Calculation of Matrix Elements in the Foldy-Wouthuysen Representation

Transcription

1 Calculatio of Matix Elemets i the Foldy-Wouthuyse Repesetatio V.P. Nezamov*, A.A.Sadovoy**, A.S.Ul yaov*** RFNC-VNIIEF, Saov, Russia Abstact The pape compaes the methods used to calculate matix elemets of the opeato of adail electo coodiates i a abitay ode i the Foldy-Wouthuyse epesetatio ad with the use of the iac equatio fo 1s-states of the hydoge-like ad helium-like ios of tasuaic elemets. The obtaied aalytical ad umeical esults fo 1s-states of the hydoge-like ad helium-like ios poves that the wave fuctio eductio equiemet is met with tasfomatio to the Foldy-Wouthuyse epesetatio ad cofims that matix elemets ca be calculated usig oly oe compoet (eithe uppe, o lowe) of the iac bispio wave fuctio. PACS umbes: W, Z. * addess: ezamov@viief.u ** addess: aa_sadovoy@viief.u *** addess: A.S.Ulyaov@viief.u

2 1. INTROUCTION Reseach ito the elativistic system popeties is impotat fom viewpoit of theoy ad applied scieces. The elemetay paticle physics studied ad cotiues studyig vaious boud systems of leptos ad bayos. Fo applied eseach, of iteest, fo example, ae the popeties of heavy ad tasuaic ios, whee elativistic effects have essetial values. Solutio of elativistic equatios associates with cosideable difficulties. So, the developmet of methods that allow easie calculatio of the elativistic system s popeties seems to be a uget poblem. The pape descibes the use of the wave fuctios i the Foldy-Wouthuyse epesetatio [1], [] ad aalytical solutios to iac equatios fo the hydoge- ad helium-like ios of heavy ad tasuaic elemets [3], [4] to demostate the Foldy-Wouthuyse epesetatio advatages i calculatios of matix elemet fo vaious opeatos, which ca be epeseted i the fom of expasio i seies i electo coodiates.

3 . SOME FEATURES OF THE FOLY-WOUTHUYSEN REPRESENTATION The Foldy-Wouthuyse (FW) epesetatio was itoduced i the pape [1]. The followig uitay tasfomatio was used to deive the FW epesetatio fom the iac epesetatio. Fo fee motio of paticles, the iac equatio looks like p xt, H xt, pm xt,. Usig the uitay tasfomatio [1] E m p U 1, E m p, we tasfom Eq. (1) to the fom E E m p x, t H x, t E x, t. () I Eq. (), FW FW FW FW H U H U E ; x, t U x, t FW FW Eqs. (1), () ad those give below ae witte i the system of uits c 1; the scala poduct of fou-vectos is take i the fom xy x y x y x y,,1,,3; 1,,3,. (1) p i ; x, t, FW x, t - ae fou-compoet wave fuctios; x i, ae iac matices; ae two-compoet Pauli matices. Solutios to Eq. (1) ae the wave fuctios with the positive ad egative eegy values: 1 1 ipx x, s U e, x, s V e, p p m. ipx FW 3 s FW 3 s (3) s I expessio (3), Us, Vs, s ad s spi fuctios. The followig othoomalizatio ad completeess coelatios ae valid fo UU s s VV s s ss; UV s svu s s ; 1 UsUs 1 ; s 1 UsUs 1. s s ae the two-compoet omalized Pauli s U s ad V s I expessios (3) ad (4), ae spio idexes, s is spi idex. Late, we ll omit the summatio symbol ad idexes themselves, while summig up with espect to spio idexes. (4) 3

