Gaphical Rsults fo Hydogn and Hlium 5 Jounal of Foundations of Physics and Chmisty,, vol (5) 5 534 ECE thoy of th Lamb shift in atomic hydogn and hlium MW Evans * and H Eckadt ** *Alpha Institut fo Advancd Studis (AIAS) (wwwaiasus); **Unifid Physics Institut of Tchnology (UPITEC) (wwwupitcog) Th ECE thoy of th Lamb shift in atomic hydogn and hlium is dvl opd using considations of th cntifugal pulsion in atomic hydogn and th Coulomb and xchang intgals in atomic hlium Radiativ coc tions a applid using th sam mthod as fo th anomalous g facto of th lcton and as dvlopd in pvious wok fo atomic hydogn without th cntifugal coction Th sults a xpssd systmatically in tms of a paamt (vac) that masus th ffct of adiativ coction on th adial and oth obitals This mthod can b xtndd systmatically in quantum chmisty packags, which can b usd fo xampl to numically gnat adial obitals fo us with this mthod Th latt may hav impotant con squncs in obtaining nw soucs of ngy fom spac-tim, (i th souc of adiativ coctions) Kywods: Einstin Catan Evans (ECE) thoy, Lamb shift in atomic hydogn and hlium, ngy fom spac-tim Intoduction Rcntly [] th Einstin Catan Evans unifid fild thoy [-] has bn applid to th dtmination of th Lamb shift in atomic hydogn using a mthod that succdd in poducing th g facto of th lcton to xpi mntal unctainty Th sults of th calculation w xpssd in tms of a adial paamt (vac) which dmonstats th way in which th adiativ coction affcts ach obital Th Lamb shift in atomic H [3] was poducd satisfactoily using this mthod, which has no f paamts In this pap th mthod is xtndd to atomic hydogn with considation of cntifugal pulsion [4] as wll as Coulombic attaction, and to atomic hlium, which consists of two lctons and two potons Th sults a again in satisfac toy agmnt with data, and can -mail: EMyon@aolcom -mail: hostck@aolcom
5 MW Evans and H Eckadt b xtndd systmatically using quantum chmisty packags such as dnsity functional cod Th ovall aim is to find mthods of amplifying th adiativ coction by sonanc, so that f lctons a lasd and usd fo pow gnation in cicuits In Sction th mthod is usd fo th complt potntial of atomic hydogn, which is wll known [4] to contain a cntifugal pulsion tm as wll as th Coulomb attaction btwn lcton and poton Th sult is xpssd in tms of (vac) and gaphical data poducd fo th lvant obitals Th oiginal Lamb shift [5] was discovd btwn th s and p obitals of hydogn, and this mthod natually movs th angula dpn dnc, so that only adial obitals nd b considd [4] In Sction 3 th mthod is xtndd to atomic hlium, and consida tions givn to th Coulomb and xchang intgals [4] in od to undstand th basic fatus of multi lcton atoms If a mthod is found to sonat (vac) to infinity, th atom will dissociat into f lctons which can b usd fo powing cicuits Th souc of this pow is spac-tim, which is also th souc of th adiativ coction in ECE thoy Thfo th mth ods of this pap complmnt pvious considations [-] of ionization of atoms using spin connction sonanc [6] Atomic hydogn with cntifugal ffcts In this pap th non-lativistic quantum limit of th ECE wav quation [- ] is usd, th Schoding quation considd in a causal and objctiv intptation of quantum mchanics ath than th Copnhagn intp tation Th Schoding quation of atomic hydogn [4] is: E m ψ 4π ψ= ψ () wh m is th ducd mass, ħ is th ducd Planck constant, є is th SI vacuum pmittivity, is th chag on th poton, is th chag on th lcton, is th adial distanc btwn lcton and poton, E is th total ngy and Ψ is th wav-function By xpssing th laplacian in sphical pola coodinats, th quation of th sphical hamonics is intoducd: ( ) Λ Y = + Y () wh Y(θ, Φ) a th sphical hamonics [4] and l is th angula momntum quantum numb Th solution of Eq () is:
(,, ) RY ( ) (, ) Gaphical Rsults fo Hydogn and Hlium 53 ψ θφ = θφ (3) wh R() a th adial wav-functions Tabl givs th fist fiv obitals of hydogn with nomalization factod out fo claity It is sn that th complt obital is xpssd in tms of th pincipal quantum numb n, th angula momntum quantum numb l and its componnts Howv, fom Eqs () to (3) th Schoding quation can b wittn [4] as: d P ( ) ff P = EP m d (4) wh th ffctiv potntial is: + = + ( ) ( ) ff 4π m (5) and wh: P = R (6) Th scond tm in Eq (5) is th cntifugal pulsion and th complt potntial has a minimum whn l is not zo (diagam (b)) Th adiativ coction that lads to th Lamb shift [] is now incopo atd by changing Eq (4) to: α d P ( ) + ff P = EP m 4π d (7) (a) (b)
54 MW Evans and H Eckadt Tabl Fist Fiv Obitals of Atomic H n m ψn( m ) (, θφ, ) s xp a s xp a a p Z cosθxp a a p X sin θxp( iφ) xp a a sin θxp φ xp a a p Y - ( i ) wh α is th fin stuctu constant This quation is quivalnt by hypoth sis [] to: d P ff P = EP m d (8) wh: ( ( )) ( + ) ( ( )) = 4π + vac m + vac ff (9) H (vac) is th xtnt to which th adiativ coction ptubs th lcton in ach obital To fist od in α: ( ) ( ) ff ff α d P = P 4πm d () wh P = P (hydogn) () to a vy good appoximation, bcaus th Lamb shift in hydogn is vy small compad with th total ngy of th unptubd obitals Thfo:
Gaphical Rsults fo Hydogn and Hlium 55 ( ) ff ( + ) ff = 4π ( vac) + m ( + ( vac) ) () and Eq () is solvd fo (vac) fo ach obital, th sult bing diff nt fo ach obital Spcifically th is a diffnc btwn th s and p obitals, giving th Lamb shift Th xpimntal valu of th latt is usd to find (vac) Comput algba is usd to solv Eq () and th sults a gaphd in Sction 4 3 Atomic Hlium This atom consists of two lctons and two potons and its Schoding quation is: HΨ = EΨ, (3) H = ( + ) + m 4π 4π 4π (4) wh is th distanc btwn lcton and th nuclus and similaly fo Th Coulombic tms a doubld bcaus th a two potons giving twic th attactiv foc Th is also an lcton lcton pulsion, wh is th distanc btwn th two lctons Th int-lcton tm can b xpandd in tms of sphical hamonics: 4π = Ym θ φ m θ φ = m = + (, ) Y (, ) (5) if >, and int-chang and whn > This givs is [4] to th Coulomb intgal: J = ψ ( ) ψ ( ) dτdτ 4π (6) Fo xampl, if w consid th intaction ngy btwn two lctons in a hydogn lik s obital [4], thn:
56 MW Evans and H Eckadt 3 Z xp Z 3 πa a ψ ( ) = (7) 3 Z xp Z 3 πa a ψ ( ) = (8) and it is found that:,, = =, (9) Ths sults can b usd to calculat J analytically: J 5 = 84π Z a () It is found that [4]: 5 6 ( ) J = E s () so th lcton lcton pulsion is a substantial faction of th unptubd total ngy of th s obital in hlium Th aim of this sction is to incopo at adiativ coctions into th hlium atom, which also contains th wll known xchang intgal lading to Fmi hap and hol thoy [4] Th Schoding quation of atomic hlium is: (, ) = ψ(, ) Hψ E () wh: ( ) nm n m ψ, = ψ ψ (3) It is assumd that th is an unptubd hamiltonian [4] which is th sum of two hydogn lik hamiltonians:
Gaphical Rsults fo Hydogn and Hlium 57 H () = H + H (4) wh: H i = m 4 π i i (5) Th facto two in th numato of th scond tm coms fom th fact that th a two lctons and two potons, so ach lcton is attactd by two potons Th assumption (4) mans that th total wav-function must b th poduct [4]: ( ) ( ) ( ) ψ, =ψ ψ (6) Th total unptubd ngy lvls a [4] E = 4hcR + n n (7) wh R is th Rydbg constant To incopoat adiativ coctions th fist stp is to dvlop Eq (5) and wit it as: d Hi = m d i ( ) ff, i (8) wh + = + ( ) ( ) ff, i πi mi (9) So th a two hydogn lik quations: d P ( ) ff, P = EP m d (3) d P ( ) ff, P = EP m d (3)
58 MW Evans and H Eckadt wh: + = + ( ) ( ) ff, π m + = + ( ) ( ) ff, π m (3) (33) Radiativ coctions a incopoatd into Eqs (3) to (3) as in Eqs (8) to () Fo ach and th will b a cosponding (vac) and (vac) and th total wav-function is th poduct as in Eq (6) Ths adiativ coctions may thn b incopoatd sytmatically into th Coulomb and xchang intgals using th xpansion (5) of in tms of and With adiativ coctions: + (vac), (34) + (vac) (35) By hypothsis, Eqs (3) and (3) bcom: d P ff, P = EP m d (36) d P ff, P = EP m d (37) with: d d α + 4 π d d (38) d d α + 4 π d d (39) wh: ( (( )) ( + ) ( ( )) = π + vac + vac ff, (4)
Gaphical Rsults fo Hydogn and Hlium 59 ( (( )) ( + ) ( ( )) = π + vac + vac ff, (4) To fist od in α: ( ) ( ) ( ) ff, ff, α d P = P 4πm d ( ) ( ) ( ) ff, ff, α d P = P 4πm d (4) (43) Fo two s lctons: 3 Z Z a 3 ( ) πa = π Z 4π Z Z a a d + d d (44) and this may b valuatd with comput algba with adiativ coctions Finally a sach may b mad numically fo conditions und which (vac) bcoms vy lag Und such conditions th atom may ioniz, giving f lctons This ida complmnts pvious wok [ ] using spin connction sonanc [7] Th ovall mthod can b systmatically xtndd by using quantum chmisty softwa to gnat adial wav-functions with which adiativ coctions can b considd 4 Gaphical Rsults fo Hydogn and Hlium In this sction w psnt som gaphical sults fo th spactim intaction of hydogn and hlium Th cosponding calculations in th pcding pap a patd with inclusion of th cntifugal tm in th potntial As statd in sction, this accounts fo all angula momntum ffcts in th limit of avagd angula dpndncis Th obital avaging usd in th pcding pap is no long quid Th mthod of calculating th quivalnt adius function (vac) fo th shift of ach obital is th sam as bfo: inst Eq () into Eq () With th wav functions of Tabl comput algba dlivs th analytical solution fo (vac) Fo th s and s obitals w obtain th sam sult as in th pcding pap sinc l = Fo th p stat th sult is (in atomic units
53 MW Evans and H Eckadt fo simplicity): ( vac), = ( + 6π 8 6π+ 8 ( ) ( ) ( ) ( 8 8 ) ( 6 8) ) ± π 8π + 6 3π + 3π 6 + π + π 3 (45) with α = = c = 4 π = (46) Th dpndnc of (vac) fom th adius coodinat was alady shown in th pcding pap, Figs c and c, fo th s and s obitals Th two solutions fo p a gaphd in Fig of this pap Th fist solution shows a pol and is thfo considd to b non-physical Th scond looks qualitativly simila to that obtaind by th ali avaging mthod (Fig 3c of th pcding pap?????) Th Lamb shift itslf is chaactizd by th diffnc of (vac) valus fo s and p This is shown in Fig fo both solutions of p Th diffnc valus fo th scond solution main positiv, indicating again th physical lvanc of this solution In th simplst appoach to H, hydogn-lik wavfunctions a bing usd (s sction 3) In this appoach th ffctiv potntials dcoupl fo both Fig acuum intaction adius (vac) fo H p obital
