Geometrical approach in atomic physics: Atoms of hydrogen and helium

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1 Aerican Journal of Physics and Applications 014; (5): Published online October 0, 014 ( doi: /j.ajpa ISSN: (Print); ISSN: (Online) Geoetrical approach in atoic physics: Atos of hydrogen and heliu Oleg Olkhov Institute of Cheical Physics, Moscow, Russia Eail address: To cite this article: Oleg Olkhov. Geoetrical Approach in Atoic Physics: Atos of Hydrogen and Heliu. Aerican Journal of Physics and Applications. Vol., No. 5, 014, pp doi: /j.ajpa Abstract: The hypothesis was earlier suggested by the author where all icro-objects are considered as specific distortions of the physical space-tie pseudo-euclidean geoetry, naely, as closed topological 4-anifolds. The foundation of the hypothesis is a geoetrical interpretation of the basic equation of quantu echanics for classical (not quantized) wave fields -- the Dirac equation for free particle. Such hypothesis does not contradict to any physical laws and experiental facts and gives firstly an opportunity to explain qualitatively within classical notions (geoetrical) the so called paradoxical properties of quantu particles such as wave-corpuscular duality, appearance of probabilities in the quantu echanics foralis, spin, EPR-paradox.To deonstrate prospects for suggested geoetrical approach the author early attepted to find new dynaic equations other than known quantu-echanical ones for atoic spectra calculations. In this work above investigation is being continued on a ore rigorous basis, representing a new geoetrical interpretation of the equation for hydrogen atos. Results of calculations of ionization potentials for heliu ato are in agreeent with experiental data. Keywords: Geoetrical Interpretation, Quantu Mechanics, Atoic Spectra, Heliu Spectru 1. Introduction The hypothesis was earlier suggested by the author where all icro-objects are considered as specific distortions of the physical space-tie pseudo-euclidean geoetry, naely as closed topological 4-anifolds. For an observer in usual -space such objects look as oving topological defects of this space. The foundation of the hypothesis is a topological interpretation of basic concepts of quantu echanics atheatical apparatus. Contrary to traditional Copenhagen interpretation, this interpretation gives an opportunity to describe quantu objects between easureent acts (with the help of geoetrical notions). Such hypothesis does not contradict to any physical laws and experiental facts and gives firstly an opportunity to explain qualitatively within classical notions (geoetrical) the so called paradoxical properties of quantu particles such as wave-corpuscular duality, appearance of probabilities in the quantu echanics foralis, spin, EPR-paradox. To becoe a physical theory any hypothesis, even internally uncontroversial, have to suggest (along with qualitative arguents) a new quantitative atheatical apparatus, having advantages over the existing one. To deonstrate prospects for suggested geoetrical approach the author attepted in a resent paper to find new dynaic equations other than known quantu-echanical ones [1]. As a topical proble we selected, as a first step, the geoetrical interpretation of known Dirac s equation for hydrogen ato and then we found new equations for spectru calculations for siplest any electron ato--heliu ato. These equations can be considered as the self-consistent field theory that differs fro the known Hartree-Fock theory. In this work above investigation is being continued on a ore rigorous basis, representing a new geoetrical interpretation of the relativistic Dirac equation for hydrogen atos. Results of calculations of ionization potentials are in agreeent with experiental data. The plan of the paper: in Section we present the ain points of suggested hypothesis, in Section we present geoetrical interpretation of known relativistic Dirac s equation for hydrogen ato, in Section 4 we present derivation of new equations for heliu ato and use the for heliu spectra calculations.