4 I case of iteactio betwee a iac paticle ad exteal static fields, the closed FW epesetatio exists if oly the equiemet of commutatio of the eve ad odd pats of the iac Hamiltoia is met [5]. By defiitio, a eve opeato does ot mix the uppe ad lowe compoets of wave fuctio. I geeal case, whe femio iteacts with a abitay boso field, the poblem of tasfomatio fom the iac epesetatio to the FW epesetatio becomes much moe complicated. The geeal fom of the exact FW tasfomatio has bee foud by Eikse [6] fo abitay static exteal fields. The Eikse tasfomatio is a oe-step tasfomatio of the iac wave fuctio ad iac Hamiltoia to the FW epesetatio. Aothe way of diect tasfomatio to the FW epesetatio offeed i the pape [7] by oe of the give pape authos (see also the eview pape [8]) fo a geeal case of iteactio with a abitay boso field is to deive the elativistic Hamiltoia i the fom of a seies i tems of the couplig costat powes. I additio to some diect methods of obtaiig Hamiltoias i the FW epesetatio, thee ae a lot of methods fo step-by-step deivatio of Hamiltoias fee of odd opeatos. Oe of them was used, i paticula, i the classic Foldy-Wouthuyse wok (see [1]) to obtai Hamiltoia i the pesece of static exteal electomagetic field i the fom of a seies i tems of 1 m powes. The papes [], [7], [9] show that step-by-step tasfomatio methods lead to the FW epesetatio oly fo the fist oe, o two steps. The pape [] studies the mai popeties of wave fuctios i the FW epesetatio ad establishes a uique elatio betwee the two wave fuctios i the iac ad Foldy-Wouthuyse epesetatios. iagoalizatio of Hamiltoia elative to the uppe ad lowe compoets of wave fuctio is a ecessay equiemet of tasfomatio fom the iac epesetatio to FW epesetatio. The secod equiemet fo the FW tasfomatio is that the uppe ad lowe compoets of x, t bispio wave fuctio xt, A must be zeoed ad the omalizatio opeato of wave x, t fuctio x must be tasfomed to the uit opeato. The pape [] poves this equiemet ad calls it the wave fuctio eductio equiemet. Fo the time-idepedet iac Hamiltoia (the case of a fee paticle, o static exteal fields), this coditio ca be epeseted i the followig fom: 4

5 x iet iet x xt, e A FW xt, e ; x x iet iet xt, e A FW xt, e. x x I this equatio, E is the paticle eegy opeato s module; A ad A ae omalizatio opeatos, which, i geeal, may be diffeet fo the positive-eegy ad egative-eegy solutios. efiitio of opeatos A ad A implies that the wave fuctios x, t, x, t ad the spios x, x ae omalized to uity: x, t x, t dv 1, x x dv 1, x x dv 1. Pluses ad miuses deote states with positive ad egative eegy, espectively. Fo a fee paticle, (5) x e ipx ad x E m p A A, e ipx E m, (6) E fo the positive ad egative eegy solutios, espectively; ad ae the two-compoet Paili s spi fuctios (see expessio (3)). Fuctios x, t ad x, t ae the appopiate solutios of the iac equatio ad the FW equatio tasfomed to the FW epesetatio fo a fee paticle ad a paticle movig i static exteal fields. The eductio coditio implies tasfomatio of the iac wave fuctio to the fom x, t with uit omalizatio opeato. I geeal, the iac ad FW Hamiltoias deped o time. I this case, the eductio coditio (5) has the same meaig. Whe solvig specific poblems of physics (at least, with the use of the petubatio theoy), we use seies expasios i the iac equatio solutios obtaied eithe fo fee motio of paticles, o fo paticle motio i the pesece of static exteal fields. I this wok i the iac ad FW epesetatios (with the use of coditio (5)) we calculate matix elemets fo the mometum opeato of electo coodiates of a abitay powe ( ) usig the iac wave fuctios of the hydoge-like ad helium-like ios detemied i [3], [4] ad [1]. Really, these examples ae additioal tests fo the wave fuctio eductio equiemet (5) ad they demostate a easie way of matix elemets calculatig usig oly oe compoet (eithe uppe, o lowe) of the iac bispio wave fuctio. FW 5

6 3. MATRIX ELEMENTS OF AN OPERATOR OF ELECTRON COORINATES IN AN ARBITRARY POWER 3.1 Hydoge-like ios I geeal, the wave fuctio the hydoge-like io is detemied [3] as f jlm 1 l l, (7) 1 g jlm whee its agula pat jlm ae the 3 spheical spios, j is the total mometum, l is the obital mometum, m is the total mometum pojectio, is the electo coodiate module. Nomalizatio fo state 1s 1/ gives us fuctios f ad g of the fom f g d 1 (8) 3/ whee EE, 1 Z of legth equals classical adius of electo E 1 f e ; 1 3/ E g e 1 1 (9), (1), E is the system s bidig eegy i uits of mc, uit e e. mc, c I case of omalizatio (8), the -th ode mometum of electo coodiate is detemied by the expessio: f g d 1 1. (11) 1 With uit omalizatio of the wave fuctio s uppe compoet f omalizatio (8) ca be witte as f d 1, (1) A f g d 1, (13) 6

