On solution of Klein-Gordon equation in scalar and vector potentials
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1 On solution of Klein-Gordon equation in salar and vetor potentials Liu Changshi Physis division, Department of mehanial and eletrial engineering, Jiaxing College, Zhejiang, , P. R. China Abstrat Based on the Coulomb gauge, the aurate Klein-Gordon equation in stati salar and vetor potentials was derived from Klein-Gordon equation in eletromagneti environment. The orret equation developed in this omment demonstrates that so-alled the Klein-Gordon equation with salar and vetor potentials is inorret; therefore, some papers published to solve Klein-Gordon equation with equal salar and vetor potential are also wrong. Keywords: Aurate Klein-Gordon equation; Salar potential; Vetor potential; Derive; Solution. PACS: , w 1. Introdution The desription of phenomena at higher energy requires the investigation of relativisti wave equation. This means equation whih is invariant under Lorentz transformation. The presene of 1
2 strong fields introdues quantum phenomena, suh as superritiality and spontaneous pair prodution that an not be desribed using existing tehniques. Perhaps beause of the diffiulties on mathematis, in relativisti quantum mehanis only a few bound state solutions of the Klein-Gordon equation with salar and vetor potentials an be solved [1]. In the last twenty years, a lot of attentions have been paid to solve these relativisti energy levels and wave funtion for various quantum systems [-7]. However, all of these works in this area are based on equation whih is expressed as [8] ( ħ = = 1, Natural unit) d dr ( l + 1) l + E V ( r) m0 + S ( r) u( r) = 0 r (1) Where vetor potential is V ( r ) and S ( r) is salar potential. However, up to now, there has hardly been one who heked whether equation (1) is absolute orret or not. Unfortunately, it will be shown in this paper that nobody an derive equation (1) from Klein-Gordon equation with an eletromagneti field. Therefore, it is interesting to know why there have been so many researhes that used equation (1) to detet Klein-Gordon equation; the reason for this question is that they have misunderstood one passage in one textbook [9],
3 ˆ α ˆ p + βˆ ( m0 + V ) ( E V1 ) Ψ = 0 Obviously, equation () is relative invariant, but equation (1) is not relative invariant.. Basi equation With the minimal oupling, the free Klein-Gordon equation was transformed into the free Klein-Gordon equation with eletromagneti field [1, 30], () Ψ = + + Ψ 1 e i ea ħ 0 ( r, t) iħ A m0 t (3) Where m 0 is rest mass, equation (3) is invariable under Lorentz transformation. When both eletroni field and magneti field is stati, a stationary state of the Klein-Gordon equation has the form E Ψ ( r, t ) = Ψ ( r ) exp i t ħ (4) Where E is the energy of the system. 1 e ( E ea0 ) Ψ ( r ) = iħ + A + m0 Ψ (5) It is not diffiultly to develop equation (3) into [31] Ψ = Ψ 1 e e e ( E ea0 ) ( r ) ħ iħ ( A) iħ A A m0 (6) 3
