m 0 are described by two-component relativistic equations. Accordingly, the noncharged

Transcription

1 Generalized Relativitic Equation of Arbitrary Ma and Spin and Bai Set of Spinor Function for It Solution in Poition, Moentu and Four-Dienional Space Abtract I.I.Gueinov Departent of Phyic, Faculty of Art and Science, Onekiz Mart Univerity, Canakkale, Turkey Uing condition of relativitic covariance, group theory and Clifford algebra the ( ) - coponent Lorentz invariance generalized relativitic wave equation for a particle with arbitrary a and pin i uggeted, where and,,,,,... It i hown that the charged calar ( e and ) and noncharged calar ( e and ) particle with are decribed by two-coponent relativitic equation. Accordingly, the noncharged calar feri particle (, e and ) can be ued a an eleentary particle of the Standard Model of particle phyic. In the cae of arbitrary integral pin,,..., the relativitic equation for lead to the equation of ale boon particle. For the olution of preented in thi work generalized relativitic equation in the linear cobination of atoic orbital approxiation, the (+)-coponent orthogonal bai et of pinor function for the arbitrary a and pin are uggeted in poition, oentu and fourdienional pace. Keyword: Relativitic covariance, Clifford algebra, Lorentz invariance, Exponential type pinor orbital, Slater type pinor orbital I. Introduction It i well known that the for of relativitic or nonrelativitic wave equation of otion depend on the pin of the particle. The uual Schrödinger equation decribe the otion of the pin- particle in the nonrelativitic doain, while the Klein-Gordon equation i the relativitic equation appropriate for pin- particle. The pin-/ particle are governed by the relativitic Dirac equation which, in the nonrelativitic liit, lead to the Schrödinger- Pauli equation [-4]. For particle with pin- or higher, only relativitic equation are uually conidered [5]. The firt higher pin equation have been propoed by Dirac in []. Thee equation in the preence of an external electroagnetic field, a wa hown by Fierz and Pauli [7], led to

2 the inconitencie. They have uggeted the equation for the pecial cae of and. Rarita and Schwinger [8] have developed theory of pin free particle which contain any of the feature of the Dirac theory. The theory of pin- free particle ha been alo developed by Proca [9], Keer [] and Bargann and Wigner []. All of thee forali for pin- free particle have any intrinic contradiction and difficultie when an electroagnetic field interaction i introduced (ee [] and reference therein). It hould be noted that the atheatical tructure of our tudy, baed on the cae of new definition of Boe-Ferion theory (ee [] and reference therein), i different fro all of the approache which are available in the literature. Therefore, the generalized relativitic equation preented in thi work can not be reduced to the. By the ue of group theory and Clifford algebra, we have hown in [4] that the generalized relativitic equation for the particle with arbitrary half-integral pin i conitent and caual in the preence of an electroagnetic field interaction. The ai of thi work i, uing the ethod et out in [4, 5], to etablih the generalized relativitic equation for ferion and boon with arbitrary value of paraeter. For the olution of thi equation by the ue of linear cobination of atoic orbital (LCAO) approach, the orthogonal bai et of pinor function are preented in poition, oentu and four-dienional pace. II. Generalized relativitic equation of arbitrary a and pin The arguent given for the olution of thi proble are baed on three copletely different point of view, naely, the application of group theory, and aking ue of the condition of relativitic covariance and Lorentz invariance. II-. Ue of group theory and Clifford algebra For a ingle particle of charge e and a the relativitic Hailton operator i given by Hˆ c ( pˆ A) c ea, (II-) e 4 c where and,,,... for ferion, and,,... for boon, A i the calar potential, A the vector potential and ˆp the oentu operator. i

3 The new arguent in preented approach baed on the ue of group theory and Clifford algebra. It i well known, in accordance with the potulate of quantu echanic the Hailton operator Ĥ ha to be linear and Heritian. One can iediately ee that the condition of linearity cannot be fulfilled, ince the quare root i not a linear operator. Therefore, the relativitic proble for arbitrary a and pin can be viewed in ter of a pecial polynoial algebra [, 7]. In a previou work [4], for the linearization of the quare root in the cae of half-integral pin we have ued the group theory and Clifford algebra. The generalized relativitic proble for arbitrary a and pin can be olved in a iilar way. Uing the ethod et out in [4], we obtain for the order of the Clifford algebraic Dirac group the following relation: g, (II-) 8( ) where,,,,,... Thi group ha g clae, therefore, repreentation. The dienion n i for thee repreentation are deterined by g irreducible g ni g, (II-) i where n i fori g. ( ) for i g (II-4) The one-dienional repreentation do not atify the condition of Clifford, therefore, only the ( ) dienional irreducible repreentation can be ued. The reult are preented in Table. II-. Generalized relativitic equation Making ue of Table obtained fro the application of group theory and condition of relativitic covariance we introduce the following () () Heritian and unitary atrice:

