APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS TO MODIFIED COULOMB POTENTIAL Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration with my advisory committee. This thesis does not include proprietary or classified information. __________________________ Julian Antolin Camarena Certificate of Approval: __________________________ __________________________ Michael S. Pindzola Eugene A. Oks, Chair Professor Professor Physics Physics __________________________ __________________________ Joseph D. Perez George T. Flowers Professor Dean Physics Graduate School APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS TO MODIFIED COULOMB POTENTIAL Julian Antolin Camarena A Thesis Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the Degree of Master of Science Auburn, Alabama December 19, 2008? iii? APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS TO MODIFIED COULOMB POTENTIAL Julian Antolin Camarena Permission is granted to Auburn University to make copies of this thesis at its discretion, upon request of individuals or institutions and at their expense. The author reserves all publication rights. ________________________ Signature of Author ________________________ Date of Graduation iv? THESIS ABSTRACT APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS TO MODIFIED COULOMB POTENTIAL Julian Antolin Camarena Master of Science, December 19, 2008 (B.S. University of Texas at El Paso, 2006) 53 Typed Pages Directed by Eugene Oks We apply Dirac?s generalized Hamiltonian dynamics (GHD), a purely classical formalism, to spinless particles under the influence of a binomial potential. The integrals of the motion for this potential were chosen as the constraints of GHD, and use Fradkin?s unit Runge vector in place of the Laplace-Runge-Lenz vector. A functional form of the unit Runge vector is derived for the binomial potential. It is shown in accordance with Oks and Uzer (2002) that a new kind of time dilation occurs for stable, nonradiating states. The primary result which is derived is that the energy of these classical stable states agrees exactly with the quantal results for the ground state and all states of odd values of the radial and angular harmonic numbers. v? ACKNOWLEDGEMENTS I would like to thank the Auburn University Department of Physics for having me as a graduate student for the past two years and for the education I have received under them. I wish to extend my heartfelt thanks to Prof. Eugene A. Oks for having been such a wonderful teacher and mentor to. You have been a very important person in my educational refinement, I thank you for this. I dedicate this thesis to my parents, Cecilia and Antonio, and Cadmiel for all their love and support throughout the time it has taken me to get this far. I thank you most of all. I never could have done this without you. Last, but not least, I thank all my friends. Whatever it is you have done, rest assured it means a lot to me. vi? This thesis has been prepared in accordance with Auburn University Graduate School?s Guide to Preparation and Submission of Thesis and Dissertations 2005, using Microsoft Word 2007. vii? ? TABLE OF CONTENTS Introduction ??????????????????????????.?......... 1 Dirac?s Generalized Hamiltonian Dynamics ?..????????????.??..... 4 Applications of the binomial potential ????????????????.??..... 8 Dynamical symmetries of Fradkin ????????????????????.. 10 Further properties of the generalized Runge-Lenz vector ???????????.. 14 Application of GHD to the binomial potential ???????????????... 15 Conclusions ?????????????????????????????. 27 Bibliography ????????????????????????????... 29 Appendix A: Derivation of the functional form of the unit Runge vector ......??????. 31 Appendix B: Derivation of the frequency of precession of the Laplace-Runge-Lenz vector ????????????????????????...??...?. 36 Appendix C: Derivation of the equations of motion via the Poisson bracket formalism ??. 43 1 ? 