The Solutions of the Classical Relativistic Two-Body Equation

Transcription

1 T. J. of Physics (998), c TÜBİTAK The Soutions of the Cassica Reativistic Two-Body Equation Coşkun ÖNEM Eciyes Univesity, Physics Depatment, 38039, Kaysei - TURKEY Received Abstact With the eativistic Kepe pobem, a two-body system is studied fo positonium, the hydogen atom and m >> m cases.. Intoduction It is we known that the eativistic two-body pobem has a ong histoy and it contains some difficuties. Thee ae two ways to undestand this system: Fist, one is via fied theoy, which the second invoves action-at-a-distance theoy [] (both of which ae discussed by Havas []). Howeve, thee have been sevea attempts to obtain the obit soution fo a cassica eativistic two-body system inteacting eectomagneticay. If it is soved exacty, then the eativistic bound state pobem in quantum mechanics wi be we undestood with it wi be possibe to stat fom coesponding pobem in the cassica mechanics and quantize it [3]. Nevetheess, it must be emphasized that these appoaches ae amost esticted to cicua obits. In Schid s wok [4], the concentic cicua motion of two cassica eativistic point chages inteacting eectomagneticay had been descibed. The simia discussion was aso made by Andeson-Baeye [5] and Codeo-Ghiadi [6]. It is known that the cassica Kepe pobem is a typica two-body pobem [7] and it was fist discussed eativisticay by Cawey [8]. On the othe hand it must be pointed out that the obita discussion of eativistic two-body system is not ony consisted of the cicua obits. Recenty, a moe genea 07

2 soution was obtained by Baut-Caig [9] in which the beginning action is same with ou action [0]. In this wok the eativistic Kepe pobem, which is a specia case of ou action, wi be studied, and the most genea obita soution wi be discussed fo the attactive and epusive potentia. Then the case of the positonium (m = m ) and the hydogen atom (m ) wi be discussed fo the eativistic two body system inteacting (e e /) potentia. Finay, the case of m >> m wi be discussed.. The Reativistic Kepe Pobem Ou one-time paamete action fo inteacting eectomagnetic two-body system is given as foows [0]. A = dt [m v + m v + e ] e ( v v ), () whee m,m ae masses coesponding to e,e chages, espectivey; v and v ae vecto veocities; and T is cente of mass time. The fist two tems ae kinetic enegy, the thid tem is the mutua inteaction tem. On the othe hand thee ae non-dynamica but hoonomic constaints on 4-veocity with x. iµ x.µ i = (i =, ; c =). This action becomes fo one patice: { A = dt m v + e e }. () The Hamitonian is obtained easiy in the poa coodinates: H = m + p ɛ K = m + p + p ϑ ɛk, (3) whee K = e e,andɛ = ±; fom which it is obvious that the potentia is the attactive fo ɛ = and the epusive fo ɛ =. The equations of motion in poa coodinates ae 08. = H p = ϑ. = H p ϑ = p m + p + p ϑ p ϑ m + p + p ϑ (4a) (4b) p = H ṙ = p ϑϑ. ɛk (4c) p. ϑ = H ϑ = p. ϑ =0, p ϑ = = Constant. (4d)

3 It can be seen easiy that the enegy is aso conseved as the angua momentum p ϑ ; E = m + p + p ϑ ɛk = Constant. (5) Now we find the obit equation. It is obvious that it is easiy obtained fom Eqns. (4a) and (4b): d dϑ = p (6) dϑ = [ K 4 whee u =(/) and afte integation it is found [ u = = ɛek + + (E m )( K ) K E K 4 du ] u + EɛK u + E m (7) ] cos K4 (θ θ 0). (8) We take c =. In ode to obtain the coect esut we have to epace c and m mc at the Eq. (8) o m mc and v (v/c) attheeq.().thenitcanbeseen that it is (/) =(/mete) inmks. If we intoduce C = ɛek K 4, β = K4, e = + (E m )( K 4 ) E K 4, (9) whee the β and e ae dimensioness constant and C has dimension of m.theneq. (8) becomes = C[ + e cos β(ϑ ϑ 0)], (0) which can be aso witten as = C[ +echβ(ϑ ϑ 0)], () whee C = ɛek K K 4 = C, β = 4 =iβ, cos iβ = chβ. () It is obvious that Eqs. (0) and () ae simia to the genea conic equations. If it is chosen β = then these equations ae educed to the cassica Kepe s equation [7]. Now et s examine the obits fo the diffeent vaues of e and foattactive potentia (ɛ =). 09

