Klein-Gordon Equation

Transcription

1 Inroducon o lnar Parcl Phscs. Lcur 5 Pag of 5 Kln-Gordon quaon 96 Schrodngr: Quanu chancal quaon for non-rlavsc chancs: V,, V V 96 Kln Rlavsc chancs (fr arcl):,, Soluons ar: ( r), whr ( r), whr ) Soluons hav boh osv ngav gnvalus for nrg. No ha n QM, on canno dro soluons ha s o b unhscal h full s of gnsas s ndd. Indd, f on sars for (), n wll volv o () ha, n gnral, can hav rojcons on h gnsas ha on would lk o g rd of ) vn wors, urnd ou ha -funcon dd no allow for sascal nrraon of as bng * * robabl dns s ngral would no ran consan. On h ohr h, h valu ρ whos ngral dd s o b consrvd, could ak ngav valus. Th caus of hs robl was rackd o h fac ha h quaon had scond-ordr drvavs. arnl, Schrodngr rd ou Kln quaon bfor rocdng wh hs non-rlavsc quaons, bu drod sng an robls nvr ublshd Ths quaon was dscardd n h coun as faul In addon, dd no lav an roo for sn sas, whl lcrons roons wr known o hav sn ½. Movad b hs robls, Drac s o fnd anohr rlavsc quanu chancal quaon. Wha hand n s: 98 Drac fnds a lnar quaon; dscrbs frons; howvr, sll has ngav nrg soluons 9 Drac nrrs h ngav nrg soluons as hols n h sa (now coonl rfrrd as Drac sa) ha us bhav as oos charg arcls 9 Posrons ar found 94 Paul Wsskof showd ha h scond robl was no rall a robl. Snc rlavsc quanu chancal quaon had o allow for craon annhlaon of arcls, h ngral robabl dd no hav o sa consan, anor. Th consrvd ρ was rnrrd as charg dns, whch, of cours, could ak ngav valus. So, non of h objcons ha ld o dscardng h Kln quaon hld u wh. s w can s now, h aarn robls wr anfsaons of h nw hscs. Lar, h quaon was rvvd s now known as Kln-Gordon quaon. I dscrbs lnar sn- bosons.

2 Inroducon o lnar Parcl Phscs. Lcur 5 Pag of 5 Drac quaon 98 Drac: sarchs for an alrnav rlavsc quaon sarng fro h gnrc for dscrbng voluon of wav funcon: H If on ks frs ordr drvav of, hn o rsrv Lorn nvaranc, h sac coordna drvavs us b of h frs ordr as wll. Havng all nrg-rlad oraors (,, ) of h sa frs ordr: ( ),, acng wh lf- rgh-h oraors wc, w g ( ) whch us b coabl wh h Kln-Gordon quaon: ( ) Ths ls ha: j j j, for Thrfor, arars canno b nubrs. Howvr, a dos work f h ar arcs, h lows ordr bng 4 4. Thrfor, us b 4-coonn vcors. Poular rrsnaons ar arcs : Paul ar whr Th quaon s usuall wrn usng γ μ -arcs, whr γ for,, γ (jus ull h abov quaon wh ar ov all rs on on sd of h quaon): whr μ μ γ γ γ Fnd soluon for arcls a rs,.. : 4 γ I has wo osv nrg soluons ha corrsond o wo sn sas of sn-½ lcrons: wo srcal ngav-nrg soluons