1. INTRODUCTION In 1950, Dirac developed a generalized Hamiltonian dynamics (hereafter GHD) [1-3]. The conventional Hamiltonian dynamics is based on the assumption that the momenta are independent functions of velocities. Dirac analyzed a more general situation where momenta are not independent functions of velocities [1-3]. Physically, the GHD is a purely classical formalism for constrained systems; it incorporates the constraints into the Hamiltonian. Dirac designed the GHD with applications to quantum field theory in mind [3]. The present work, where GHD is applied to atomic and molecular systems by choosing integrals of the motion as the constraints of the system, stems from a paper in which this idea was applied to hydrogenic atoms treated non-relativistically on the basis of the Coulomb potential [4]. Using this purely classical formalism, Oks and Uzer demonstrated the existence of non-radiating states and found their energy to be in exact agreement with the corresponding results of quantum mechanics. They employed two fundamental experimental facts, but did not ?forcefully? quantize any physical quantity describing the atom. In particular, this amounted to classically deriving Bohr?s postulate on the quantization of the angular momentum rather than accepting it on an axiomatic basis. 2 ? It important to point out that the physics behind classical non-radiating states is a new kind of time-dilation found by Oks and Uzer. The content of this thesis differs from the above mentioned paper by Oks and Uzer in that the dynamics analyzed are of a more general nature: a term proportional to 1/r 2 is added to the Coulomb potential. This more complicated potential we call here the binomial potential. Then the generalized unit Laplace-Runge-Lenz vector [5,6], or as named by Fradkin, the unit Runge vector [5], is utilized instead of the classical Laplace- Runge-Lenz vector. This binomial potential has interesting applications. The primary application considered here is to pionic atoms. We will classically obtain results corresponding to the solution of the quantal (relativistic) Klein-Gordon equation, which is appropriate because pions are spinless particles. Another application concerns the precession of planetary orbits: for this phenomenon Einstein?s equations of general relativity are equivalent to non-relativistic equations for the motion in the binomial potential [7]. We shall also briefly mention an application furnished by the description of the energy of nonradiating states of the so-called nanoplasmas [14]. An outline of the remainder of the thesis is in order: In section 2, we briefly outline Dirac?s generalized Hamiltonian dynamics. Section 3 serves to describe with more detail the applications of the binomial potential given in the above paragraph. In sections 4 and 5 we discuss the dynamical symmetries or Fradkin and the generalization of the Laplace-Runge-Lenz vector. 3 ? We present our new results in section 6 and appendices A, B, and C. Section 7, contains the conclusions. 2. DIRAC?S GENERALIZED HAMILTONIAN DYNAMICS. Dirac [1-3] considered a dynamical system of N degrees of freedom characterized by generalized coordinates q n and velocities dt dq v n n = , where n = 1, 2, ..., N. If the Lagrangian of the system is ( )vqLL ,= , (2.1) then momenta are defined as n n v L p ? ? = . (2.2) Each of the quantities q n , v n , p n can be varied by ?q n , ?v n , ?p n , respectively. The latter small quantities are of the order of ?, the variation being worked to the accuracy of ?. As a result of the variation, eq. (2.2) would not be satisfied any more, since their right-hand side would differ from the corresponding left side by a quantity of the order of ? as can be seen from: ()0=?= ? ? ? ? ? ? ? ? ? ? ?= nnnn n n pvvp v H vL ??? for an arbitrary variation in the momenta. In the above, Hamilton?s canonical equations of motion were invoked. Further, Dirac distinguished between two types of equations. To one type belong equations such as eqs. (2.2), which does not hold after the variation (he 4 ? called them "weak" equations). In what follows, for weak equations, adopting Dirac?s nomenclature, we use a different equality sign ? from the usual. Another type constitute equations such as eq. (2.1), which holds exactly even after the variation (he called them "strong" equations). If quantities ?L/?v n are not independent functions of velocities, one can exclude velocities v n from Eqs. (2.2) and obtain one or several weak equations ( ) 0, ?pq? , (2.3) containing only q and p. In his formalism, Dirac [1-3] used the following complete system of independent equations of the type (3): ( ) 0, ?pq m ? , ( )Mm ,...2,1= . (2.4) Here the word "independent" means that neither of the ??s can be expressed as a linear combination of the other ??s with coefficient depending on q and p. The word "complete" means that any function of q and p, which would become zero allowing for eqs. (2.2) and which would change by ? under the variation, should be a linear combination of the functions ? m (q, p) from (4) with coefficients depending on q and p. Finally, proceeding from the Lagrangian to a Hamiltonian, Dirac [1-3] obtained the following central result: ( ) ( )pqupqHH mmg ,, ?