2 109 Oleg Olkhov: Geoetrical Approach in Atoic Physics: Atos of Hydrogen and Heliu. Foundations of the Hypothesis Let us reind at first ain points of the suggested geoetrical interpretation of quantu echanics [,]. The starting oent was the topological interpretation of Dirac s relativistic equation for free particle with spin ½. Syetric for of this equation is [4] ˆp γ, ik ψ = ψ (1) k i is the ass of the particle, ˆp = i = (i / t, i ), γ ( = 0,1,,) are four-row Dirac s atrices, ψ (x)- i Dirac s byspinor, i = 1,,,4. We use in (1) relativistic units where ħ = с = 1. Here and later we use 4-etrics with signature ( + ). With such units the eleentary charge square is e = 1/17. Solution of (1) for free particle s states with definite values of 4-oentu p has the for of plane wave ψ = u πix λ () p 1 pexp( ), where u p-- definitely noralized byspinor and λ λ λ λ = λ λ = πp λ = π () ,,. Within traditional interpretation (1) describes only possible results of easureents over quantu object, naely, possible values of its 4-oentu. But this equation does not include no another inforation about the object before or between easureents (for exaple, what is it within classical notions). It is supposed in our hypothesis that (1) includes such inforation. The starting point of the hypothesis is the fact that above solution of (1) can be interpreted as a basic vector of a representation for the infinity translation group, including all translation of the for s = n λ n λ n λ n λ (4) n Here λ0, λ1, λ, λ - four basic orthogonal vectors into four-diensional pseudo-euclidean space, n0, n1, n, n - integers. The physical pseudo-euclidean space-tie does not have such syetry (corresponding to a syetry of infinite crystal), and our ain suggestion is that above group works into the definite auxiliary space -- the universal covering space of soe closed 4-anifold described by (1). Such spaces are used in topology for description of closed anifolds because discrete groups, working into such spaces, isoorphic to so called anifold s fundaental groups. Eleents of this group are different classes of closed paths that starts and finishes at sae point (Poincare group π 1[4,5]). In particular, an infinite translation group operating in one-diensional Euclidean space isoorphic to a fundaental group of the one-diensional closed anifold hoeoorphic to a circle. An infinite translation group operating in two-diensional Euclidean space isoorphic to a fundaental group of the two-diensional closed anifold hoeoorphic to a torus [5,6,7]. In addition, the wave function () is a byspinor -- tensor realizing two-sign representation of a rotation group and therefore can be considered as a description of nonorientable geoetric objects [8,9]. Above consideration leads to a hypothesis that (1) can be considered as a description of a closed anifold (by coordinates of its universal covering space), naely, as a description of the closed nonorientable topological space-tie 4-anifold, where spin ½ corresponds to index of the two-sign rotation group and relation () is a etric restriction on possible values of closed paths λ in the anifold. It can be shown that due to pseudo-euclidean etrics of the physical space-tie such four-diensional object represents oving topological defect of three-diensional space and that this defect has wave-corpuscular and stochastic properties. And this gives an opportunity to identify such geoetrical object with the quantu object, described by (1)[,]. Dirac Equation for Hydrogen Ato We start application of the hypothesis to atoic physics (to atoic spectra theory) with the siplest case---with the geoetrical interpretation of the Dirac relativistic equation for hydrogen ato [4] (pˆ A ) γ ψ = ψ, (5) ik k Where A -- nuclear 4-potential. It is supposed within traditional interpretation that (5) describes (as (1) for free particle) only possible results of easureents with atos but says nothing about (does not explain) what the ato is before and between easureents. In particular, it does not follow fro (5) that hydrogen ato contains inside any point-like particle (electron). Our topological interpretation of (5) supposes that this equation describe not only possible results of easureents but also describes specific distortion of space-tie Euclidean geoetry (existing between easureents) naely the closed nonorientable anifold. Then function ψ (x) i appears to be a basic vector for representation of a group of syetry of this object (radial and angle coponents of wave function realize representation of corresponding subgroups). If we express all notions in (5) through notions with diensionality of length we obtain instead (5) 1 ( i + Θ ) γ k Ψ k( x ) = Ψ ( x ), L x ( t,x, y, z ) i = (6) a a a a wherel =ħ /c, Θ 0 = α /r, α = 1/17 = e / ħ c. According to our geoetrical interpretation, coordinatesx in (6) are not coordinates of the physical space-tie and equation (6) itself does not connect any events in this space. We suppose that (6) and (1) are written through coordinates in auxiliary space -- through coordinatest,x a, y, z in universal covering space a a a