7 whee 3/ 1 f e (14) 1 ad E A. It is clea that i this case expessio (11) fo mometum emais uchaged. Accodig to coditio(5), the wave fuctio i FW epesetatio is fo ou case the wave fuctio s uppe compoet (14) omalized to uity. Oe may calculate mometum of electo coodiate with omalizatio (1) ad with the use of the wave fuctio s uppe compoet aloe f d e d 1. (15) 1 Oe ca see that expessio (15) coicides with expessio (11). The absolute values of momet of electo coodiates fo tasuaic elemets with,...,3 ae give i Table 1. Table 1 Radial momet of the hydoge-like ios of tasuaic elemets i state 1s1/. Name of elemet Nuclea chage Z 1 1 U Np Pu Am Cm Bk Cf Es Fm Mv Helium-like ios Fo the helium-like ios, we use the iac equatio solutio with miimum appoximatio by the method of multidimesioal agula Coulomb fuctios [13]. The wave fuctio of state + of the helium-like ios coespodig to cofiguatio 1s has the fom 3 7

8 detemiats 1, 1 M U 1, 1,, 5 (16) 1, N W1, whee 1 is a collective vaiable ad multidimesioal agula fuctios ae Slate U , ; W (17)!! fom the basic fuctios espectively, whee jlm i , 1 i 1 i 1 i 1 i m m m 3 m ae the thee-dimesioal spheical spios., (18) The magetic quatum umbe is m 1, the phase facto of the lowe compoet equals lj 1 1 1, the spio idexes coespod to the quatum umbes j l j lowe spios ae homogeeous polyomials of oe ad the same powe, K. The equatio system fo the helium-like ios i state О + has the fom [4],, z. The uppe ad Z E i ip i; (19) i1 i 1 i1 Z E4 i ip i, () i1 i 1 i1 whee is the total eegy of system, i uits of mc ; E A is the bidig eegy of e system; uit of legth equals classical adius of electo, Multiplyig Eqs. (19) ad () by the mc * * complex-cojugate ow U W ad itegatig with espect to each agula vaiables give us the followig expessios fo amplitudes of wave fuctio (16) expasio i seies i tems of twocompoets agula fuctios M M E 4 Z N ; 16 (1) N N E Z 16 M () The pocess of solvig Eqs. (1) () descibed i details i the pape [4] simila to that descibed i the pape [3] fo the hydoge-like ios esulted i the equatio fo the bidig eegy of the helium-like ios 8

9 5 E 1 Z 1. (3) 16 The bidig eegy values fo some helium-like ios of tasuaic elemets calculated fom Eq. (3) ae give i the Table. Table The bidig eegy of the helium-like ios i state of tasuaic elemets. Bidig eegy Е, kev Quatum umbe, 1 Name of Nuclea chage elemet Z Helium-like ios U Np Pu Am Cm Bk Cf Es Fm Mv With egad to omalizatio M N d 1, (4) amplitudes of expasio i seies of the uppe ad lowe compoets of the wave fuctio fo the mai ( ) state of the helium-like io of tasuaic elemet, which ae solutios to the system of Eqs. (1) (), have the fom M E 4 1 e 1 ; (5) whee EE 4. N E 1 e 1, (6) 9

10 It should be oted that the expasio amplitudes above ae popotioal (the popotioality costat is E 4 ). E us The -th ode mometum fo the helium-like ios i the mai state is calculated as 1 1 MU MUd d NW NWd d. 1 5 i 1 5 i 1 i1 i1 Itegatio with espect to agula vaiables fo the fist tem of the -th ode mometum gives Simila to (8), we obtai (8) M d (9) N d As a esult, we obtai the followig expessio fo the electo adius mometum of the heliumlike ios (3) M N d Use amplitudes (5), (6) ad obtai fom expessio (3) the aalytical expessio fo the -th ode mometum (31) With uit omalizatio of the wave fuctio s uppe compoet omalizatio (4) ca be witte i the fom M M d 1, (3) A M N d (7) 1, (33) whee 4 E A ad 4 1 e. (34) M 1 Appaetly, expessio (31) fo mometum emais uchaged. 1