+= (2.5) (here and below, the summation over a twice repeated suffix is understood). Equation (2.5) is a strong equation expressing a relation between the generalized Hamiltonian H g and the conventional Hamiltonian H(q, p). Quantities u m are coefficients to be determined. Generally, they are functions of q, v, and p; by using Eqs. (2.2), they could 5 ? be made functions of q and p. It should be emphasized that H g ? H(q, p) would be only a weak equation - in distinction to Eq. (2.5). Equation (2.5) shows that the Hamiltonian is not uniquely determined, because a linear combination of ??s may be added to it. Equations (2.4) are called constraints. The above distinction between constraints (i.e., weak equations) and strong equations can be reformulated as follows. Constraints must be employed in accordance to certain rules. Constraints can be added. Constraints can be multiplied by factors (depending on q and p), but only on the left side, so that these factors must not be used inside Poisson brackets. If f is some function of q and p, then dt df (i.e., a general equation of motion) in the Dirac's GHD is [][ mm fuHf dt df ?,, += ], (2.6) where [f, g] is the Poisson bracket defined for two functions f and g of the canonical variables p and q as: [] rrrr q g p f p g q f gf ? ? ? ? ? ? ? ? ? =, . (2.7) where r is an index put to stress the fact that in general there will be several generalized coordinates and momenta. Here and throughout we adopt the summation convention so that a sum is understood over any repeated index unless it is explicitly stated otherwise. Substituting ? m' in (2.6) instead of f and taking into account eqs. (2.4), one obtains: [][ ] 0,, mmm ?+ ?? ??? m uH. (m? = 1, 2, ?, M). (2.8) 6 ? 7 ? These consistency conditions allow determining the coefficients u m . Last of all, we note that the GHD was designed by Dirac specifically for applications to quantum field theory [3], that is, for the purpose totally different from our purpose. 3. APPLICATIONS OF THE BINOMIAL POTENTIAL A. Pionic atoms described by the Klein-Gordon equation of relativistic quantum mechanics. Relativistic treatments of the hydrogenic atoms are typically presented working with the Dirac equation, which is a relativistic wave equation that is particularly suited well for spin-1/2 particles. However, in the literature one may also find a treatment of hydrogen and hydrogenlike atoms ignoring spin; that is, working with the Klein-Gordon equation (hereafter, the KG equation) [8,10-13]. The radial KG equation for the problem of the hydrogenic atom is given by: ( ) 0 )1( 4 12 2 2 2 2 = ? ? ? ? ? ? ?+ ??++ R Zll d dR d Rd ? ? ? ? ??? . (3.1) where Z is the atomic number and 137 1 2 ?= hc e ? is the fine structure constant. Thus, the radial KG equation for the Coulomb potential is equivalent to the radial Schr?dinger equation for the binomial potential - ?/? - ? 2 /? 2 . For usual hydrogenic atoms, the fine structure splitting predicted by the KG equation is greater than what is observed experimentally [8]. However, for pionic atoms, the KG equation becomes exact. Indeed, the pionic atom is an exotic hydrogenic atom, 8 ? where the atomic electron is substituted by a negative pion. Negative pions are spinless particles of the same charge as electrons, but 273 times heavier than electrons. Due to the spinless nature of pions, the KG equation for pionic atoms becomes exact. B. Precession of planetary orbits In his seminal paper, Die Grundlange der allgemeinen Relativit?sthoerie [7], Einstein showed that general relativistic effects perturb the Kepler potential by an additive term proportional to 1/r 2 and used it to calculate the precession of Mercury?s orbit around the sun. His calculations for the precession yielded 43??