Gravitational Corrections to Energy-Levels of a Hydrogen Atom

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1 Commun. Theor. Phys. Beijing, China pp c Internationa Academic Pubishers Vo. 47, No. 4, Apri 5, 27 Gravitationa Corrections to Energy-Leves of a Hydrogen Atom ZHAO Zhen-Hua,, LIU Yu-Xiao, 2 LI Xi-Guo Institute of Modern Physics, the Chinese Academy of Sciences, Lanzhou 73, China 2 Institute of Theoretica Physics, Lanzhou University, Lanzhou 73, China Received Apri 24, 26 Abstract The first-order perturbations of the energy eves of a hydrogen atom in centra interna gravitationa fied are investigated. The interna gravitationa fied is produced by the mass of the atomic nuceus. The energy shifts are cacuated for the reativistic S, 2S, 2P, 3S, 3P, 3D, 4S, 4P eves with Schwarzschid metric. The cacuated resuts show that the gravitationa corrections are sensitive to the tota anguar momentum quantum number. PACS numbers: 4.9.+e, 3..+z Key words: hydrogen atom, gravitationa perturbation, generay covariant Dirac equation Introduction The study of gravitationa fieds interacting with spinor fieds constitutes an important eement in constructing a theory that combines quantum physics gravity. For this reason, the investigation of the behavior of reativistic partices in this context is of considerabe interest. It has been known that the energy eves of an atom paced in an externa gravitationa fied wi be shifted as a resut of the interaction of the atom with space-time curvature see Refs. [] [4] for exampes. And the geometric topoogica effects ead to shifts in the energy eves of a hydrogen atom were considered in Ref. [5]. Recenty, there has been a dramatic increase in the accuracy of experiments that measure the transition frequencies in hydrogen. The most accuratey measured transition is the S-2S frequency in hydrogen. It has been measured with a reative uncertainty of 25 Hz f/f =. 4, f = 2466 THz, [6,7] an order of magnitude arger than the natura inewidth of.3 Hz natura width of the 2S eve. [8,9] Indeed, it is ikey that transitions in hydrogen wi eventuay be measured with an uncertainty beow Hz. [] Though that accuracy cannot expore the gravitationa effect produced by the hydrogen atom nuceus, with the progress of experiments we can detect the gravitationa effect. In this paper we investigate another previousy negected gravitationa effect on the energy-eve shifts of a hydrogen atom. This is to give some expicit vaues for energy-eve shifts of a hydrogen atom by the genera reativistic effect with Schwarzschid metric. And the difference with Refs. [] [5] is that the gravitationa fied in this paper is not an externa fied but is produced by the mass of hydrogen atom nuceus. To our knowedge no one has given expicit vaues for energy-eve shifts of a hydrogen atom with gravitationa corrections. Athough the effect is very sma, it aso has the physica significance as a test of genera reativity at the quantum eve. This paper is organized as foows. In Sec. 2 we review the formaism of the generay covariant Dirac equation in curved space-time. In Sec. 3 we give the tetrad spinor connections with Schwarzschid metric. The gravitationa perturbation of reativistic S eve is cacuated in Sec. 4. The summary discussion are given in Sec Generay Covariant Dirac Equation in Curved Space-Time To write the generay covariant Dirac equation in curved space-time with metric g µν, one first introduces the spinor affine connections ω µ = /2ωµ ab I ab, where I ab are the generators of SO4 group, whose spinor representation is I ab = 4 γ aγ b γ b γ a. Here γ a are the Dirac Paui matrices with the foowing reation, γ a γ b + γ b γ a = 2η ab, 2 γ = γ, γ i = γ i, i =, 2, 3, 3 γ = i β, γ i = i βα i, 4 σi I α i =, β =, 5 σ i I where I is the 2 2 identity matrix, η ab = η ab = diag,,, is the Minkowski metric tensor, σ i are the stard Paui matrices σ =, σ 2 = i, i The project supported by Nationa Natura Science Foundation of China under Grant Nos the Chinese Academy of Sciences Knowedge Innovation Project under Grant Nos. KJCX2-SW-Nb KJC-SYW-N2 Corresponding author, E-mai: zhaozhenhua@impcas.ac.cn