4 (II-5) and I,,,... ( ) for and II I for,,... and. ( II 7) Thee atrice atify (II-8) k k. (II-9) k l l k kl I It hould be noted that, in the cae of integral value of the atrice in the for I I do not atify the condition (II-8), i.e., k k for,,... (II-) Therefore, Eq. (II-7) correpondence to the cae of ale particle for,,.... The generalized relativitic equation correponding to the atrice (II-5), (II-) and (II-7) i defined a i ˆ H t (II-) Hˆ c ( pˆ A) c ea (II-) e c 4

5 5, (II-) where for integral pin (=,,, ),,... (II-4a),,.,.. (II-4b) for half-integral pin 5,,,...,,... (II-5a)

6 ...,, (II-5b) Here,, and, are the ingle- and two-coponent atrice, repectively. The two-coponent atrice are defined a u u, (II-a) u u,, (II-b) where and for =,,, and 5,,..., repectively. By the ue of procedure decribed in Dirac paper [8, 9] it i eay to how that the generalized relativitic equation (II-) atifie the condition of Lorentz invariance. In the pecial cae of calar particle (=), the generalized relativitic equation (II-) ha the for: i t Hˆ (II-7) Hˆ c ( pˆ A) c ea (II-8) e c, (II-9) where α σ, β. (II-)

7 Here, i fored by the Pauli atrice,, and pˆ i z x y. (II-) i i x y z In the cae of free particle for e=, = and, the two-coponent relativitic equation becoe H (II-), H c p c (II-). A can eaily be een that the Eq.(II-7) for relativitic calar particle i a firt-order differential equation, while the Klein-Cordon equation for a econd-order differential equation. Therefore, one ha to arrive iediately at the concluion that the Klein-Cordon equation doe not eet the requireent of the condition of relativitic covariance, naely, the condition of linearity for a relativitic Hailtonian. The Klein-Gordon equation only partially atifie the potulate of (relativitic) quantu echanic. III. Bai et of pinor function in poition, oentu and four-dienional pace The elaboration of algorith for the olution of the generalized relativitic equation for the particle with arbitrary a and pin in linear cobination of atoic orbital (LCAO) approach [-] neceitate progre in the developent of theory for coplete orthonoral bai et of relativitic pinor function of ultiple order. The ethod for contructing in poition, oentu and four-dienional pace the coplete orthonoral bai et for (+)-coponent relativitic tenor wave function and Slater tenor orbital ha been uggeted in previou article []. Extending thi approach to the cae of pinor of ultiple order and uing the ethod et out in [4], we contruct in thi tudy the relevant coplete orthonoral bai et of (+)-coponent relativitic -exponential type pinor orbital ( -ETSO) for particle with arbitrary a and pin in poition, oentu and four-dienional pace through the et of one- and two-coponent pinor type tenor pherical haronic and radial part of the coplete orthonoral et of nonrelativitic - exponential type orbital ( -ETO) [5] the angular part of which are the calar pherical 7

8 haronic. The indice occurring in the radial part of quantu nuber []. It hould be noted that the nonrelativitic cae of -ETO i the elf-frictional -ETO are the pecial -ETSO for =, i.e.,. The bai et of relativitic pinor of ultiple order obtained ight be ueful for olution of generalized relativitic equation of arbitrary a and pin particle when the coplete orthonoral relativitic -ETSO bai et in LCAO approxiation are eployed. We notice that the definition of phae in thi work for the calar pherical haronic Y * l l the ing factor Yl i Yl. l l * ll Yl l III-. Relativitic pinor type tenor pherical haronic differ fro the Condon-Shortley phae [7] by In order to contruct the coplete orthonoral bai et of relativitic -ETSO and X -Slater type pinor orbital ( X -STSO) of (+) order in poition, oentu and fourdienional pace we introduce the following forulae for the independent pinor type tenor (STT) pherical haronic of (+) order (ee Ref. []): for integral pin: H H H H, H H H H, H H,,,,,,,,,,,, (III-a) (III-b) for half-integral pin 8