4 4 E ea0 e m0 [ + A ] Ψ ħ ħ ħ e + iħ ( A + A ) Ψ ( r ) = 0 ( r ) (7) Stati magneti field always preserves the Coulomb gauge and this meaning an be expressed by A = 0 (8) Introduing this expression into equation (7) yields 4 E ea0 e m0 [( ) + ( A) ħ ħ ħ 4 e + iħ A ] Ψ( r ) = 0 Equation (9) is ustomarily alled the general form of Klein- Gordon equation with stati eletromagneti field. The author of ref. [8] neither told reader who had alulated equation (1) nor derived this equation by himself, even if there are not any approximations in Ref. [8] to obtain Eq. (1); therefore, reader an not know how to obtain equation (1) in ref. [8]. It is easy for anyone who master quantum mehanis to reognize that the finally result of equation (1) is d dr ( + 1) l l + E m0 + V S ( EV + m0 S) u( r) = 0 r (9) (10) Beause equation (9) is one partial differential equation in form of omplex, there is no reason to say that the real part of equation (9) is zero, meanwhile, the imaginary part of equation (9) an not equal to zero, and therefore, equation (10) an not be
5 derived from equation (9). Beause another form of equation (1) expressed by equation (10) an not be derived from equation (9) whih is Klein-Gordon equation in stati salar and vetor potential, equation (1) is inorret. Moreover, some papers [-7] whih are so-alled solution of Klein-Gordon equation in equal salar and vetor potentials are wrong too. Referenes [1] W. Greiner, Relativisti Quantum Mehanis Wave Equations, Third edition, Springer-Verlag, : P.5-41 [] Gang Chen, Zi-dong Chen, Pei-ai Xuan, Physis Letters A, 35, 317 (006) [3] Gang Chen, Zi-dong Chen, Pei-ai Xuan, Phys. Sr., (006) (in Chinese) [4] Gang Chen, Ata Physia Sinia, 50 (9), 1651 (001) (in Chinese) [5] Zhang Min-ang, Wang Zhen-bang, Ata Physia Sinia, 55 (), 51 (006) (in Chinese) [6] Gang Chen, Chinese Journal of Atomi And Moleular Physis, 0(4), 456 (003) (in Chinese) [7] Gang Chen, Lou Zhi-mei, Ata Physia Sinia, 5 (5), 1075 (003) (in Chinese) 5
6 [8] Hou Chun-feng, Li Yan, Zhou Zhong-xiang, Ata Physia Sinia, 48 (11), 1999 (1999) (in Chinese) [9] Gang Chen, Lou Zhi-mei, Ata Physia Sinia, 5 (5), 1071 (003) (in Chinese) [10] Gang Chen, Ata Physia Sinia, 53 (3), 684(004) (in Chinese) [11] Qiang Wen-hao, Chinese Physis, 13(5),571,(004) [1] Chen Chang-yuan, Liu Cheng-lin, Lu Fa-lin, et al, Ata Physia Sinia, 5 (7), 1679 (003) (in Chinese) [13] Gang Chen, Zi-dong Chen,Lou Zhi-mei, Chinese Physis, 13(3),79,(004) [14] Qiang Wen-hao, Chinese Physis, 1(10),1054,(003) [15] Guo Jian-you, Xu Fu-xin, Chinese Journal of Atomi And Moleular Physis, 19(3), 313 (00) (in Chinese) [16] Gang Chen, Zhao Ding-feng, Ata Physia Sinia, 5 (1), 954(003) (in Chinese) [17] Qiang Wen-hao, Chinese Physis, 13(3),83, (004) [18] Qiang Wen-hao, Chinese Physis, 11(8), 757, (00) [19] Qiang Wen-hao, Chinese Physis, 1 (), 136, (003) [0] Zhang Xue-ao, Chen Ke and Duan Zheng-lu, Chinese Physis, 14 (1), 4, (005) [1] Qiang Wen-hao, Chinese Physis, 13(5), 575, (004) 6
7 [] Lu Fa-lin, Chen Chanf-yuan and Sun Dong-sheng, Chinese Physis, 14 (3), 463, (005) [3] Lu Fa-lin, Chen Hong-xia and Chen Chanf-yuan, Chinese Journal of Atomi And Moleular Physis, (3), 443 (005) (in Chinese) [4] Lu Fa-lin, Chen Chanf-yuan, Ata Physia Sinia, 53 (6), 165 (004) (in Chinese) [5] Zhao Ding-feng, Gang Chen, Ata Physia Sinia, 54 (6), 54 (005) (in Chinese) [6] Liang Zhong-yi, Yong Feng-diao, Jian Yi-liu, et al, Physis Letters A, 333, 1 (004) [7] Yong Feng-diao, Liang Zhong-yi, Chun Sheng-jia, Physis Letters A, 33,156 (004) [8] F. Dominguez-Adame, Bound States of The Klein-Gordon Equation with Vetor And Salar Hulthen-type Potentials, Phys. Lett. A, 136, 175 (1989) [9] W. Greiner, Relativisti Quantum Mehanis Wave Equations, Third edition, Springer-Verlag, : p34 [30] J. D. Bjorken and S. D. Drell, Relativisti Quantum Mehanis, MGraw-Hill Book Company, 1964 : p [31] L. I. Shiff, Quantum Mehanis, Third edition, MGraw- Hill Book Company, 1968 : p
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