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A 2D Formulation for the Helium Atom Using a Four- Spinor Dirac-Like Equation and the Discussion of an Approximate Ground State Solution

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A 2D Formulation for the Helium Atom Using a Four- Spinor Dirac-Like Equation and the Discussion of an Approximate Ground State Solution Physical Science International Journal 9(4): 1-12, 2016, Article no.PSIJ.24355 ISSN: 2348-0130 SCIENCEDOMAIN international www.sciencedomain.org A 2D Formulation for the Helium Atom Using a Four- Spinor Dirac-like Equation and the Discussion of an Approximate Ground State Solution D. L. Nascimento 1* and A. L. A. Fonseca 1 1Institute of Physics and International Center of Condensed Matter, University of Brasília, 70919-970, Brasília, DF, Brazil. Authors’ contributions This work was carried out in collaboration between both authors. Both authors read and approved the final manuscript. Article Information DOI: 10.9734/PSIJ/2016/24355 Editor(s): (1) Shi-Hai Dong, Department of Physics School of Physics and Mathematics National Polytechnic Institute, Mexico. (2) Christian Brosseau, Distinguished Professor, Department of Physics, Université de Bretagne Occidentale, France. Reviewers: (1) Balwant Singh Rajput, Kumaun University, Nainital, India. (2) S. B. Ota, Institute of Physics, Bhubaneswar, India. Complete Peer review History: http://sciencedomain.org/review-history/13496 Received 16 th January 2016 Accepted 14 th February 2016 Original Research Article Published 29 th February 2016 ABSTRACT We present a two-dimensional analysis of the two-electron problem which comes from the classical conservation theorems and from which we obtain a version of the Dirac equation for the helium atom. Approximate solutions for this equation are discussed in two different methods, although in principle it can be solved analytically. One method is variational, of the Hylleraas type, the execution of which is left for a later communication. In contrast, the other method will have a more complete treatment, in which the set of equations will be separated into its angular and radial components. Furthermore, an exact solution for the angular component will be displayed as well as an approximate solution for the radial component, valid only for the fundamental state of the atom. Keywords: Helium atom; Dirac equation; relativistic quantum mechanics; variational methods; Hylleraas method; semi analytic solutions. _____________________________________________________________________________________________________ *Corresponding author: E-mail: [email protected]; Nascimento and Fonseca; PSIJ, 9(4): 1-12, 2016; Article no.PSIJ.24355 1. INTRODUCTION parameter. The gradual superposition of the corresponding Hamiltonians yields a system of Since the beginning of quantum chemistry in the differential equations that is dependent on the 1920s the implementation of purely parameter of penetration, which should be used computational calculations of the Hartree-Fock at the end of the calculation to obtain the type [1] has prevailed, due to their high minimum energy of the two-electron system. performance and easy implementation, over Considering now the Hylleraas-like procedure, more analytical structures such as those of the the Hamiltonian is used in the traditional way in Hylleraas type [2]. Nevertheless some authors which the system is treated as a whole, without have tried to give an analytical basis to their distinction of individual equations for each iterative calculations, some of which have electron. For both procedures we try to express become the source of inspiration to start this line the system of equations in a truly covariant form, of work [3-6]. in which we can introduce later the retardation effects without breaking this fundamental Following this analytical aim [7,8], we try to requirement of the Theory of Relativity. explore the Classical Theorems of Conservation before any quantization procedure is performed. In the last section of the paper we separate the We believe that they can reduce the dimensions angular and radial components of the Dirac-like of the coordinate systems that are necessary to system of partial differential equations for the formulate the problem in the quantum domain. helium atom [13]. We find the angular More recently [9], we demonstrated that it is eigenfunctions that allow us to separate the possible to use, in the analytical solution of the system of radial equations and an asymptotic Dirac equation [10] for the hydrogen atom, Dirac form of the wave function that is a solution of this 2x2 matrices rather than the usual 4x4 matrices, system for the ground state of the atom. From which leads to a considerable reduction in this we get a determination of the atom energy complexity of the problem. Moreover, our method eigenvalue