9 Y Y Y, Y Y,,,,,,,,,,,,, (III-a) (III-b) Thee STT pherical haronic are eigenfunction of operator ˆj, ˆj, lˆ and ŝ. The one- z and two-coponent bai et of STT pherical haronic H,, H, Y,,, occurring in Eq. (III-a), (III-b) and (III-a), (III-b), repectively, can be expreed through the calar pherical haronic: and, ( ), H a Y H Y l, ia ( ) Y, l, a l a( ) Y, l Y, ia( ), l,, ia( ), l Y Y (III-a) (III-b) (III-4a) (III-4b) where for integral pin (), l j j, j j, / and, l j l t, t ( j l),,...,, for half-integral pin 9

10 (), l j j, j j, j l t, t ( j l),,...,, l /,. Here, a ( ) are the odified Clebch-Gordan coefficient defined a a ( ) l( ). (III-5) See Ref. [7] for the definition of Clebch-Gordan coefficient ll l. The STT pherical haronic H,, H, and Y,,, atify the following orthonorality relation: for fixed H * *, H, Sin d d H, H, Sin d d l j l j ll jj (III-a) * *,, Sin d d,, Sin d d l j l j ll jj H H H H (III-b) Y, Yl j, Sin dd Y, Yl j, Sin d d ll jj (III-7a), l j, Sin d d, l j, Sin d d ll jj. (III-7b) III-. Bai et of relativitic -ETSO and X -STSO function To contruct the bai et of (+)-coponent relativitic pinor fro STT pherical haronic and radial part of nonrelativitic orbital we ue the ethod et out in a previou paper [4]. Then, we obtain for the coplete bai et of relativitic pinor wave function, and Slater pinor orbital X in poition pace the following relation: for integral pin

11 nl() r,, r,, a a nl() a nl( ) r,, R r H, nl( ),, nl a r n r,, R r H, ( ),, nl ia nl r ia nl( ) r,, ia nl() r,, ia nl() r,, nl() r,, r,, a a nl() a nl( ) r,, R r H, ( ),, nl a nl r n r,, R r, ia nl( ) r,, nl H ia nl( ) r,, ia nl() r,, ia nl() r,, (III-8a) (III-8b) X nl() r,, r,, a a nl() a nl( ) r,, R r H,,, a r H,,,, (III-9) ia nl() r,, ia nl() r,, ia () r,, nl nl( ) n n r,, R r n ia nl( ) r for half-integral pin

12 a a R ry, nl n r,, R r, nl ia ia nl() r,, r,, nl() a nl( ) r,, a nl( ) r,, ia nl( ) r,, ia r,, nl() nl() r,, r,, nl() nl() r,, r,, a a nl() R ry,,, nl a nl r n r,, R r, nl nl() ia nl() r,, ia nl() r,, a nl( ) r,, ( ) ia nl( ) r,, ia r,, (III-a) (III-b) X n r,, a a R ry, n R r, n ia ia nl() r,, r,, nl() a nl( ) r,, a nl( ) r,, ia nl( ) r,, ia r,, nl() nl() r,, r,, nl() (III-), where n, j n, j l in( j, n ) and

13 n n ( ) r Rnr Rn, r r e. (III-) ( n)! The relativitic pinor wave function ( K, n K n poition, oentu and four-dienional pace are defined a ) and Slater pinor orbital K in K (, r), (, k), Z (, ) (III-) n n n n n K (, r), (, k), Z (, ) (III-4) n n n n Kn X (, r), Un(, k ), Vn (, ). (III-5) n Here, the k nl, nl k and k nl are the nonrelativitic coplete bai et of orbital. They are deterined through the correponding nonrelativitic function in poition, oentu and four-dienional pace by k (, r), (, k), z (, ) (III-) nl nl nl nl k (, r), (, k), z (, ) (III-7) nl nl nl nl k (, r), u (, k), v (, ). (III-8) nl nl nl nl See Ref. [8] for the exact definition of function occurring in Eq. (III-) - (III-8). The relativitic pinor orbital atify the following orthogonality relation:,, Kn x Kn l j x dx nn ll jj (III-9) n n! n n n nlj ll jj K, x K, x dx.!! (III-) Uing the relation a and forulae jl l H (, ) Y (, ) (III-a) l ll H (, ) i Y (, ) (III-b) l ll for the calar particle it i eay to how that the relativitic pinor function

