is nsud Substituting th ximntally masud valus o intinsic magntic momnts µ β + µ n µ β + µ n 95 µ ( + n

35 o th lcton and oton into Eqn (8) oducs a mo accuat solution bcaus this contains th anomalous magntic momnts Thn w would hav n + n 95 _ µ µ µ µ µ β h (8) + n Th only simultanous solution o Eqns (8) and (8) o which _ and _ a both ositiv a _ 857 x _ (8) Th valus o th ctiv units o action o th oton and lcton givn in Eqn (8) show that angula momntum is consvd duing th dcay o a nuton whn th nuton is considd to b a oton in obit aound an lcton und th nocoulombic oc Th thid mao agumnt against a nuton bing a stat o lcton and a oton obits stms om th ximntal vidnc on th violation o Nwton's Thid Law duing dcay That is th ngy o th lcton mging at dcay is inconsistnt with th qual and oosit columbic ocs btwn an lcton and a oton H w ind that th nocoulombic ocs a unqual in magnitud and oosit in diction; thus th ngy o an lcton mging as th sult o cossing om such an obit cannot b consistnt with Nwton's thid law Th now xists a outh agumnt against this ictu o a nuton: th ossibl xistnc o th nutino Th abov ictu o th nuton oducs no nd to ostulat th xistnc o nutinos What thn can b said about th ximntal vidnc that has bn ut owad in suot o th catu o nutinos? A conclusiv answ will nd to await uth invstigation Nucla Masss Th diiculty oducd by th asymmty o ocs that aiss in th intaction o an lcton with a oton may b avoidd i two otons a considd to b in obit about th singl lcton I w thin o a snashot o such a cas w would ind that th situation dictd in Fig 9 allows us to visualiz th ocs Figu 9 Two otons in obit about a singl lcton Th oc on th lcton would b zo bcaus it has a oton on ach sid diamtically oosd to on anoth Th oc on ach oton will b mad u o two ats; on th oc that is du to th snc o th lcton and th oth owing to th oth oton Th symmty

36 5 guaants that ach oton will xinc an idntical oc i cicula obits a assumd towad th cnt o otation Th oc on th oton on th lt would b To b su quantum mchanical ocdu should b usd; howv it may b bnicial to bgin by assuming cicula obits simila to Boh's initial aoach to atomic stuctu This may indicat th otntial utility o th oc in Eqn (8) as wll as has idntiying ocdus to b usd lat Any nucla obits should obably b lativistic; tho in cylindical coodinats wh th vlocity o motion in a lan is givn by thn w hav Fo cicula obits this bcoms Thus th lativistic quations o motion o th oton bcom Equation (85) saats into two quations and Th scond o ths quations says that th angula momntum is givn by () ) ( + F (8) + v θ θ ˆ ˆ & & c ) + ( c v θ & & c θ & (8) ) ( + m dt d & & & θ θ ˆ ˆ (8 m dt d ˆ & (86) m dt d θ &

37 6 wh _ indicats that whas th unit o angula momntum will b a constant o a givn obit it may b dint o dint obits Th ist o Eqn (86) is but o cicula motion & tho Substituting om Eqn (87) into Eqn (88) w hav Th otntial ngy o on o th otons can b ound by intgating th oc and is Thn th total ngy o th thbody systm including st ngy would b Howv by substituting Eqn (87) into Eqn (8) and solving o w ind n_ L m θ & (87) dt d m ) M ( m m dt d θ & & && & m θ & (88) m ) ( n h (89) d F() d V() (9) c + m c m + E T (9)