3 Aerican Journal of Physics and Applications 014; (5): of above anifold. All physical notions in (6) are expressed through variables with diensionality of length so Planck constantħand light velocityсare happens to be siply coefficients of translation fro conventional variables to geoetrical ones. For exaple, energy can be expressed through inverse centieters -- units that are often used in atoic physics. For above topological interpretation of equation (6) will be possible it is necessary to show that this equation can be considered (like (1)) as an equation written in soe auxiliary space playing the role of universal covering space of soe closed anifold. To prove this we need to show that 4-potentials in (6) can be considered as connectivities in soe non-euclidean space so that derivatives in (6) can be considered as covariant derivatives in this space. But we need not in fact to prove this in the scope of our work because this was proved earlier within the theory of fiber spaces where it was forally shown (without any connection with suggested hypothesis) that electroagnetic potentials in the Dirac equation (5) can be considered as connectivities in the space of gauge group of this equations (group of phase transforation of wave functions)[10,11]. So we have shown that suggested topological interpretation of quantu objects can be applied not only to equation (1) for free particle but also to equation (5) for particle interacting with the external field. This eans that two fundaentally different odels for hydrogen ato are possible. They both are described forally by the sae equation (with different designations) and therefore they both in the sae way corresponds to experiental facts. Within one of the (traditional) the hydrogen ato is considered as a syste containing a specific point-like particle -- electron. The wave function in this case is considered as a functions of the physical space-tie coordinates. Within another odel (geoetrical) this ato is considered as a specific stretched icroscopic deforation of the physical space-tie. The wave function is considered here as a function of coordinates in the definite auxiliary space and these coordinates have nothing in coon with coordinates of any point-like particle. The wave function in this case realizes a representation of the object syetry group. We see that theoretical calculation of the hydrogen ato spectru eans within both above odels solution of the sae equations (or (5) or (6)). Therefore, suggested geoetrical approach has in this case no practical advantages. But for any-electron atos geoetrical interpretation leads to a principally ore siple atheatical forulation of the proble of spectra calculation in copare with the traditional quantu foralis. This follows fro the fact that wave functions of any-electron atos are considered within geoetrical approach as tensors realizing a representation for the group of syetry of closed anifolds describing these atos. It is naturally to suppose that these tensors will be (as for hydrogen ato in (6)) functions of only one space-tie coordinate in corresponding universal covering spaces of above anifolds whereas such functions depend within traditional approach on coordinates of all atoic electrons. Just this late circustance akes it ipossible to solve Schrodinger s equation with good accuracy for any-electron atos even with the help of odern coputers. Therefore, geoetrical approach (being approved) should lead to principally ore siple calculations of atoic spectra because equations for functions of only one variable can be usually solved without difficulties by nuerical ethods. And we will obtain below within geoetrical approach the new equations for calculations of the spectru for siplest any-electron ato -- the heliu ato. 4. Spectru of the Heliu Ato The spectru of heliu atos is defined in nonrelativistic liit by the known Schrodinger equation. In atoic units this equation is (see, for exaple, [1]) 1u + u + (E + /r1 + /r 1/r 1)u = 0 (7) Wherer 1 andr -- distances between nuclear and first and second electrons, r1 -- distance between electrons = / x + / y + / z -- Laplacian in the space of first electron, u -- function of six coordinates x 1,y 1,z 1 and x,y,z. Equation (7) is not relativistic invariant. It is ipossible to write here (to contrast to (5)) the closed equation that takes into account relativistic corrections for heliu ato. The reason is the electron-electron