4 I- e>: a- When (E m K ) > 0, 4 < the soution is Eq. (0) which is simia to the noneativistic motion. The obit is a hypeboa when cos β(ϑ ϑ 0 ) > 0 and the minimum distance that the patice can appoach is min =[/C( + e)]. When cos β(ϑ ϑ 0 ) < 0 the soution is unphysica. b- When (E m K ) < 0, 4 > the soution is eativistic and given by Eq. (). The obit is a spia which comes fom max =[/C( +e)] and fas to the oigine. The numbe of it s evoution depends on e. II- e<: a- When (E m K ) < 0, 4 < the soution is Eq. (0) which is simia to the noneativistic soution. The obit is a pecessing eipse by about ϑ K4 π pe evoution. b- When (E m ) > 0, K4 > the soution is Eq. (), which is the eativistic soution. III- e =,E m =0: a- When K < the soution is Eq. (), coesponding to the noneativistic motion: = C[ + cos β(ϑ ϑ 0)]. (3) The obit is a paaboa and the min distance that the patice can come is min =(/C). b- When K4 > the eativistic soution is = C[ +chβ(ϑ ϑ 0)]. (4) The obit is a semipaaboa, oiginating fom and faing to the oigin. IV- e =0: a- When K4 < the soution is = C and the obit is a cice as with noneativistic motion. b- When K4 > thee is no soution. On the othe hand, we can aso discuss the antipatice soutions. In ode to obtain anti-patice soutions, we must epace E E in the patice soutions. As it wi be easiy seen in Tabe, thee is a coss eationship between the patice and anti-patice soutions. Fo exampe, the patice soution when e>, ɛ= + becomes the anti-patice soution fo e>, ɛ=. Thee ae aso othe simia soutions. This shoud not come as a supise as we expect it to confom with quantum mechanics. If the potentia is epusive (e = ) we must epace C C and C C in the above soutions. Thus the a soutions fo the eativistic Kepe pobem can be odeed as in Tabe. It is obvious that these soutions ae moe genea than Cawey s soutions [8]. 0

5 3. Two-Body System Now et s etun to the action descibed by Eqn. () and negect the inteacting tem between the veocities and the sef inteaction tems. Eqn. () then becomes A = dt [m v + m v + e ] e. (5) It is known that the tota momentum of the system is a constant in the COM [0]. If it is chosen identicay zeo then the Hamitonian is obtained easiy: H = m + p + m + p ɛ K, (6) whee p is the eative momentum ( p = p = p when p = 0), is the eative coodinate K = e e,andɛ = ±. Binomia expansion of this Hamitonian contains the Beit Hamitonian in quantum mechanics []. It is obvious that the enegy is a constant of motion. In poa coodinates, E = m + p + p ϑ + m + p + p ϑ ɛk = Constant. (7) The equations of motion ae aso obtained easiy. = H = p + p m + p + p ϑ m + p + p ϑ ϑ. = H = p ϑ + p ϑ m + p + p ϑ m + p + p ϑ p = H ṙ = p ϑϑ. ɛk (8a) (8b) (8c) p. ϑ = H ϑ =0, p ϑ = = Constant. (8d) As is expected, the tota angua momentum is a constant of motion. The obits equation can be found afte itte ageba: dϑ = (E + ɛ K ( ) E + ɛ K d ) 4 ( ) E + ɛ K [ ]. (9) (m + m 4 )+ +(m m )

