arXiv:1803.09238v1 [physics.atom-ph] 25 Mar 2018 oeo esabtaiy orpeetteaslt positio absolute the represent to syst X arbitrarily, the of less mass e or of this center more solve the to of used coordinates was the approach determining r following the The Feynman, derived. of s been formalism two-particle S-matrix The relativistic the [2]. Using work well-known the in investigated ubr ontdpn ntemto fteao etro mass. of center atom the sq t of the motion result, to the a on proportional As depend inversely not mass. are do of which number, center atom the m hydrogen of relative the the motion atom the hydrogen the from the is introduce separates for This Hamiltonian to the [1]. possible in velocities is Then, their it of on approximation levels depend relativistic not energy do the atom, mechanics hydrogen quantum non-relativistic In Introduction 1 shifts atom-motion-induced shift; Lamb the yrgneeg-eessit nue yteao oin rsoe rmthe from Crossover motion. atom the by induced shifts energy-levels Hydrogen µ nqatmeetoyai h on-tt rbe o w i two for problem bound-state the electrodynamic quantum In 31.30.jf 31.30.J-, numbers:03.65.Pm, PACS ewrs yrgnlk tm h nrylvldegenerac energy-level the atom; hydrogen-like Keywords: = constant, momentum, number, quantum principal same the h yrgnmmnu n h lcrni fodrof order t of that is shows electron the analyze and simple momentum A hydrogen electron. the hydrogen the with interacts o h yrgnvelocity opera hydrogen velocity-dependent the two for the equation, Bethe-Salpeter the aitv orcin teLm hf)ta r rprinlto proportional are that shift) Lamb (the corrections radiative hoydeveloped. theory hi re fmgiue h rsoe rmteLm hfst t to shifts velocity Lamb atom the the from or at crossover their occur the in should magnitude, different of essentially are order which their perturbations degene two mutually these the the between of to per corrections correction velocity-dependent energy diagonal the off-diagonal The the the is ene states. shift degenerate the the Lamb and of The shifts each motion. Lamb atom the the between by difference important an degen ever, this remove interactions perturbation velocity-dependent on βx n µ hntehdoe tmmvs h rtncretgnrtsa generates current proton the moves, atom hydrogen the When 1 and +(1 v − j neprmn sdteobtlmto fteErh spooe t proposed is Earth, the of motion orbital the used experiment An . steao velocity, atom the is β j ) u h ieetobtlaglrmomenta angular orbital different the but , x µ 2 ainlRsac ete”ucao Institute”, ”Kurchatov Centre Research National with absit otemto-nue shifts. motion-induced the to shifts Lamb β v c n rirr osat n h eaiecoordinate relative the and constant, arbitrary any <α << Agafonov ocw138,Russia 123182, Moscow c swl nw,tedgnrc fteeeg eeswith levels energy the of degeneracy the known, well As . v stesedo ih,and light, of speed the is = n ..Agafonov A.I. n h aeqatmnme ftettlangular total the of number quantum same the and , kα Abstract 2 c [email protected] where , 1 >> k α ,i ieadsae ftesystem, the of space, and time in n, 3 aisvco ftecne fmass. of center the of vector radius se a osdrdt emoving. be to considered was ystem sanmrclfco depended factor numerical a is 1 v m twsslce oecoordinate, some selected was it em, c ltvsi on-tt qainhas equation bound-state elativistic m l ubto neatosrsl in result interactions turbation m uto.Ntn h iclisin difficulties the Noting quation. tm n,i atclr fthe of particular, in and, atoms = where , u otefc hti h non- the in that fact the to due ;teBteSlee equation; Bethe-Salpeter the y; oso hsodraederived are order this of tors to fteeeto n proton and electron the of otion rc swl.Teei,how- is, There well. as eracy aeo h rnia quantum principal the of uare steeeto as Using mass. electron the is aesae.Tejiteffect joint The states. rate j edgnrt nrylvl of levels energy degenerate he α 5 trcigfrin a been has fermions nteracting ± emto-nue shifts motion-induced he m g-eessit induced shifts rgy-levels gn saaye.Given analyzed. is igin, i neato between interaction his 1 ti hw htthe that shown is It . / α antcfil which field magnetic nrysprtl for separately energy srmvdb the by removed is 2 stefiestructure fine the is x etthe test o µ = x µ 1 − x µ 2 . Then, the solutions of the equation was sought for the special form of the wave function (see Eq. (13) [2]), ψ(1, 2) = exp (iKµXµ)ψ(xµ), where Kµ is an arbitrary constant four-dimensional vector