Coulob potential in (7) whose relativistic corrections can be represented only as a series expansion over powers of the fine structure constant [1]. Unlike (5,6), the solution of (7) cannot be expressed in a closed analytical for, and even approxiate solutions can be rarely obtained without nuerical calculations [1]. Nevertheless theoretical values for the spectru of heliu ato obtained fro (7) are in good agreeent with experiental data (with accuracy to relativistic corrections). The heliu ato (as a hydrogen ato) is considered within suggested geoetrical approach as soe localized distortion of the space Euclidean geoetry and there is now no necessity for its representation as a syste containing two identical point-like particles-electrons being in external field. But for finding out new equations for heliu atos (within geoetrical odel) it is necessary to take into account two new factors that are absent in equation (6) for hydrogen atos. Firstly, heliu ato has a syetry with respect to perutation of two identical electrons. Secondly, heliu ato has an additional energy due to electron Coulob interaction. These two factors are direct consequence of the atoic odel with two point-like electrons, and both these factors should be presented in atheatical description within geoetrical approach that does not consider such particles. To take into account perutation invariance we suppose that the wave function is a basis for representation of the group of perutations of second rank and that the wave function depends within geoetrical approach on only one space-tie coordinate. This eans that we consider a perutation group

4 111 Oleg Olkhov: Geoetrical Approach in Atoic Physics: Atos of Hydrogen and Heliu of second rank as a group isoorphic to the group of soe syetric transforations of the geoetrical object, representing the heliu ato. It is well known that a perutation group can be isoorphic to a group of syetry transforations of soe geoetrical object: the classical exaple is a isoorphis of perutation group of third rank and the group of syetry transforations of the plane equilateral triangle [1]. Therefore, inclusion of a perutation group in syetry groups of the geoetrical object representing heliu atos eans that its description within geoetrical approach has to use a tensor, depending (firstly) on one space-tie coordinate and (secondly) realizing a representation of the perutation group of second rank. The siplest such tensor is a 4-vecor B that realizes representation of the inversion group isoorphic to a perutation group of second rank [4]. Then initial equations for the vector B (without taking into account interaction of electrons to one another and with nuclear) should be relativistic invariant relations obtained with the help of operatorsˆp and containing electron ass. The siplest such relation relations are (in relativistic units) where pˆ = ˆ ˆ pˆ B B, = (8) p p,-- ass of an electron. The next step is inclusion in (8) interactions of electrons to one another and with nuclear. Nuclear potential is introduced in (8) in the sae anner as this potential was introduced in (1) for obtaining equation (5) for hydrogen ato, that is, by lengthening of derivatives. So after taking nuclear potential into account equation (8) take the for ˆ P B B, = (9) Where,, A -- 4-potential of the nuclear (for exaple, A0 = / r). There is no any guidance how to introduce in (8) interaction corresponding to Coulob electron-electron interaction. We suppose that this interaction can be taken into account by introducing in (9) additional potential ( ), that is defined, asa,fro Maxwell s relativistic equations for the tensor of electroagnetic field F ( ) α [14] α F ( ) = j ( ), F ( ) = V ( ) V ( ). (10) V α α α α Here in brackets indicates on belonging to corresponding equation in (9). Currents j ( ) in the right side part of (10) are defined as conserving currents corresponding to different coponents of 4-vector fro (9). Before to define these currents we suggest that the equation forb0describes atoic states with spin zero corresponding to para-heliu and equations for other coponents in (9) B( B1, B, B ) describe states with spin one corresponding to orto-heliu. Using known expressions for currents of wave fields with spin one or zero we obtain for scalar and vector 4-currents [4] B c 0 B0 j(0) = j = i( B0 B0 ), B k k j(1) = j() = j() = j = i( B B B B ),( k = 1,,) k k (11) ib = pˆ B pˆ B, (1) α β α β β α Where coefficient is an electron s doubled charge. Above currents do not depend on tie because their expressions contain binary cobinations of coplex conjugate functions and V ( ) satisfied the Poisson equation (this follows fro (10))[14] V ( ) = j ( ), V ( ) = j d ( ) r1 r r1. (1) Finally, the required syste of relativistic