/century, which was later confirmed by observations. There are many good textbooks on general relativity that derive this result [15-17]. C. Radiation of nonrelativistic particles in a central field Karnakov et al. [14] derive the spectrum and expressions for the intensity of dipole radiation for a classical nonrelativistic particle executing nonperiodic motion. The potential in which the particles under consideration move is of the form () 2 rr rU ?? +?= . The authors of this paper apply their results to the description of the radiation and the absorption of a classical collisionless electron plasma in nanoparticles irradiated by an intense laser field. Also, they find the rate of collisionless absorption of electromagnetic wave energy in equilibrium isotropic nanoplasma. 9 ? 4. DYNAMICAL SYMMETRIES OF FRADKIN Fradkin [5] has shown that all classical dynamical problems of both the relativistic and non-relativistic type, dealing with a central potential, necessarily possess O(4) and SU(3). This led him to a generalization of the Runge-Lenz vector in the Kepler problem. Fradkin also found a generalization of the conserved symmetric tensor for the harmonic oscillator problem, and constructs a systematic way of imbedding the Lorentz and the SU(3) group in and infinite-dimensional Lie algebra. Here we will only be concerned with the results relating to the generalization of the Runge-Lenz vector and the construction of the elements of the Lie algebra of O(4) and SU(3) in terms of canonical variables. In the non-relativistic Kepler problem the force on the affected particle is an inverse square force given by: r r ? 2 ? ?= ? p ; , ? = rp m r r r =? (4.1) and the overdot denotes total differentiation with respect to time. In the Kepler problem, the Hamiltonian and the angular momentum (vector L and magnitude L 2 ) are the conserved quantities. There also exists another conserved vector quantity, namely the Laplace-Runge-Lenz vector, or simply the Runge-Lenz vector. It is defined to be: ()(rmmE ? 2 2 1 ????= ? LpA ) (4.2) 10 ? For negative energies (E<0) A is a real vector. This vector, which is a constant of the motion, lies in the plane of the orbit and points from the center of motion to perihelion (that is, along the major axis from one focus to the closest point of the orbit); some authors refer to it as the eccentricity vector [10]. Fradkin found, by differentiation via the standard Poisson bracket formalism, that for the Kepler problem, and indeed for all central potential problems, that A, L, and H satisfy the following closed Lie algebra: [ ] [ ] [] [] [] kijkji kijkji kijkji ii LAA AAL LLL HLHA ? ? ? = = = == , , , 0,, (4.3) It is seen that the Lie algebra given above is isomorphic to that of the generator of the O(4) symmetry group, which is the group of orthogonal transformations representing rotations in four dimensions. Fradkin also concluded that if the existence of the Runge- Lenz vector is simply to ensure that the plane of the motion is conserved, then it should always be possible to find a vector analogous to the Runge-Lenz vector for all central potentials. Fradkin proposed a generalization for the Runge-Lenz vector choosing as a mutually orthogonal triad of unit vectors. This unit Runge vector is LrLr ? ? and , ? ,? ? LrLrkLLkrrkk ? ?) ? ? ? ( ? ) ?? (?)? ? ( ? ???+?+?= , (4.4) 11 ? but since the unit Runge vector is in the plane of the orbit and the angular momentum vector is perpendicular to the plane of motion, then the second term is identically zero ( ). may be chosen to be the direction from which the azimuthal angle ? is L?k ? k ? measured (with the positive sense given by a right-handed rotation about L ? ), then we have: ?? sin ? ? ? and cos ? ? =??=? Lrkkr (4.5) thus Lrrk ? ?)(sin?)(cos ? ?+= ?? (4.6) Defining u=1/r, we may write the following differential equation for u and the azimuthal angle ? in terms of the energy E, potential V and angular momentum L: () 2 2 2 2 uVE L m d du ?? ? ? ? ? ? ? = ? ? ? ? ? ? ? (4.7) At this point we note the following relations and definition: () L r u f ELuf p? ? ? ? ? ? ? ? ? = = ? sin ),,(cos 2 ? ? . (4.8) Further, the putting V=-?u for the potential of the Kepler problem, the orbit equation becomes: [ ] ( )muLmmELf ??? ?+== ? 2 2 1 22 )(2cos . (4.9) The unit Runge vector may be expressed as: Lp? ? ? + ? ? ? ? ? ? ? ? ?= ? u f Lr u f ufk 2 ? ? (4.10) and it?s Poisson bracket with the total energy E (or, more importantly, the Hamiltonian) vanishes. 