Gaphical Rsults fo Hydogn and Hlium 53 Fig Effctiv Lamb shift adius (vac)(s) - (vac)(p) lctons and (vac) is th sam in both cass Fo th s stat th Hydognlik wavfunction 3 Z ψ = π a a ( s)( ) 3 Z (47) is usd which contains a paamt Z fo th co chag Solving Eqs (4/43) fo (vac) thn givs ( vac) Z Za = πmc Z + 4 a Za (48) Th cuvs of this function fo both H and H a plottd in Fig 3 It can b sn that th H cuv is compssd in adial diction du to th high co chag of H Highly intsting is th ffct of (vac) on th xchang intgal (6) Its dfinition thn is to b modifid to ψ( ) ψ( ) ( )( ) ( )( ) J = dτdτ 4π + vac vac (49) Fo two s lctons this givs in analogy to Eq (44):
53 MW Evans and H Eckadt Fig 3 acuum intaction adius (vac) fo H s and H s obitals 3 Z Z a 3 ( 4 ) ( ( vac) ( )) πa J = π + 4π + ( vac)( ) ( ( vac)( ) ) ( vac)( ) + + ( ( vac)( )) Z a + + a + ( vac)( ) d Z d d (5) This is a complicatd function with intgation limits dpnding on (vac) Evaluation of J is fasibl fo a constant (vac) Th sult is gaphd in Fig 4 Fo lativly small valus of (vac), J gows naly xponntially ov sval ods of magnitud Anoth fasibl fom of (vac) is ( )( ) vac = i i (5) Sinc this is a singl function, w do not obtain a cuv but a singl valu fo J in this cas: J 5 = 7 4π Z a (5)
Gaphical Rsults fo Hydogn and Hlium 533 Fig 4 (vac) dpndnc of xchang intgal J fom fo H s - s Again this valu is much gat than th oiginal valu of Eq (): J 5 = 84π Z a (53) Ths sults indicat that th intaction of atoms with th quantum back gound of spactim has an nomous potntial Th intaction should b xploitabl via spin connction sonanc in od to dliv ngy fom spactim Acknowldgmnts Th Bitish Govnmnt is thankd fo a Civil List Pnsion to MWE and th staff of AIAS and oths fo many intsting discussions Rfncs [] M W Evans and H, Eckadt, wwwaiasus, pap 85 [] MWEvans, "Gnally Covaiant Unifid Fild Thoy" (Abamis Acadmic, 5), volum on [3] ibid, volums two and th (6) and volum fou (7), volum fiv, in pp, paps 7 to 86 on wwwaiasus [4] L Flk "Th Evans Equations of Unifid Fild Thoy" (Abamis Acadmic, 7) [5] L Flk, H Eckadt, S Coths, D Indanu and K Pndgast, paps and slids on wwwaiasus [6] M W Evans, paps on Omnia Opa on th pcuso gaug thois of ECE (99 to 3) [7] M W Evans and L B Cowll, "Classical and Quantum Elctodynamics and th B(3) Fild" (Wold Scintific, ) [8] M W Evans (d), "Modn Non-lina Optics", a spcial topical issu in th pats of I Pigogin and S A Ric (Sis Editos), "Advancs in Chmical Physics" (Wily Intscinc, Nw Yok,
534 MW Evans and H Eckadt, scond dition), vols 9() to 9(3), ndosd by th Royal Swdish Acadmy; M W Evans and S Kilich (ds), ibid, fist dition (Wily Intscinc, Nw Yok, 99, 993, 997), vols 85() to 85(3), Polish Gov, Awad fo xcllnc [9] M W Evans and J- P igi, "Th Enigmatic Photon" (Kluw, 994 to hadback and softback), in fiv volums [] M W Evans and A A Hasanin, "Th Photomagnton in Quantum Fild Thoy" (Wold Scintific, 994) [] M W Evans, "Th Photons Magntic Fild : Optical NMR Spctoscopy" (Wold Scintific, 99) [] M W Evans, Physica B, 8, 7, 37 (99), fist paps on th B(3) fild [3] L H Ryd, "Quantum Fild Thoy" (Cambidg Univsity Pss, " Ed, 996) [4] P W Atkins, "Molcula Quantum Mchanics" (Oxfod Univ Pss, 983, " and subsqunt ditions) [5] H A Bth, Phys Rv, 7, 339 (947) [6] M W Evans and H Eckadt, pap 63 of wwwaiasus [7] M W Evans, Acta Phys Polon, B Publishd Jun 7 and sval paps on wwwaiasus