represented the momentum and energy of the moving system. Note that this wave function ψ(xµ) describing the relative motion of the two particles, was assumed to be independent on the atom four momentum Kµ. As noted in [2], although the equations for ψ(xµ) are different in non-relativistic quantum mechanics and QED, the situation is the same in principle, namely, the integral equation (see Eq. (16) [2]) for the momentum- space wave function ψ(pµ) depends on only four variables that are the components of the relative momentum of the two interacting particles. It differs from the ordinary Schrodinger momentum wave function by the appearance of the ”relative energy” p as a fourth independent variable. − 4 The foundations of the work [2] were then used to study many-electron systems, nuclei and quarks [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Meanwhile, in the relativistic problem the discrete atomic energies can depend on the mo- mentum of the atom center of mass. To demonstrate it, we present here a non-rigorous analysis, which allows us to understand the origin of some perturbation induced by the finite velocity of the atom. Note that this analysis is close to qualitative consideration of the spin-orbit interac- tion in atoms. Let the atom be moving with the constant velocity v. Both the electron and proton have two velocity components. The first components are related to the motion of the atom center of mass, are the same for both the particles and equal to v. The second components are due to the relative motion of the particles. e r The moving proton with the velocity v creates both the electric field E = r2 r and the e magnetic field H = c [v, E], where e is the proton charge, r is the relative radius vector between the electron and proton, and c is the speed of light in vacuum. Here, neglecting the terms of the order of v2/c2, the Lorentz factor equal to 1 is assumed. Then the magnetic field is: e r v e H = [v, ] = [ , ] = [ , A]. c r3 ∇ c r ∇ Hence the vector potential is v e A = . (1) c r This expression for the vector potential can be obtained by another way. The moving proton creates the current density j = evδ(r vt). Then, the vector potential is 1 − ev δ(r vt)dr A(r)= 1 − 1 , c r Z where we take into account the move of the electron with the same velocity and, hence, the relative radius vector between the electron and proton does not depend on v. The integration over r1 leads to the same expression for the vector potential. Substituting (1) in the Dirac Hamiltonian for the electron in the external field, H = α(pˆ + eA)+ βm eΦ, we obtain an additional perturbation caused by the interaction of the electron − spin with the magnetic field, generated by the moving proton,: α v Wˆ = e V (r). (2) e − c
Here the αe matrix is used in the standard representation, the subscript e means the action of 2 the Pauli matrices on the electron spin, and V (r) = e is the Coulomb interaction between − r the electron and proton.
2 The similar consideration of the fields generated by the moving electron with the velocity v leads to the interaction of the proton spin with the magnetic field, created by the moving electron,: α v Wˆ = p V (r), p − c where the subscript p means the action of the Pauli matrices on the proton spin. The commutator [H,ˆ r]= α, and, hence, the operators αe and αp can be considered as the operators of the relative velocities of the electron and the proton in the bound state. Since the relative velocity of the proton is 1836 times less than that for the electron, the interaction Wˆ p can be neglected in comparison with the interaction (2). 3 v The interaction (2) is proportional to α c m, where α is the fine structure constant and we take into account that the electron velocity < α > αc. Perturbations of this order will | e | ∝ play a key role in the present paper. The origin of the other perturbation that is of the same order as (2), can be understood from the following considerations. The Dirac Hamiltonian for the electron in the Coulomb field can be reduced to the Schr¨odinger one in the known reasonable approximations. Replacing the electron momentum in the kinetic energy operator by its canonical momentum with the vector-potential (1), we find the electron perturbation operator: 1 v Qˆ = [pˆ, V (r)] . (3) e 2mc c +