11 Similaity to expessio (15) mometums ae calculated with the use of the uppe compoet (34) of bispio wave fuctio (16) omalized to uity. The electo coodiate momet with omalizatio (3) ae M d ad expessio (35) coicides with expessio (31) like i case of hydoge-like ios (see expessio (15) ad (11)). The absolute values of momet of vaious odes fo heavy elemets (fom uaium to (35) medelevium) i state ae give i the Table 3. Table 3 Radial momet of the heavy elemets helium-like ios i state Name elemet of Nuclea chage Z 1 1 U Np Pu Am Cm Bk Cf Es Fm Mv CONCLUSION The pape compaes the methods used to calculate matix elemets of the opeato of adail electo coodiates i a abitay ode i the Foldy-Wouthuyse epesetatio ad with the use of the iac equatio fo 1s-states of the hydoge-like ad helium-like ios of tasuaic elemets. By meas of the diect calculatios we obtai that oe ad the same opeato ca be used as a opeato of coodiate i iac ad FW epesetatio fo 1s - states of hydoge like ad helium like ios. opeato As is kow, i geeal case, these opeatos ae diffeet i these two epesetatios. If the FW is used i FW epesetatio, the i iac epesetatio it is UFWFWUFW, whee U FW 11

12 is tasfomatio matix with a complex depedece o the exteal fields. I the fee case is a opeato of Newto Wige 14. The obtaied aalytical ad umeical esults fo 1s-states of the hydoge-like ad helium-like ios poves that the wave fuctio eductio equiemet is met with tasfomatio to the Foldy- Wouthuyse epesetatio ad cofims that matix elemets ca be calculated usig oly oe compoet (eithe uppe, o lowe) of the iac bispio wave fuctio. 1

13 Refeeces 1 L.L Foldy, S.A Wouthuyse. Phys. Rev. 78, 9 (195). Nezamov V.P., Sileko A.J.. Joual of Mathematical Physics (9), axiv: 96.69v1 [math.-ph.]. 3 V.B.Beestetskii, E.M.Lifshitz, L.P.Pitayevskii. Quatum electodyamics. IV M.: FIZMATLIT Publishes, 1 (i Russia). 4 A.A.Sadovoy, A.S.Ul yaov. Voposy Atomoi Nauki I Tekhiki. Seiya: Teoeticheskaya i Pikladaya Fizika. -3, (7) (i Russia). 5 A.J.Sileko. J. Math. Phys. 44, 95 (3). 6 E.Eikse. Phys. Rev. 111, 111 (1958). 7 V.P.Nezamov. Voposy Atomoj Nauki i Tekhiki. Seiya: Teoeticheskaya I Pikladaya Fizika., 1 (1988) (i Russia). 8 V.P.Nezamov. Fiz. Elem. Chastits At. Yada. 37, 15 (6) (i Russia) Phys. Pat. Nucl. 37, 86 (6). 9 E. de Vies, J.E. Joke. Nucl. Phys. B6, 13 (1968). 1 V.G.Bagov ad.m.gitma, Exact Solutios of Relativistic Wave Equatios (Kluwe Academic, odecht, 199). 11 A.J.Sileko, Pis ma v Jual Fizika Elemetaykh Chastiz I Atomogo Yada. 5, 84 (8) (i Russia) Phys. Pat. Nucl. Lett. 5, 51 (8). 1 A.A.Sadovoy, A.S.Ul yaov. The electo desity of highly ioized ios i tasuaic elemets. VI-th Scietific-Techical Cofeece «Youth i Sciece» (7) (i Russia). 13 A.A.Sadovoy. Multidimesioal agula fuctio methods i theoetical ad applied physics. Azamas-16, VNIIEF, 1994 (i Russia). 14 T..Newto ad E.P.Wige, Rev. Mod. Phys. 1, 4 (1949). 13