invariant equations for calculation of the energy levels of heliu are P ˆ ( ) B = B, ˆ ˆ ˆ ˆ ˆ P ( ) = P ( ) P ( ), P ( ) = p A V ( ), (14) j ( ) d r1. r r1 V ( ) = Let us consider above equations in nonrelativistic liit. The quantu object s relativistic energy contains the rest energy. Therefore we have to exclude this energy for transition to nonrelativstic liit by introducing in (14) ' functionsb insteadb [ ] ' B B i E t v it = exp ( + ) = exp( ). (15) This expression contains the ass of only one electron (but not two electrons). It eans that zero ionization energye = 0 corresponds to excitation in the continuous spectru of only one electron. In other world our equations describe one-particle excitations with ass (therefore the initial equation (9) contain a unit ass). Let us insert now (15) in (14) and take into account that differentiation with respect to tie leads to appearance of the large factor. Then we neglect of ters of the order E/. In result we obtain the following equations for calculation of heliu spectru (r 1) r 1 v d E ( ) = 0. v r r r r1 (16) These equations are siilar to equations of the theory of self-consistent field suggested by Hartree and Fock [1], but they are ore siple that akes its solution by nuerical ethods ore easy. Let us consider soe solutions of above equations for the case of syetrical s-states. In this case

5 Aerican Journal of Physics and Applications 014; (5): (16) takes the for (in atoic units) d d 4 v(r 1) d r1 ( E )v 0 dr + r dr + r r r = 1 d d 4 u(r 1) d r1 ( + + ε )u 0, dr r dr r = r r r1 (17) (18) Where u = ux + uy + uz. Notice that equation (18) for orto-heliu differs fro equation (17) for para-heliu by presence of centrifugal potential that reflects the syetry of this state corresponding to spin1. The nuerical solution of (17) with accuracy to second decial place gives for ionization potential of para-heliu E = 0.9. The experiental value iseexp = 0.91, and the value fro the Hartree-Fock theory ise = 0.86 [1]. Nuerical solution H F of (18) gives for ionization potential of orto-heliu ε = 0.19, while the experiental value is ε exp = 0.18[1]. 5. Conclusion We see that an accuracy of results obtained fro the new equations is the sae as in Hartree-Fock theory (several percent). This eans that new equations (14) cannot be used for calculation of relativistic corrections having the order 5 of α 10, where α- the fine structure constant [1]. Notice that relativistic derivation of the equations was required only for correct accounting of the considered syetries. It should be stressed in conclusion that the ain goal of the work is not to iprove the accuracy of Schrodinger s equation for heliu ato but to find out the possibility for reflection of physical reality within principally new approach in quantu physics. As it was here shown (at least approxiately) the suggested geoetrical description of the heliu ato reflects reality correctly. Acknowledgeents It is y pleasure to thank V.L.Bodneva and A.S.Prostnev for their help in nuerical calculations. References [1] O. A. Olkhov, Geoetrical approach to the atoic spectra theory. The heliu ato, Russian J.of Phys.Cheistry B, vol. 8, pp.0 4, February 014 [Chi.Fis., p.6, vol.,, 014]. [] O. A. Olkhov, Geoetrization of Quantu Mechanics, Journal of Rhys.:Conf.Ser. vol. 67, p [] O. A. Olkhov, Geoetrization of Classical Wave fields, Mellwill, Ney York, AIP Conference Proceedings p. 16, vol.96,008(proc.int.conf, Quantu theory: Reconsideration of Foundations, Vaxjo, Sweden, June, 007, 007); arxiv: [4] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics. vol.4: V. B. Berestetzki, E. M. Lifshitz, L. P. Pitaevski, Quantu Electrodynaics, Butterworth-Heineann, 198. [5] B. A. Dubrovin, A. T. Foenko, S. P. Novikov, Modern geoetry Methods and Applications, Part : The Geoetry and Topology of Manifolds, Springer, [6] A. S. Schwartz, Quantu field theory and topology. Grundlehren der Math. Wissen.07, Springer, 199. [7] H. S. M. Coxeter, Introduction to geoetry. New York, London: John Wiley&Sons, [8] P. K. Rashevski, Rieann geoetry and tensor analysis (in Russian). Moscow, Nauka, [9] V. A. Zelnorovich, Theory of spinors and its applications (in Russian). Moscow, August-Print, 001. [10] Ta-Pei Cheng, Ling-Fong Li, Gauge theory of eleentary particle physics. Clarendon Press, Oxford, [11] V. Popov, N. Konopleva, Gauge fields. Gordon and Breach Publishing Group, 198. [1] H. A. Bethe and E. E. Salpeter, Quantu echanics of one- and two-electron atos. Berlin-Gottingen-Heidelberg, Springer-Verlag, [1] V. Heine, Group theory in quantu echanics. London-Oxford-New York-Paris, Pergaon Press, [14] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics, vol., L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, Butterworth Heineann, 1975.