The similar expression for the electron perturbation operator Qˆp can be written by the replace- ments p p and m M in (3). For the above reason Qˆ can be omitted. →− → p We emphasize that it would be wrong to introduce directly the particle canonical momenta with the vector-potential (1) in the Bethe-Salpeter equation and, as well, in the Schr¨odinger equation. The operators (2) and (3) do not have the exact forms that will be obtained below. They only give an idea of the considering effect of relativistic corrections to the hydrogen discrete levels due to the atom momentum. The effect under investigation should manifest itself when the solutions of the relativistic two-fermion equation are sought in the form:
ψ(1, 2) = exp (iKµXµ)ψ(xµ; K), where the relative wave function ψ(xµ; K) is considered to depend on the four-dimensional vector K represented the atom momentum. As well known [16], the spin-orbit interaction in the hydrogen atom removes the degeneracy of the energy levels inherent in the non-relativistic theory, but not completely. The energy levels with the same n and j, but different l = j 1/2, remain mutually degenerate. This remaining ± degeneracy is removed by the so-called radiation corrections (the Lamb shift) [17, 18, 19, 20, 21], which are not taken into account in the Dirac equation for the one-electron problem. 5 2 The Lamb shift for the hydrogen levels 2s1/2 and 2p1/2 is equal to δEL = 0.41α mc [16]. Looking ahead, we note here that the atom-momentum-perturbation operators that have forms closer to (2) and (3), also remove this remaining energy level degeneracy with the same n and j, but different l = j 1/2. As will be shown below, for moving hydrogen atoms the energy ± shift due to these interactions turn out to be proportional to δE α3 v mc2 for the same levels. v ∝ c For this reason, there must exist the region of the atom velocities in which
δEv > δEL.
In this case, the levels degeneracy considered will mostly be removed by the perturbation being studied. The aim of the work is to investigate the effect of the hydrogen atom velocity on its discrete energy levels. Our analysis is carried out for the atomic velocities much less than the velocity
3 of light in vacuum, v << c. A more detailed restriction on the atom velocities will appear below. Using the Bethe-Salpeter equation we deduce the operator of the atom-momentum- induced perturbations for the hydrogen-like atoms. Then we study the shifts of the mutually degenerated energy levels due to both the radiative corrections and the interactions induced by the atom motion. Also we provide estimations for experimentally verifications of the theory developed. Natural units (¯h = c = 1) will be used throughout.
2 The Bethe-Salpeter equation for the electron-proton system
The two-fermion system considered is the electron and the proton each of which is described by the Dirac theory. Therefore, we must use the free particle propagator, which follows from the Dirac equation for the free fermion. Considering this case, Feynman proposed the free fermion propagator that can be written as [22]:
e 1 + iεp(t2 t1) iεp(t2 t1) ip(r2 r1) K0(2, 1) = Λe e− − + Λe−e − e − θ(t2 t1) (4) p 2εp − X h i for the electron and
p 1 + iωq(t4 t3) iωq(t4 t3) iq(r4 r3) K0 (4, 3) = Λp e− − + Λp−e − e − θ(t4 t3) (5) q 2ωq − X h i for the proton. Here we use following notations. For the electron propagator (4),
+ Λ = ε + mβ + α p, Λ− = ε mβ α p, (6) e p e e e p − e − e 2 2 2 α and εp = m + p , m is the electron mass, βe and e is the matrices in the standard represen- 1 0 0 σ tation, β = , α = e , σ is the Pauli matrices. The subscript e of the e 0 1 e σ 0 − ! e ! Pauli matrices means their action on the electron spin. For the proton propagator (5),
+ Λ = ω + Mβ + α q, Λ− = ω Mβ α q. (7) p q p p p q − p − p 1 0 0 σ Here ω2 = M 2 + q2, M is the proton mass, β = , α = p . The lower q p 0 1 p σ 0 − ! p ! index p of the Pauli matrices means their action on the proton spin. In the ladder approximation their reaction can be presented by the interaction function [22]:
G(1)(3, 4; 5, 6) = e2(1 α α )δ (s2 )δ(3, 5)δ(4, 6). (8) − − e p + 56 2 Here s56 is the invariant distance between these particles, and δ+(s56) is the virtual photon propagation function, 1 eik(r5 r6) ik0(t5 t6) δ (s2 )= − − − d4k. (9) + 56 4π3 k2 k2 iδ Z − 0 − In the ladder approximation the Bethe-Salpeter equation for the electron-proton system is:
ψ(1, 2) = i dτ dτ dτ dτ Ke(1, 3)Kp(2, 4)G(1)(3, 4; 5, 6)ψ(5, 6), (10) − 3 4 5 6 0 0 ZZZZ where dτi = dridti.