that agrees with the experimental led us to a new approach to the relativistic data within 0.1% of accuracy and we also check Hylleraas procedure in which the Dirac equation that it tends to the exact value of the ion energy is derived from an extremum problem. This when the outer electron is displaced to infinity. procedure was used to carry out numeric calculations for hydrogen-like atoms that resulted 2. THEORY in extremely accurate energy eigenvalues with respect to the exact values, which are well In the infinity mass nucleus rest frame, the known [11]. relativistic classical Hamiltonians for the individual electrons of the Helium atom in natural = = 2 In this paper we treat the problem of the helium units h c 1 and α =e ≅ 1 / 137 are atom in a similar way to what we did in the case of the hydrogen atom. This treatment has 2α α allowed us to use Dirac 4x4 matrices instead of H=p2 + m 2 − + 1 1 r r the 16x16 matrices of the Breit theory for the 1 12 , same atom [12], although it should be mentioned 2α α that we do not consider here the time retardation H=p2 + m 2 − + 2 2 r r effects. Therefore we have developed a dual 2 12 , (1a,b) procedure: on the one hand we obtain a Dirac- like system of partial differential equations and =2 + 2 − θ on the other a Lagrangian density to carry out a where r12 r 1 r 22 rr 1 2 cos . We see that the variational calculation of the Hylleraas type, repulsion energy entries fully for each electron in whose execution is however left to an upcoming this case, on the other hand, if we consider the article. energy of the whole system, not taking into account the electrons individually, we arrive at For the Dirac-like procedure we take into account the usual classical Hamiltonian the trivial fact of the Theory of Relativity that we cannot add together the geodesics of individual α α α =2 ++ 2 2 +−−+ 2 2 2 particles. Then we consider a system formed by Hp1 m p 2 m , (1c) + a single electron plus the nucleus, i.e., the He r1 r 2 r 12 ion, as a substrate on which an outer electron is introduced gradually through a penetration in which the repulsion energy entries only once. 2 Nascimento and Fonseca; PSIJ, 9(4): 1-12, 2016; Article no.PSIJ.24355 Still in the infinity mass nucleus rest frame, we on the other hand, σ =1 correspond to the limit choose to use a coordinate system in which the when the two electrons form a single system with motion occurs in the plane defined by the perfectly symmetric positions so that the system = Hamiltonian becomes two times the Hamiltonian nucleus and the two electrons, i.e., pz 0 , whose z axis may be moving at constant velocity of one of the electrons, which was chosen by with respect to the z axis of another inertial convenience to be the electron 2. In fact, it will be σ system, so that the system is invariant against seen that r12 becomes a function of , so that space translations in this direction. In this frame, the equations of the system are solved for r = the only non vanishing components of the 12 classical angular momentum of each electron constant and then, at the final of the calculation, this constant is varied through σ in order the J= r × p and J= r × p and of the total 1 1 1 2 2 2 equilibrium configuration may be obtained. = + angular momentum J J1 J 2 are their z components, namely, J= xp − yp , We now search for a Dirac equation 1z 11 y 11 x corresponding to the quantization of the classical = − = + J2z xp 22 y yp 22 x and Jz J1 z J 2 z Eqs. (1c) and (1d), in the infinity mass nucleus respectively. Now, we know from Classical rest frame. The quantization is done in a way Mechanics [14] that the Poisson Bracket for similar to that performed by Breit [12], in which each electron angular momentum with respect to each square root is “linearized” individually: the Classical Hamiltonian (1c) is not null and that 2 they are symmetric with respect to each other, 225+=γ − γ 3 + γ 0 p1m( p 1y pm 1 x ) { } =−3 θ = − { } , (2a) i.e., HJ,1z rrr 1212 sin HJ , 2 z , in which θ = θ− θ , so that the summation of them is null 2 2 1 p221+=m(γ p − γ 2 pm + γ 0 ) 2 2y 2 x . (2b) and hence the total angular momentum J z becomes a constant of the motion. This happens We need five 4× 4 anticommuting matrices, because the repulsion force between the µ electrons is a non central force and hence it which are the four usual γ Dirac matrices produces a torque in each electron that makes it 5 together with the γ matrix which always oscillating about the axis that join the nucleus to the other electron. In the Poisson Bracket it appears connected with Dirac´s theory: appears due to the implicit derivatives of 1/ r 12 1 0 0 iσ γ 0 = = with respect to x1, y 1 or x2, y 2 , which produces γ 0− 1 −iσ 0 the symmetric terms because , , 2 2 r=()() xx − +− yy in Cartesian 0 1 12 1 2 1 2 γ5= − i γγγγ 0123 = coordinates.
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