12 ? Lastly, all of its entries have mutually vanishing Poisson brackets and it satisfies the following relation with the angular momentum: [ ] ;0, ? =Hk i [ ] ;0 ? , ? = ji kk [ ] 3,2,1,,for ; ?? , == kjikkL kijkji ? . (4.11) 13 ? 5. FURTHER TOPICS ON THE GENERALIZATION OF THE LAPLACE- RUNGE-LENZ VECTOR We now turn to a brief discussion of further results that were utilized in our work. They are the results of Holas and March [6] on a further treatment of the unit Runge vector of Fradkin discussed in the previous section. These results, however, are centered on the construction and time dependence of the vector itself rather than on the dynamical symmetries of central potentials or the algebras satisfied by the unit Runge vector. Holas and March using ( ) rL rpr Lp ? ? ?=? Lrr L 22 2 (5.1) they rewrite the unit Runge vector, eq. (4.10), as: ( ) rL u f Lr rfk ? ? ? ? ? ? ?? ?= rp (5.2) where the function f is specified in the next section. This is the form of the unit Runge vector with which we shall work. 14 ? 6. APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS TO THE BINOMIAL POTENTIAL In our work, the angular momentum vector and the unit Runge vector are constants of the motion for a centrally symmetric potential and consequently have vanishing Poisson brackets with the Hamiltonian for the system and are thus suitable constraints for the application of GHD. Following Oks and Uzer [4], the Hamiltonian for this system is: ()( 00 2 22 ?? 22 kk rr Zep H g ??+??+ ? +?= wLLu ?? ), (6.1) where ? is the strength of the binomial potential, Ze is the nuclear charge, e is the charge of an electron, ? is the reduced mass, u and w are the yet unknown constant vectors (to be determined later) of the GHD formalism, L 0 and are the values of the angular momentum and unit Runge vector in a particular state of the motion so that in those states 0 ? k 0 LL ? (6.2) and . (6.3) 0 ?? kk ? 15 ? We may define the following quantities: 2 0 22 0 2 2 r HH r Zep H B ? ? ? += ?= (6.4) where the subscript B is for binomial. The consistency conditions for this system are: [ ] []0, ? 0, ? ? g g Hk HL . (6.5) First we must derive the form of the unit Runge vector in this problem. It is derived in Appendix A. We arrive at the result: ( ) () rL uu gg f gg gg gg fg Lr r gggg gg k ? ? 1 1 1 ? 1 1 ? 3 3 3 2 0 2 0 0 0 2 0 22 0 2 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? + ? ? +++ + = rp . (6.6) where 2 0 22 0 2 0 1 1 gggg gg f +++ + = . (6.7) and ()( ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +++ ++ ? +++ = ? ? u g gggg ggggg gggg g u f 2 3 2 0 22 0 2 2 00 2 0 22 0 2 0 1 1 1 , (6.8) 16 ? The functions and are defined in Appendix A. The unit Runge vector as appears in eq. (6.6) is a general result, valid for any value of the parameter g 0 g ?. Hereafter, however, we only consider a small perturbation in the binomial potential such that . We therefore perform a Taylor series expansion about 2 L<> . ? At this point, in keeping with the development of the problem as in [4], we should introduce Planck?s hypothesis, whereby we assume that the smallest possible change in energy is proportional to the frequency of the motion, and the proportionality constant is the reduced Planck?s constant Jsh 34 1005.1 2 ? ??? ? h in SI units. In our particular problem, however, this is not so simple because, as is established in Holas and March [6], the unit Runge vector is only piecewise continuous reflecting the well-known fact that the motion in the modified Coulomb potential is only conditionally periodic (as opposed to periodic). Given this fact, the relation between changes of the energy and of the angular momentum should be refined as follows: ??? ?=?=? ?? LdtLdEdt 00 ? TT r (6.36) where T r is the period of radial motion and T ? is the period angular motion. Eq. is justified as the change in energy correlates with the change, in this case a decrease, of the size of the orbit. Therefore, the integral containing the energy is over the period of radial motion. On the right-hand side of eq. (6.36), the integral contains the angular momentum which is the variable canonically conjugate to the angular variable ?, therefore the integration is performed over the period of angular motion. Combining eq. (6.36) with Planck?s hypothesis we get: ???? ? ? hLE T T LdtLdEdt r TT r =?=???=?=? ??? 00 (6.37) In eq. (6.37), the change in energy must, of course, satisfy ( SSgSS LH h hHHHE , 2 00 ?? ==??