14 K n, K n and relativitic Slater pinor orbital K n for particle with pin = are reduced to the correponding quantitie for nonrelativitic coplete bai et in poiton, oentu and four-dienional pace, i.e., K k, K k and n nl n nl l l K n k, where nl l, j l, t, ( ) l and l. Thu, the nonrelativitic and relativitic calar particle can be alo decribed by wave function K, K n n and Kn for =t=, j l and l, i.e., K k nll iknl l nll (III-a) K nl l knl l ik nll (III-b) K k nll. iknl l nll (III-) The -, -,- and 4-, 8-coponent coplete orthonoral bai et of relativitic through the nonrelativitic -ETO in poition pace for repectively, are given in Table,, 4 and 5,. III-. Derivative of -ETSO in poition pace -ETSO,, and,, Now, we evaluate the derivative of -ETSO with repect to Carteian coordinate that can be ued in the olution of reduced relativitic equation when the LCAO approach i eployed. For thi purpoe we ue the -ETSO in the following for: for integral pin 4

15 , n, n, n n n, n n, n, n, (III-4) for half integral pin, n, n, n n n, n n, n, n, (III-5) where,, and,,, are the one- and two-coponent pinor, repectively, are, defined by r H, R (III-a), n nl r, R H (III-b), n nl ry, R (III-7a), n nl r R,. (III-7b), n nl Here, () and () for integral and half-integral pin, repectively. 5

16 To obtain the derivative of ' df f l ( f Yl ) k, kl bk Yl k, z k dr r -ETSO we ue the following relation [5]: (III-8) ' df f l i f Yl k, kl dk Yl k, x y k dr r (III-9) ' df f l i f Yl k, kl ck Yl k, x y k dr r, (III-) where f i any function of the radial ditance r and l bk ( l k)( l k) / ( l ) k ( l k) (III-) l k / ( k)( ) / ( )( ) (III-) d k l k l k l k l k l / l, k k k c k ( l k )( l k ) / ( l ) k ( l k) d. (III-) The ybol / ' in Eq. (III-8), (III-9) and (III-) indicate that the uation i to be perfored in tep of two. Thee forulae can be obtained by the ue of ethod et out in Ref.[9]. Uing Eq. (III-8), (III-9) and (III-) we obtain for the derivative of two-coponent pinor of half-integral pin the following relation: ˆ ˆ c ' drnl R nl c( p) c( p) Rnl Y k, kl i k dr r B ;, D ;, ;, B ;, k k k C k (III-4) ˆ ˆ ' drnl R nl c( p) c( p) Rnl c k, kl k dr r B ;, D ;, ;, B ;, k k k C k, (III-5)

17 where, ( ;, ), B a b Y (III-) l k k ( ) lk, ( ) ( ;, ), C a c Y (III-7) l k k ( ) lk, ( ) ( ) D ;, a d Y,. (III-8) l k k ( ) lk, The forulae preented in thi work how that all of the (+)-coponent relativitic bai pinor wave function and Slater bai pinor orbital are expreed through the et of one- and two-coponent bai pinor. The radial part of thee bai pinor are deterined fro the correponding nonrelativitic bai function defined in poition, oentu and four-dienional pace. Thu, the expanion and one-range addition theore etablihed in [8] for the nonrelativitic k nl and k l nl bai et in poition, oentu and four- l dienional pace can be alo ued in the cae of relativitic bai pinor function K nl l and K nl l. Accordingly, the electronic tructure propertie of arbitrary a and pin relativitic yte can be invetigated with the help of correponding nonrelativitic calculation. IV. Concluion In thi tudy, we have generalized the Dirac pin theory to a relativitic theory for particle with arbitrary a and integral and half-integral pin. The relativitic bai et of pinor orbital for the arbitrary pin particle in poition, oentu and four-dienional pace are alo contructed. It i hown that thi theory ha the following propertie: () The generalized relativitic atrice are irreducible and Clifford algebraic. () The generalized relativitic wave function and atrice poe the independent coponent. () The generalized relativitic equation atifie the condition of Lorentz invariance. (4a) The relativitic calar particle for e and atify the two-coponent relativitic equation. (4b) The free particle with e=, = and i decribed by two-coponent relativitic equation. 7