38 (9) n_ + m c Thus substituting Eqn (9) into Eqn (89) oducs a tanscndntal quation whos solution givs (n) which may thn b usd in Eqn (9) to obtain th total ngy o th systm Th mass o th systm should thn b ound om E T c M (9) Bcaus this systm has on lcton and two otons it has a total lctic chag o + and would hav a mass o aoximatly amu This is th sam chaactistic xhibitd by th dutium nuclus I this is th stuctu o th H nuclus thn th mass givn by Eqn (9) should cosond to th mass o th goundstat nucla mass o n I th + cas xisting wh n is th goundstat H nuclus thn is th xcitd stat sntd by two otons in th n stat o can it b sntd by on oton in an n obit and on in an n obit? Th quations dvlod h consid only th cas whn both otons a in th sam obit Any considation o th otons bing in dint obits intoducs an asymmty in th ocs and a simila diiculty acd in th nuton cas Tho o th momnt w will consid only th siml cass wh symmty ducs th comlxity o th solution Notic though that vn in th siml symmtic cas no analytical solution xists o Eqn (89) o (n) bcaus th oc contains a tanscndntal unction By allowing h 5to b dint o ach n thn o th gound and ist xcitd stats o th H th a ou quantitis to b dtmind: _ () 5 and _ 5 O cous in thoy w could dtmin both 5 and 55 om scatting ximnts: thn w would only hav two _ () 56and _ () 57 But i w discov xactly how th _ 58 dnds uon th obit thn a solution o Eqn (89) would snt a u thotical diction o both th gound stat mass m(n ) and th xcitd stat mass m(n ) o thn w would b abl to xss _ _ () 59 I w thin about th ossibility o adding an additional oton to th + cas w a acd with a qustion Can th additional oton b lacd in th n obit without considing th asymmty thus intoducd into th systm o must w consid a singl obit with th otons symmtically sacd? Th answ lis atly in th solution o th aoiat Schoding o Diac quation bcaus this would inom us o th numb o otons that a allowd in a givn obit This howv would not answ th qustion concning how th asymmty intoducd by a singl oton in th n obit acts th oblm To 7

39 obtain an answ to this qustion th situation should b addssd by both mthods and both sults should b comad with th ximntal vidnc Whichv aoach ovs to b coct th sult would b a nuclus with a total chag o + and a mass numb o o Z and A This cosonds to th H nucli I th thid oton can b lacd in th n obit o th solution o a singllcton cas and dtmining 6 and 6 by scatting ximnts lus dtmining _ () and _ () 6 by th bound and xcitd stats o th H nuclus thn th mass o th H is a u thotical diction W shall consid this a littl lat Th act that th xcitd stat o th H nuclus is a vitual stat imlis that adding two otons may itsl b naly a vitual stat and thus a cuto would occu in th numb o otons that can xist in a bound stat o a onlcton co To ossibly loo at oth nucli suos ist that w go bac and consid th li aticl oc btwn two lctons F (9) This oc is ulsiv until th saation btwn th lctons bcoms lss than and thn it bcoms attactiv But can th b bound stats o two lctons? Th answ lis in th ct o th gaug on _ 6 I th _ 65 o unli aticls is lss than _ 66 thn th sign vsal btwn th oc o unli aticls and th oc o li aticls should man that th _ 67 o li aticls should b lag than _ 68 Thus bound stats o li aticls would b obiddn As an xaml o th sign vsal ct on _ 69 suos that th gaug o th unli lcton oton cas is givn by ln o x (95) Thn (dσ ) gˆ x ( dq ) g dx d dx dx o g gˆ 8

40 Suos gˆ 7 δ 75; thn Eqn (78) would bcom _ δ _ gˆ _ δ l [ +] δ l (96) Thn om Eqn (96) w ind _ _ Now < < 78; tho whn th gaug unction om Eqn (95) is lss than unity bcaus < Thus th _ 79 givn by this unction is always lss than o qual to On th oth hand o li aticls wh only on is involvd th _ gaug unction would b givn by x (97) Thn bcaus > quiing _ _ 8 Although liaticl bound stats may b obiddn by th unctainty incil o lag _ 8 bound stats o unli aticls o nucla dimnsions a allowd by an _ 8that may b much much lss than _ 85 Nxt w might consid ossibl bound stats btwn lctons and ositons with subnucla dimnsions I th o th ositon + is lss than th o th lcton thn w could hav bound stats with two lctons in obit about a singl ositon that would b givn by quations xactly li th quations o th + stats wh th ositon lacs th lcton in th + quations and th two lctons in th + cas lac th two otons in th + shll It is ossibl now to consid a + co thus intoducing qustions concning asymmty ascts and oth ossibl qustions Howv owing to th + < << th co stuctu is on wh in lctons obit about ositons and th lcton obits a << Fo th otons in a shll obit th intio co stuctu may b ngligibl ust as th intnal stuctu o th nuclus has almost no ct on th atomic lcton obits W shall not xlo th co stuctu h but shall consid only th ct o dintco xcss lcton chag uon allowd oton shll obits I w dnot th xcss lcton chag o th co by th intg Y by which w man th total numb o co lctons lss th numb o co ositons thn by dnoting th numb o shll otons in obit aound this nucla co w ind that th chag on th nuclus Z is givn by Z A Y (98) 9