=? )?????????????????????????????(6.38) or ( SSS LH h H , 2 0 ?? ). (6.39) We note that on both sides of the eq. (6.39) only physical quantities pertaining to the stable states are present. Also, in eq. (6.37) we have 24 ? 2 1 ; 1 LT T r r ? +=== ? ?? ? ? ? (6.40) (note that as ,0?? 11 2 ? ? += L ? , which implies that ? TT r = , as known from the Coulomb potential) and therefore () ( ) 2 3 42 0 2 1 2222 , 2 1 S rr SSS H eZ m nh m n hmnh LH h HE ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? += ? ? ? ? ? ? ? ? += + =?=? (6.41) where n,m = 1,2,3? . In the third step of eq. (6.41) we used the relation between the frequencies given in eq. (6.40) and we have substituted ( ) SS LH , 0 ? for the term () 2 ? ?? mn r + , which is the average of the two frequencies throughout the motion (hence the 1/2); and, further, the expression must be valid not only for the first harmonic, but for all occurring harmonics of the radial and angular frequencies, hence the integer factors n and m. We have also used: 2 2 3 42 0 0 2 1 8 L Ze H eZ S r ??? ? ? ? + == . (6.42) We now take notice that from eq. (6.41) we may obtain an expression for the Hamiltonian in the radiationless state of motion in terms of the integers n and m, we find: () ...2,1 2 22 42 = + = ; n,m mnh eZ H S ? ? . (6.43) 25 ? We compare this classically-derived result with the known quantal result as may be found in, say, Quantum Mechanics: Nonrelativistic Theory of Landau and Lifschitz [18] in problem 3 after section 36: ()( ) ...2,1,0 1212 2 2 2 42 = +++ = l l ,; n nh eZ H r r quantal ? ? . (6.44) where n r and ? are the radial and angular momentum quantum numbers, respectively. We see that in the quantal result, the ground state ( 0, = r nl ), agrees exactly with our derived expression for n,m=1. Furthermore, the correspondence between the quantal result and ours agrees for all odd n and m, i.e. when these integers are of the form n=2k+1 and m=2q+1, q,k=0,1,2?. We may identify n and m as the radial and angular harmonic numbers. 26 ? 27 ? 7. CONCLUSIONS We close with a brief recapitulation of the work put forth in the preceding. In the section of application, motivation was given for the use and interest of the binomial potential. The well-known and interesting applications mentioned were that of the solution to the Klein-Gordon equation governing the dynamics of pionic atoms; radiation of particles in nanoplasmas; and the advance of the perihelion of planets orbiting in a central potential as can be shown by means of general relativity. The main new results obtained for the binomial potential are as follows. 1. We obtained an explicit expression for the additional (to the angular momentum) vector integral of the motion: the unit Runge-Lenz vector. 2. Beginning with Dirac?s generalized Hamiltonian dynamics, a purely classical formalism, a (generalized) Hamiltonian was set up that described the dynamics of a spinless particle in a Coulomb potential perturbed by the presence of a binomial potential, i.e. one that varies inversely with the square of the distance from the center of force. With this Hamiltonian and the use of consistency conditions, in this case the necessity that the angular momentum, energy (Hamiltonian), and the unit Runge-Lenz vector be the seven conserved quantities of the central potential it was shown that the use of GHD leads to an effective time dilation. 3. We derived classical energies of radiationless states in the system of bound spinless particles and found that they agree with quantum theory for the ground state and with all states of odd principle and angular momentum quantum numbers. 4. We derived the explicit expression for the generalized Hamiltonian. It leads to a dynamics that is much richer than the usual classical dynamics. This can be seen from the many additional terms in the equations of motion derived in Appendix C. It is worth emphasizing some interesting physics of classical nonradiating stable states following Oks and Uzer [4]: In those states, 0== dt d dt d pr , so that 0 )( rr =t and , where and are some constant vectors. Thus, the particle (for example, the pion) is motionless, but its momentum is nonzero. This is not surprising: for example, for a charge in an electromagnetic field characterized by a vector potential A, it is also possible to have 0 )( pp =t 0 r 0 p 0= ? =v m mc e Ap , while 0?