4 3 Separation of the atom total momentum
As known [16], the Dirac equation for the electron in the external Coulomb field gives the correct values of the binding energy up to α4. The terms which are proportional to αn with n > 4, should be analyzed together with perturbations which are not contained in the Dirac equation. In Eq. (10) the effects of both the propagation function of the virtual photon that leads to retardation interaction between the particles, and the interaction through the vector potential created by the relative motion of the particles in the hydrogen atom, result in the energy corrections of order of αn with n 5 [16]. In our consideration both these effects do not ≥ make sense to hold in this equation because the level shifts due to the proposed interaction is of order of α3, as was noted in Introduction (see (2) and (3)). Then, using (4)-(9), Eq. (10) is reduced to the form:
2 1 ip(r1 r3)+iq(r2 r4) ψ(1, 2) = ie dr3 dt3 dr4 dt4 e − − pq 4εpωq Z Z Z Z X
+ iεp(t1 t3) iεp(t1 t3) + iωq (t2 t4) iωq (t2 t4) 1 Λe e− − + Λe−e − Λp e− − + Λp−e − ψ(3, 4). (11) r3 r4 h i h i | − | In (11) it is convenient to separate the motion of the atom center of mass from the relative motion of the electron and proton. In relativity theory the coordinate of the center of mass cannot be defined, and the absolute position, in time and space, of the system considered, can be chosen quite arbitrarily [2]. We consider the velocities of atoms which are much smaller than the speed of light in vacuum. Taking into account the non-relativistic limit, the choice is obvious: R = (mr + Mr )/(m + M), r = r r , T = t = t . (12) 1 2 1 − 2 1 2 In (12) the first part is the absolute position of the atom, the second one describes the relative motion of the electron and proton, and latter, that is the equal time approach, is due to the fact that the retardation effect is absent in (11). The solutions of Eq. (11) for the wave function are sought in the form:
igR iET ψ(1, 2) = ψ(r; g)e − , (13) where g is the momentum of the atom, E is the total energy of the moving system. Changing to new variables of integration (12) and using (13), Eq. (11) is rewritten as:
T ′ Mp−mq ′ igR iET 2 1 i(p+q)(R R )+i (r r ) ψ(r; g)e − = ie dr′ dR′ dT ′ e − M+m − pq 4εpωq Z Z Z−∞ X
′ ′ ′ ′ ′ ′ + iεp(T T ) iεp(T T ) + iωq (T T ) iωq (T T ) 1 igR iET Λe e− − + Λe−e − Λp e− − + Λp−e − ψ(r′; g)e − (14) r′ h i h i Integrating over R′ in (14) that gives us p + q = g, and then integrating over T ′, we obtain the following integral equation:
′ 2 i(p m g)(r r ) 1 ψ(r; g)= e dr′ e − M+m − F (p, g p) ψ(r′; g), (15) − p − r′ Z X where + + + 1 Λ Λ Λ−Λ F (p, g p)= e p + e p + − 4εpωg p E εp ωg p + iδ E + εp ωg p + iδ − − − − − − + Λ Λ− Λ−Λ− e p + e p (16) E εp + ωg p + iδ E + εp + ωg p + iδ − − −
5 In (15) we introduce a new summation variable, m f = p g, (17) − M + m m and, further, we use the notations: ε = ε with p = f + g, and ω = ωq with q = g p = p M+m − f + M g. Then, using (6) and (7), the right-hand side of (16) is reduced to: − M+m 1 2E E2 + ε2 ω2 E2 ε2 + ω2 F = + mβe + αep + − Mβp + αpq + − . (18) 2E ∆ " 2E # " 2E #
Here 2 ∆= E4 2E2 ε2 + ω2 + ε2 ω2 . (19) − − As a result, the integral equation(15) is rewritten as:
2 if(r r′) 1 ψ(r; g)= e dr′ e − F (f, g, E) ψ(r′; g). (20) − r′ Z Xf It is easy to verify that the integral equation (20) corresponds to the following differential equation: 2 e2 E4 2E2 ε2 + ω2 + ε2 ω2 1+ ψ(r; g)= − − 2Er E2 + ε2 ω2 E2 ε2 + ω2 e2 2E mβe + αep + − Mβp + αpq + − − ψ(r; g). (21) " 2E # " 2E # r Here ˆf = i r and − ∇ m M ε2 = m2 + (ˆf + g)2,ω2 = M 2 + ( ˆf + g)2 (22) M + m − M + m In the equations (20) and (21) E is the total energy of the moving hydrogen atom. We present E as E = (M + m + + δ )2 + g2 + iδ. (23) E Eg Here presents the energy levelsq of the hydrogen atom at g = 0, and the value is the g- E Eg dependent relativistic energy-levels shifts induced by the atom motion. The representation (22) is convenient since in non-relativistic quantum mechanics the energy levels do not depend on g. Further, the infinitesimal positive δ can be omitted since the bound states are investigated. The integral equation (20) and the differential equation (21) can be reduced to the Dirac equation for the hydrogen atom in the limit m/M 0, and can be transformed to the → Schr¨odinger Hamiltonian in the non-relativistic limit, as shown in Appendix A.
4 The g-dependent perturbation operators
The radiative corrections that are not be included in (21), are proportional to α5m. As was 3 v shown in the Introduction, the g-dependent perturbation operators is of order of α c , where v is the atom velocity, c is the speed of light in vacuum. Terms of this order must be deduced from Eq. (21), but firstly we must determine what atomic velocities are considered. To observe crossover from the Lamb shifts to the motion-induced shifts for the hydrogen atoms it turns out enough that the atom velocities v << α. (24) c
6 Appendix B provides the corresponding transformations of the Eq. (21) in detail. As a result, it was obtained that the atom-motion-induced shift of the atomic-energy levels for hydrogen atoms is given by: δ =< ψ Wˆ + Qˆ ψ > (25) Eg | | where the velocity dependent perturbation operators are α σ ˆ ev α 0 v e W = V = σ (26) 2c −2cr v e 0 ! and vˆf v α Qˆ = V 2 = i ∇ + 2 (27) 2mc − E 2mc r E In Eq. (26) σe is the Pauli matrices acting on the electron bispinor. In Eq. (27) the operator ˆf is the momentum operator of the electron. The perturbation operators (26) and (27) exist only for the moving atoms. The represented expressions for the perturbation operators are correct only for the atom’s velocities v << αc. In Eq. (25) ψ is the electron wave functions of stationary states. Since M = 1836m the proton mass corrections to the wave functions and the energy eigenvalues are small. Hence we can use the well-known stationary wave functions obtained for the relativistic Coulomb problem [16]. The spin-orbit interaction does not completely remove the degeneracy of the hydrogen energy levels. The levels with the same principal quantum number, n, and the same quantum number of the total angular momentum, j, but different orbital angular momenta l = j 1/2 remain ± mutually degenerate with the exception of the levels with the maximal possible number j = n 1/2. That is, for the same n the hydrogen levels s and p , p and d , d and f , − 1/2 1/2 3/2 3/2 5/2 5/2 and so on, are mutually degenerate. This remaining degeneracy is removed by the radiative corrections (the Lamb shift), which are not taken into account in the Dirac equation. The Lamb shift for the hydrogen is proportional to α5m. In the next Section we show that the perturbation interactions (25)-(27) remove this degeneracy as well.