2 No. 4 Gravitationa Corrections to Energy-Leves of a Hydrogen Atom 659 σ 3 =. 6 ω ab µ is defined by the vanish of the generaized covariant derivative [,2] of the tetrad or vierbein fied e a µx, [3] D µ e a ν = µ e a ν Γ λ µνe a λ η bcω ab µ e c ν = µ e a ν η bc ω ab µ e c ν, 7 where the tetrad fied e a µx it is inverse e satisfy the foowing equations: µ a x g µν x = η ab e a µxe b νx, 8 µ e a µxeb x = δa b, µ, ν, a, b =,, 2, 3, 9 µ ν are the space-time indices owered with the metric g µν, a b are the Lorentz group indices owered with η ab. One aso needs to introduce generaized Dirac Paui matrices Γ µ x = e a µxγ a, which satisfy the equation [2] Γ µ xγ ν x + Γ ν xγ µ x = 2g µν x. The covariant derivative acting on a spinor fied ψ is then D µ ψ = µ ψ ω µ ψ, the generay covariant form of the Dirac equation [4] in pure gravitationa fied is Γ µ xd µ ψx + mc ψx =, 2 h where Γ µ x = g µν Γ ν x, m is the mass of spinor partices. For an eectron near the atomic nuceus one needs to consider the effect of the eectromagnetic vector potentia A µ, here A µ satisfies the Maxwe equations [2,4] g λσ λ σ A µ R ν µ A ν = 4πJ µ, 3 where J µ is the current vector. So the covariant derivative acting on a spinor fied shoud be rewritten as D µ ψ = µ ω µ i qa µ ψ. 4 Then the generay covariant form of the Dirac equation in gravitationa eectromagnetic fieds is Γ µ µ ω µ i qa µ ψx + mc ψx =. 5 h 3 Spinor Connections in Schwarzschid Space-Time In what foows, we wi cacuate the spinor connections in a Schwarzschid spacetime. The ine eement corresponding to the spacetime is given by dt 2 ds 2 = g µν dx µ dx ν = c 2 R s r where R s = 2GM/r. With the time gauge conditions, [5,6] e i = e i foows: e a µ = R s r R s /r dr2 r 2 dθ 2 r 2 sin 2 θd φ 2, 6 =, the tetrad fied e a µ is given as sin θ cos φ r cos θ cos φ r sin θ sin φ sin θ sin φ. 7 r cos θ sin φ r sin θ cos φ cos θ r sin θ Taking the approximation R s /r = R s /2r, we have R s 2r sin θ cos φ r cos θ cos φ r sin θ sin φ e a µ = R s /2r sin θ sin φ. 8 r cos θ sin φ r sin θ cos φ R s /2r cos θ r sin θ R s /2r From Eq. 7, it foows that ω ab µ = µ e a νe b λ gλν, 9 ω µ = 2 ωab µ I ab = 2 I ab µ e a νe b λ gλν 2 I ab Γ ρ µνe a ρe b λ gλν. 2

3 66 ZHAO Zhen-Hua, LIU Yu-Xiao, LI Xi-Guo Vo. 47 Thus using Eqs. 6, 8, 9, 2, we obtain the expicit expressions of the nonzero components of spinor connections, R s cos θ R s sin θ e iφ R s sin θ e iφ R s cos θ ω = R s cos θ R s sin θ e iφ, 2 R s sin θ e iφ R s cos θ r R s e iφ 2r R s r R s e iφ ω 2 = 2r R s r R s e iφ, 22 2r R s r R s e iφ 2r R s ic R s sin 2θ i e iφ 8r 4R s 8r 4R s ir s sin 2θ e iφ ic ω 3 = 8r 4R s 8r 4R s ic R s sin 2θ i e iφ, 23 8r 4R s 8r 4R s ir s sin 2θ e iφ ic 8r 4R s 8r 4R s where C = 4r 3R s + R s cos2θ Gravitationa Perturbation of Reativistic Hydrogen Atom: S /2 States From Eq. 5 the corresponding Hamitonian in curved space-time reads, The Dirac Hamitonian in fat space is H = i hcγ Γ i i ω i i qa i + i hcω + i qa imc 2 Γ. 25 H = i hcγ γ i i i qa i hcqa imc 2 γ, 26 where A µ are the eectromagnetic vector potentias in fat spacetime. Here we can take the approximation A i = A i = A = A = er, the detaied discussions of this probem is contained in Ref. [2]. So the Hamitonian of the gravitationa perturbation is given by H I = H H = i hcγ Γ i i ω i + i hcω im e c 2 Γ + i hcγ γ i i + im e c 2 γ. 27 The exact soutions of the Dirac equation for a hydrogen atom in fat space-time serve as the basis for perturbation theory. The energy eigenvaues of a hydrogen atom are E nκ = m e c 2 ζ 2 +, 28 n κ + s where ζ = Ze 2, s = κ 2 ζ 2, n =, 2,... is the principa quantum number. The bound state functions of a hydrogen atom can be written in stard representation [7,8] as ψ = ψκ M grχ M κ = ifrχ M, 29 κ where M is the eigenvaue of J z, κ is the eigenvaue of K = β σ L + I, the functions fr, gr spinors χ M κ, χ M κ are given by fr = 2s /2 λ s+3/2 Γ2s + n r + W c r s e λr Γ2s + n r!ζ ζ λκ