= Ap mc e . 28 ? 29 ? BIBLIOGRAPHY 1. Dirac, P. A. M 1950 Canad. J. Math. 2, 129 2. Dirac, P. A. M. 1958 Proc. R. Soc. A 246 326 3. Dirac, P. A. M 1964 Lectures on Quantum Mechanics (New York: Academic). Reprinted by Dover Publications, 2001. 4. Oks E. and Uzer T. 2002 J. Phys. B: At. Mol. Opt. Phys 35 165 5. Fradkin D. M. 1967 Prog. Theor. Phys. 37 798 6. Holas A. and March N. H. 1990 J. Phys. A: Math. Gen. 23 735 7. Einstein A. 1916 Annalen der Physik 49. It is reprinted in The Principle of Relativity (Dover 1952) with other landmark papers by Weyl H., Lorentz H., and Minkowski H. 8. Josephson J. 1980 Found. of Phys. 10 243 9. Landau L. D. and Lifschitz E. M. 1982 Mechanics 3 rd Edition Butterworth- Heinneman 10. Sokolov A. A., Ternov I. M., and Zhukovskii V. Ch. 1984 1 st edition Quantum Mechanics Mir Publishers. 11. Greiner W. 1990 Relativistic Quantum Mechanics: Wave Equations 12. Schiff L. I. 1968 Quantum Mechanics (International Pure and Applied Physics Series) 3rd edition McGraw-Hill Companies. 30 ? 13. Capri A. 2002 Relativistic Quantum Mechanics and Introduction to Quantum Field Theory 1 st edition World Scientific Publishing Company. 14. B. M. Karnakov, Ph. A. Korneev, and S. V. Popruzhenko 2008 J. of Exp. and Theor. Phys., 106, No. 4, 650 15. Landau L. D. and Lifschitz E. M. 1980 Classical Theory of Fields 2 nd Edition Butterworth-Heinneman 16. Walecka J. D. 2007 Introduction to General Relativity 1 st edition World Scientific Publishing Company 17. Schutz S. 1985 A First Course in General Relativity Cambridge University Press. 18. Landau L. D. and Lifschitz E. M. 1981 Quantum Mechanics: Nonrelativistic Theory 3 rd Edition Butterworth-Heinneman? ? ? ? ? ? ? ? ? APPENDIX A DERIVATION OF THE FUNCTIONAL FORM OF THE UNIT RUNGE VECTOR The function f, given by ()() ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?== ? u u eff ud uL u VE L f 0 2 1 2 1 2 ;cos ??? ?? (A.1) where ??= 22 LL eff (A.2) is the effective angular momentum and shows a correction due to the binomial potential. The integral in eq. (A.1), upon the substitution of the Coulomb potential, may be rewritten as: ()() () ?? ? ? ? ? ? ? ? ? ? ??+ ? ? ? ? ? ? ? ? ? =? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? u u u u eff uL uZeE ud ud uL u VE 00 2 2 2 1 2 2 1 2 ?? ?? . (A.3) If we now introduce the substitutions 31 ? ? ? ? ? ? ? ? ? ?= ? ? ? ? ? ? ? ? += ? ? ? ? 42 2 2 2 2 42 2 2 2 1 2 1 2 1 eZ EL L Ze u eZ EL L Ze u , (A.4) then the left-hand side of eq. (A.3), in the indefinite form of the integral, becomes: ()() ()() ? ? ? ? ? ? ? ? ? ? ? ? ?? + ? = ?? ? ? ? 21 21 1 21 2 tan uuuu uu u uuuu ud (A.5) after some simplifications. It is convenient to define 2 2 21 3 2 L Zeuu u ? = + = (A.6) and thus eq. (A.5) reduces to: ()() ()() ? ? ? ? ? ? ? ? ?? ? = ? ? ? ? ? ? ? ? ? ? ? ? ?? + ? ?? 21 31 21 21 1 tan 2 tan uuuu uu uuuu uu u . (A.7) Putting in the limits of integration yields: ()() ()() ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?? ? = ?? 2001 301 21 31 tantancos uuuu uu uuuu uu f . (A.8) It is convenient to define: ()() 00 21 3 )( ;)( gug uuuu uu ugg ? ?? ? ?= . (A.9) Using the identity () 2222 11 1 1 )(tan)(tancos BABA AB BA +++ + =? ?? , (A.10) 32 ? we may then write 2 0 22 0 2 0 1 1 gggg gg f +++ + = . (A.11) Consequently, the partial derivative in the unit Runge vector becomes: ()( ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +++ ++ ? +++ = ? ? u g gggg ggggg gggg g u f 2 3 2 0 22 0 2 2 00 2 0 22 0 2 0 1 1 1 , (A.12) where ()() () ()() 2 3 21 21 21 21 2 2 2 11 uuuu uuu uu u uuuuu g ?? ++?? ? ? ? ? ? + ? ? ?? = ? ? . (A.13) We may use the definitions (A.9) and (A.11) to rewrite eq. (A.13) and put it into eq. (A.12) to get the following compact form: ( ) () ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? + = ? ? 3 3 3 2 0 2 0 0 0 1 1 1 uu gg f gg gg gg fg u f . (A.14) where the term in the second set of parenthesis is the simplification of u g ? ? . We thus arrive at: ( ) () rL uu gg f gg gg gg fg Lr r gggg gg k ? ? 1 1 1 ? 1 1 ? 3 3 3 2 0 2 0 0 0 2 0 22 0 2 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? + ? ? +++ + = rp . (A.15) This is a general result valid for any value of ?. However, since we are considering a small perturbation in the binomial potential, such that , then we may perform a Taylor series expansion of the unit Runge vector with respect to 2 L<