6 I- When m = m = m (The positonium case) It wi be seen easiy that the ast tem at the denominato is zeo. Thus the Eq. (9) becomes du dϑ = [ K 4 ], (0) u 4 + EɛK u + E 4m 4 4 whee u =/. This equation is same with the Eq. (7) when = and m =m. Theefoe the soutions given in the Tabe can be adapted to this case easiy. II- When m (The case of hydogen atom): To cacuate the imit of the Eq. (9) when m we must epace E E +m. Atfe some ageba it becomes dϑ = du [ K 4 ]. () u + E ɛk u + E m ageba: It is aso obvious that this equation is same as the Eq. (7) when E = E. III- When m >> m : Afte we epaced E E + m at the Eq. (9), Eq. () is obtained with itte dϑ d = () (m + V ) (m + m ) 4 + (m m ) (m +V ) whee V = E + e K. If we ommit the second and highe ode tems with espect to (/m ), and afte expanding the denominato of the ast tem in the squae oot in binomia seies, we obtain: dϑ = du ( K 4 ) ( ), (3) u E + ɛk + m ɛk m u + E m + m E m whee u = /. As is seen, this intega contains the coection tems accoding to one patice system, which ae m ɛk m u + m E m. Nevetheess, it is obvious that this intega is simia to Eq. (7) if the constants in the squae oot ae edefined. Consequenty obits ae aso simia to the eativistic Kepe obits.

7 Tabe. The eativistic Kepe soution of patice and antipatice fo attactive and epusive potentia whee (*): Noneativistic soutions, (#): Unphysica soutions, (x): Reativistic soutions Patice Antipatice K 4 < K 4 > K 4 < K 4 > (*) (cos βϕ > 0) (x) (cos βϑ < 0) e> e m > H. boa ε = (at.) = c[ + e cos βϕ ] = C [ + e cos βϑ ] (x) (#) e m < 0 - = C[ +echβϑ ] - = C[ +echβϑ ] (*) (cos βϑ < 0) (x) (cos βϑ > 0) ɛ = e m > (ep.) = C[ + e cos βϑ ] = C [ + e cos βϑ ] (#) (x) e m < = C[ +echβϑ ] = C [ + e chβϑ ] (x) (#) e< ɛ = e m > Eipse (att.) = C[ +echβϑ ] = C [ + e chβϑ ] ( ) (#) e m < = C[ + e cos βϑ ] = C [ + e cos βϑ ] (#) (X) ε = - - (ep.) e m > 0 = C[ +echβϑ ] = C[ + e chβϑ ] (#) (X) e m < = C[ + e cos βϑ ] = C [ + e cos βϑ ] e =0 E m =0 ɛ = (*) (x) (#) (#) Paaboa (att.) = C[ + cos βϑ ] = C[ +echβϑ ] = C [ + cos βϑ ] ɛ = (#) (#) (x) (x) (ep.) = C[ +chβϑ ] = C [ + cos βϑ ] = C[ +chβϑ ] = C [ + cos βϑ ] = C[ +chβϑ ] e =0 E = m ( K4 ) ɛ = (*) (#) - - cice (att.) = C = C ɛ = (#) (x) - - (ep.) = C = C

8 Refeences [] A.O. Baut, Eectodynamics and Cassica Theoy of Fieds and Patices (Second Edition Dove, New Yok 980) Ch.6. [] P. Havas, Phys. Rev., 74 (948) 456. [3] A. Degaspeis, Phys. Rev. D., 3 (97) 73. [4] A. Schid, Phys. Rev., 3 (963) 767. [5] C.M. Andesen and Hans C. von Baeye, Ann. Phys., 60 (970) 67. C.M. Andesen and Hans C. von Baeye, Phys. Rev., D. 5 (97) 80. [6] P. Codeo and G.-C. Ghiadi, J. Math. Phys., 7 (973) 85. [7] H. Godstein, Cassica Mechanics (Second Edition, Addison-Wesey Pubishing Company, 980) Ch. 3. [8] Robet G. Cawey, J. Math. Phys., 8 (967) 09. [9] A.O. Baut and G. Caig, Physica A, 97 (993) 75. [0] C. Önem, The Cass Rea. Two-Body Pob. with Sef Inteaction, N. Cimento B, 0 (995) 0. [] H.A. Bethe and E.E. Sapete, Quantum Mechanics of One and Two Eecton Atoms (Spinge-Veag, 957) p.9. 4