18 (5) The integral pin,,... atifie the equation for ale particle. coponent generalized relativitic () The half-integral pin,,... relativitic equation for particle with. atifie the (+)-coponent generalize (7) The relativitic bai et of pinor orbital for the arbitrary a and pin particle in poition, oentu and four-dienional pace are expreed through the correponding quantitie of nonrelativitic pinor function. The generalized relativitic theory preented in thi work can be ued in the olution of different proble of decribing particle with arbitrary a and pin within the fraework of relativitic quantu echanic when the poition, oentu and the four-dienional pace are eployed. Reference. A. S. Davydov, Quantu Mechanic, nd ed., Pergaon, Oxford, 95.. A. Meiah, Quantu Mechanic, Vol. II, Wiley, New York, 98.. I. P. Grant, Relativitic Quantu Theory of Ato and Molecule, Springer, New York,. 4. K. G. Dyall and K. Faegri, Introduction to Relativitic Quantu Cheitry, Oxford Univerity Pre, New York, V. B. Beretelkii, E. M.Lifhitz and L.P. Pitaevkii, Quantu Electrodynaic, nd., Pergaon, Pre, Oxford, P. A. M. Dirac, Proc. R. Soc. Lond. A 55 (9) M. Fierz and W. Pauli, Proc. R. Soc. Lond. A 7 (99). 8. W. Rarita and J. Schwinger, Phy. Rev. (94). 9. A. Proca, Le Journal de Phyique et le Radiu, 7 (9) 47.. N. Keer, Proc.Roy. Soc. A 77 (99) 9.. V. Bargann and E. Wigner, Proc. Nat. Sci. (USA), 4 (948).. V. V. Varlaov, Int. J. Theor. Phy., 4 (7) 74.. B. A. Maedov, Int. J.Quantu Che., 4 (4). 8

19 4. I. I. Gueinov, Phil Mag., 9 () I. I. Gueinov, Phy. Scr. 8 () 555 (pp).. I. V. V. Raghavaryulua and N. B. J. Menon, Math. Phy. (97) P. Weinberger, Phil. Mag., 88 (8) P. A. M. Dirac, Proc. Roy. Soc. A7 (98). 9. P. A. M. Dirac, Proc. Roy. Soc. A (9).. W. Greiner, Relativitic Quantu Mechanic, Springer, 99.. I.P. Grant, Relativitic Quantu Theory of Ato and Molecule, Springer,.. K.G. Dyall, K. Fǽgri, Introduction to Relativitic Quantu Cheitry, Oxford Univerity Pre, 7.. I.I. Gueinov, Phy. Lett. A, 7 (7) 44; 7(9) I. I. Gueinov, J. Math. Che., 47 () I. I.Gueinov, Int. J. Quantu Che., 9 () 4.. I. I. Gueinov, Aerican Intitute of Phyic Conference Proceeding, 889 (7) E. U. Condon, G.H. Shortley, The Theory of Atoic Spectra, Cabridge Univerity Pre, Cabridge, I. I. Gueinov, J. Math. Che., 4 (7) H.A. Bethe, E.E. Salpeter, Quantu Mechanic of One- and Two- Electron Ato, 957, Berlin, Springer. 9

20 Table. Suary of the generalized Dirac group propertie for 7 No. of No. of No. of No. Group No. of No. of No. of No. of No. of No. of -D -D 4-D -D Order Clae -D irred -D irred 4-D irred -D irred 8-D irred irred irred irred irred of Note: irred-irreducible repreentation Table. The exponential type pinor orbital in poition pace for, n, l n, and l l l n l l nl l i i i - i i

21 Table. The exponential type pinor orbital in poition pace for, n, l n, l j l, j j, t j l and n n t n l j t n n i i - i 4 i 4 i 4 - i 4 i i i i - i 4 i 4 i 4 i 4 i 4 4 i 4 i - 4 i - i 4

22 Table 4. The exponential type pinor orbital in poition pace for l j l, j j, t j l and n n t, n, l n, n l j t n n 4 5 i 5 i 5 i 5 - i 5 - i 5 4 i i i - i - i 4 i 4 4 i

23 i 4 4 i 4 i i 4 i 4 - i 4 i 4 i 4 - i 4 i 4

24 Table 5. The exponential type pinor orbital in poition pace for t j l n, l n, l j l, j j,, and n n t n l j t n / n / / -/ / / -/ i i i i / - / -/ i i i i / / i / -/ i i i i -/ i 4

25 Table. The exponential type pinor orbital in poition pace for, n, l n, l j l, j j, t j l and n n t n l j t n / n / 4 / i 4 / i 4 -/ i 4 -/ / 5 / / i 4 5 i 5 i -/ i 5 -/ i 5 / / 5 i 5 i 5

26 / 5 i i i 5 5 -/ -/ i i i 5 5 i 5 i i 5 5 5/ 5 5/ i 5 / i i 5 / 5 i i i 5 -/ i i 5 -/ i i 5-5/ 5 i