41 Equation (98) indicats that th xcss co lcton numb bhavs idntically with th nuton numbs in cunt nucla thoy although th a no nutons as such in this nucla modl Indd th nuton in this ictu is simly anoth stat namly Y and A This suggsts a ictu o th nuclus in which th a otons in obits about a nucla co Th numb o otons a givn by th cunt mass numb A Th adii o th oton shll obits a aoximatly th valu o ; that is about mi Th co may b mad u o lctons in obit about ositons and is sizd aoximatly th sam as which is much much lss than This viw o th nuclus is simila to that o th atomic viw but h th nucla co lays th ol o th atomic lctons Th oc law o th shll oton obits would thn b givn om Eqn (85) by F Y (99) Th quations sciying th oton shll obits a n (h ) m Y () and n_ + m c Th total ngy o th nucli would b givn by E(AY) Y n R(n) A(n) R(n) R(n) A(n)m + + Ec(Y) [R(n) () wh A(n) is th numb o otons with th quantum numb n; R(n) is th adius o th oton obit with th numb n; E c (Y) is th ngy o th nucla co o which Y is th xcss lcton chag; and [R(n)] is th lativistic valuatd o R(n) Th mass o th nucli with ngis givn by Eqn () would thn b 5

42 M(A Y) E(AY) c () This aoach has a siml loo to it Fo instanc i Y thn E c 5 MV th st ngy o th lcton Thn th gound stat o H would b E( ) E + E c whas th xcitd stat is * E( ) E + E c Using E c 5 M V w ind that th ngy o a singl oton in th n obit would b * [ E( ) Ec] Tabl I Eximntal and dictd nucla masss E Eximntal Pdictd M E E E Pdictd MassMass M/A BE/A Y Z A (MV) (MV) (MV) (MV) (MV)

43 Thus th H nucli ngy would b E() E()+ * [ E( ) Ec] Now using th tabulatd ximntal data w ind that th dictd nucla mass o th H should b comad to th tabulatd valu o 88 amu Similaly th dictd mass o Li should thn b 888 comad to th tabulatd 5 Th dinc btwn th dictd and th tabulatd valus a 8 and 7 MV/nucla sctivly o th H and Li nucli Bcaus th co ngy and obital ngy lvls should chang [ 6 59 ] E() 86 amu + 56 amu Tabl II Engylvl avag valus x amu Y E +E c E +6E +E c E E (amu) (amu) (amu) (amu) whn th xcss lcton numb o th co changs w may constuct Tabl I wh slctd nucli a usd to stablish th co and shll ngy lvls o dint Y Pdictions o th mass o oth nucli a mad using th ngy Eqn () and assuming that th numb o otons in a ull shll cosonds to th numb o lctons in th atomic shlls i 8 8 Fo ach Y som o th ximntal masss a usd to stablish an ngy valu; tho th dictd valu aas th sam as th ximntal In ach cas th ngy valu stablishd by this data oint aas in th aoiat column Th RMS o in th dictd valus o all nucli was MV with an abitay slction o which nucli w usd to stablish an ngy lvl A btt way o aoaching th stablishmnt o th ngis would b to ta th avag valu o all ossibl ways to ind a aticula ngy Tabl II lists th ngylvl avag valus ndd in th total ngy quation o th sam nucli By using th avag valus om this 5