3 Inroducon o lnar Parcl Phscs. Lcur 5 Pag of 5 9 Drac s nrraon of h ngav nrg soluon If ngav nrg sas do s, on had o lan wh lcrons dd no fall down o lowr nrg lvls. To hl hs robl, Drac aald o h fac ha lcrons ar frons no wo lcrons could ak h sa quanu sa (h Paul cluson Prncl). Thrfor, Drac argud, hr would b no robl f on could agn ha all h ngav nrg sas wr alrad flld wh lcrons, forng a so-calld Drac sa of lcrons wh ngav nrgs. Thn lcrons wh osv nrg would no b abl o fall down o h lowr lvl nrg lvls. Snc hs ngav nrg lvls wr naccssbl o h fw ral lcrons, h snc of such sa would no b obsrvabl Howvr, Drac hslf soon nocd ha hr would b a las on obsrvabl anfsaon of such sa. On could c an lcron fro h ngav nrg sa o h osv lvl b ransfrrng nough nrg, nu bng abou MV (wo asss of an lcron). rsulng hol crad n h ngav nrg sa, whch s rl an absnc of ngav charg, would aar o b a osvl chargd arcl. h whn Drac nvnd hs quaon h lcron roon wr assud o b h onl wo subaoc arcls (nuron was no dscovrd a ha ). Drac hrfor assud ha hs osv arcl, roducd b an lcron lavng h sa of ngav nrg sas, would b a roon. Thr was a robl wh hs hor howvr. Svral hscss showd ha h hol n h Drac sa would hav o bhav as a arcl wh h sa ass as an lcron h roon of cours was uch oo hav Is ossbl hn ha hr could b anohr osvl chargd arcl of lcron ass? No on dard a ha o suggs such a bold da ll on would navl nd s o h suffcnl hard h sa (vacuu) wh h nrg > MV. nnhlaon of an lcron wh a hol would also b ossbl wh ng hoons n h sa wa as ransons bwn nrg lvls n aos. nohr foral nrraon of h ngav nrg soluons: h look lk arcls wh osv nrg, bu ovng backward n ll hs nrraons ar awkward no as o hl. Morovr, h da of h da of h Drac sa no work for bosons a all vn a hs surfcal lvl. Th fnal forals of dalng wh h rlavsc quanu quaons has bn dvlod snc ha s now calld h Quanu Fld Thor. Th soluons for arcls wh ngav nrgs ar rnrrd as an-arcls wh osv nrg ll h rul lnar arcls fundanal forcs as w know h now fall no hr cagors: Sn- arcls: Kln-Gordon quaons Sn-½ arcls: Drac quaons Sn- arcls: Proca quaons Th Drac quaon rdcs ha h lcron agnc on s sn ar rlad as μμ S, whl for noral orbal oon μμ L. Dald calculaons gv sall corrcons o h facor g s rcs asurns/calculaons agr wh ach ohr a h lvl of ~ - a scacular confraon of h hscs sng bhnd h Drac quaon. 9 Drac rcvs h Nobl Pr (oghr wh Schrodngr)

4 Inroducon o lnar Parcl Phscs. Lcur 5 Pag 4 of 5 Dscovr of osron 9 Carl D. ndrson (Calch) found vdnc for an lcron wh a osv charg, or osron. ndrson dscovrd h osron whl usng a cloud chabr o nvsga cosc ras. ndrson's frs cur of a osron rack Th osron gos uwards loss so nrg as asss hrough a lad la n h ddl of h chabr. Is rack s curvd bcaus hr s a agnc fld n h chabr. Th fac ha s curvd or n h ur half lls ha s onu s sallr hr hrfor ravls uward. 9 Ths work was connud n h Cavndsh Lab b Parck lack Gus P.S. Occhaln who showd ha a osron was roducd oghr wh an lcron, n ln wh an arlr hor of Paul Drac's. ccordng o hs hor, a osron was a hol n a sa of ordnar lcrons. Th osron was h anar quvaln o h lcron. s h aaranc of cosc ras s unrdcabl was sard racc o s u h chabr o ak a hoograh vr 5 sconds, roducng a vas aoun of worhlss aral for analss. To avod hs lack nroducd n 9 h counr-conrolld chabr. Ggr counrs wr arrangd abov blow h chabr ha whn a cosc ra assd hrough boh, acvad h anson of h chabr hoograhng h on racks roducd b h ra. Usng hs dvc h confrd n Carl ndrson's dscovr of h osron. Th also suggsd ha h osron was roducd b h nracon of gaa ras wh ar, n whch a hoon s convrd no an lcron-osron ar. Th hnonon s known as ar roducon. 96 ndrson, a ag, bca h scond oungs Nobl laura for hs dscovr of anar whn h obsrvd osrons n a cloud chabr. 948 lack wns Nobl Pr for hs dvlon of h Wlson cloud chabr hod hs dscovrs hrwh n h fld of nuclar hscs cosc radaon.

5 Inroducon o lnar Parcl Phscs. Lcur 5 Pag 5 of rkl varon (6 GV roons) lo Sgrè, Own Chabrlan 959 Nobl Pr "for hr dscovr of h anroon" Dscovr of anroon buld a ba ln for.9 GV/c onu slc ngavl chargd arcls (osl π ) rjc fas π b Črnkov ffc: lgh sson n ransarn du f arcl vloc v > c / n (n: rfracon nd) anroons hav v < c / n no Črnkov lgh asur of flgh bwn counrs S S ( ah): 4 ns for π, 5 ns for anroons