4 No. 4 Gravitationa Corrections to Energy-Leves of a Hydrogen Atom 66 κ ζk c F n r, 2s +, 2λr n r F n r +, 2s +, 2λr, 3 λ gr = 2s /2 λ s+3/2 Γ2s + κ ζ λ χ M κ = C /2 Y M /2 χ M κ = C /2 Y M /2 Γ2s + n r + n r!ζ ζ λκ W c r s e λr F n r, 2s +, 2λr + n r F n r +, 2s +, 2λr + C /2 Y M+/2 cos θ e iφ sin θ + C /2 Y M+/2, 3, 32 e iφ sin θ cos θ, 33 where W c = W nκ /m e c 2, = m e c 2 / hc, λ = m 2 ec 4 Enκ/ hc, 2 C /2 C /2 are the C-G coefficients. For a hydrogen atom there are two S /2 n =, =, J = /2, κ = states, which correspond to M = ±/2. The states can be written as fr ψ = igr sin θ e iφ, 34 ψ 2 = where ψ corresponds to M = /2 ψ 2 to M = /2, igr cos θ fr igr cos θ igr sin θ e iφ, 35 fr = 2 3/2+s e rλ r +s λ /2+s + W c Kc ζ + λ πζγ + 2s, 36 gr = 2 3/2+s e rλ r +s λ /2+s W c Kc ζ + λ πζγ + 2s, 37 Γ + 2s is the Γ function. The gravitationa perturbation matrix eements are H I ab ψ a, H I ψ b, 38 where the subscripts a b take the vaues, 2. Because we take the gravitationa fied metric as the Schwarzschid metric, so we need to confirm the range of the integration. Here it is taken from R n to, R n =.3 5 m is the atomic nuceus radius. With the computer agebra system Mathematica, we obtain the foowing resuts for those perturbation matrix eements δ ab H I ab = Kc 2 R n ζγ + 2s 2 +2s cr s λr n λ 2s ζ + λ cmr n W c E 2s 2R n λ + Kc 2 Wc 2 he 2 2s 2R n λ, 39 where E n z = ezt /t n dt is the exponentia integra function. Using the equation [2] det[ψ a, H I ψ b E i δ ab ] =, 4 from the usua perturbation theory of a degenerate energy eigenvaue, it foows that both of the degenerate S /2 eves are shifted by the same perturbation: E S /2 = Kc 2 R n ζγ + 2s 2 +2s cr s λr n λ 2s ζ + λ cmr n W c E 2s 2R n λ + Kc 2 Wc 2 he 2 2s 2R n λ. 4 Substituting the constant vaues in Tabe into Eq. 4, we get E S /2 = ev. 42

5 662 ZHAO Zhen-Hua, LIU Yu-Xiao, LI Xi-Guo Vo. 47 Tabe The constants tabe. [9] Quantity Symbo Vaue Units Eectron charge magnitude e C Speed of ight in vacuum c m s Eectron mass m e kg Panck constant, reduced h s Permittivity of free space ɛ s 4 A 2 kg m 3 Proton mass M p kg Gravitation constant G m 3 kg s 2 5 Summary Discussion In a simiar cacuation as the S /2 state, we find that a the reativistic S, 2S, 2P, 3S, 3P, 3D, 4S, 4P energy eves are respectivey shifted as the same amount isted in Tabe 2. This means that the first-order gravitationa perturbations can party remove the degeneracy of the hydrogen atom states. Athough the effect is very sma, from Tabe 2 we find that the quantity of corrections of the energy eves with same principa quantum number n tota anguar momentum quantum number J, ike 2S /2 2P /2, 3S /2, 3P /2, 3P 3/2 3D 3/2, are very cose. But for the eves with same principa quantum number different tota anguar momentum quantum number, ike 3S /2 3P 3/2, their corrections have obvious difference. Those cacuations show that the gravitationa corrections are sensitive to the tota anguar momentum quantum number. It is a very important feature of the interaction between gravitationa fieds spinor fieds. With this feature we can find the gravitationa effect in other system, make a test of genera reativity at the quantum eve. State S /2 2S /2 2P /2 2P 3/2 3S /2 3P /2 3P 3/2 3D 3/2 4S /2 4P /2 Tabe 2 The energy-eve shifts. The energy-eve shift Unit: ev References [] L. Parker, Phys. Rev. Lett [2] L. Parker, Phys. Rev. D [3] L. Parker L.O. Pimente, Phys. Rev. D [4] Y.S. Duan, J. Expt. Theor. Phys. U.S.S.R ; E. Fischbach B.S. Freeman, Phys. Rev. D [5] G. de A. Marques V.B. Bezerra, Phys. Rev. D [6] M. Fischer, et a., Phys. Rev. Lett [7] N. Koachevsky, J. Anis, S.D. Bergeson, T.W. Hänsch, Phys. Rev. A [8] C.L. Cesar, et a., Phys. Rev. Lett [9] T.C. Kiian, et a., Phys. Rev. Lett [] U.D. Jentschura, P.J. Mohr, G. Soff, Phys. Rev. Lett [] T.W.B. Kibbe, J. Math. Phys [2] R. Lo, Discrete Approaches to Quantum Gravity in Four Dimensions, [3] E. Poisson, The Motion of Point Partices in Curved Spacetime, [4] C.W. Misner, K.S. Thorne, J.A. Wheeer, Gravitation, Freeman, San Francisco 973 p [5] J. Schwinger, Phys. Rev [6] J.W. Mauf, J.F. da Rocha-Neto, T.M.L. Toribio, K.H. Casteo-Branco, Phys. Rev. D [7] M.E. Rose, Reativistic Eectron Theory, Wiey, New York 96 p. 77. [8] P. Strange, Reativistic Quantum Mechanics, Cambridge University Press, Cambridge 998 p [9] S. Eideman, et a., Phys. Lett. B