5 The splitting of the hydrogen energy levels
In the relativistic case, there are no separately conservation laws of spin and orbital angular momentum. Consequently, a definite value of the orbital angular momentum and its projection on any axis are absent in the stationary states. Since there is no dedicated direction, the x ,y and z axes must be equally represented. The interaction operators (26) and (27) are − − − proportional to the atom velocity, v. We choose the coordinate system such that vx = vy = v = v . Then the linear superposition of the mutually-degenerate states can be written in the z √3 following form: ψ = amψ 1 + bm′ ψ 1 ′ , (28) nj,j 2 ,m nj,j+ 2 ,m m − ′ X Xm where summations are taken over the secondary total angular momentum quantum numbers m and m′ that are the allowed values of the projection of the total angular momentum on the z axis. − Our choice simplifies essentially the calculations, since in the general case, in (28) instead 1 of the functions ψ 1 , one should use (ψ 1 + ψ 1 + ψ 1 ), where mx nj,j 2 ,m √3 nj,j 2 ,mx nj,j 2 ,my nj,j 2 ,m and m are the projection± of the total angular momentum± on the± x and y axis.± In this case y − − v//z axis can be chosen. − In the superposition (28), for example, for the principal quantum number n = 2 and the quantum number of the total angular momentum j = 1/2 the function ψnj,j 1 ,m can be treated − 2 as the wave function of the 2s1/2 state with the orbital momentum l = 0, and ψ 1 ′ - as nj,j+ 2 ,m the 2p1/2 wave function with the orbital momentum l = 1.
7 In the standard representation, this superposition (28) is rewritten as:
j 1 j+ 1 − 2 2 Fnj Ωj,j 1 ,m Fnj Ωj,j+ 1 ,m′ − 2 ′ 2 ψ = am j 1 + bm j+ 1 . (29) − 2 ′ 2 m Gnj Ωj,j+ 1 ,m m Gnj Ωj,j 1 ,m′ X 2 X − − 2 j 1 j 1 ± 2 ± 2 Here Fnj (r), Gnj (r) are the well-known real radial wave functions (see [16]), the functions Ω with l = j 1 are the spherical spinors which are formed by combining the spherical jlm ± 2 harmonics Yl,m 1 (m,m′ = j, j + 1, , j 1, j), ± 2 − − · · · − j+m j m+1 2j Yj 1 ,m 1 −2j+2 Yj+ 1 ,m 1 Ω 1 = − 2 − 2 , Ω 1 = − 2 − 2 . (30) j,j 2 ,m j m j,j+ 2 ,m j+m+1 − q 1 1 q 1 1 −2j Yj ,m+ 2j+2 Yj+ ,m+ − 2 2 2 2 q q The components of both the bispinors in (29) contain the spherical harmonics (30) of the both orders l = j 1/2 that expresses the absence of a definite value of the orbital angular ± momentum and its z projection in the stationary states. − The matrix elements of the perturbations (26) and (27) are conveniently calculated sepa- rately. Using (29) and (30), it is easy to see that the non-zero matrix elements of (26)
α dr + 0 σi W = Wi = ψ vi ψ (31) −2c r σi 0 ! i=Xx,y,z Z hi=Xx,y,z i are only the off-diagonal matrix elements between the states nl and n(2j l) . Moreover, the j − j matrix elements of the operator Wz are non-zero only if the quantum numbers m and m′ are the same, m = m′: 1 1 1 1 α vz m j j+ 2m j j+ mm mm ∞ − 2 2 − 2 2 Wj 1 ,j+ 1 = Wj+ 1 ,j 1 = rdr Fnj Gnj + Gnj Fnj . (32) − 2 2 2 − 2 2 c 0 j 2j + 2 Z The off-diagonal matrix elements of the operator W +W are non-zero only if m m = 1: x y − ′ ± ′ α 1 j 1 j+ 1 1 j 1 j+ 1 mm ∞ − 2 2 − 2 2 Wj 1 ,j+ 1 = rdr Fnj Gnj + Gnj Fnj − 2 2 2 0 2j 2j + 2 Z vx ivy vx + ivy − (j + m)(j m + 1)δm′,m 1 + (j m)(j + m + 1)δm′,m+1 , (33) c − − c − q q and ′ ′ m m mm ∗ Wj+ 1 ,j 1 = Wj 1 ,j+ 1 2 − 2 − 2 2 Now consider the perturbation (27) whose matrix elements are: i α Q = drψ+ v + 2 ψ . 2mc ∇ r E Z Taking into account that the wave functions ψ decrease exponentially with increasing r, the above expression can be reduced to a form that is more convenient for calculations: i α Q = dr + 2 (vgradψ+)ψ. (34) −2mc r E Z We will show that the diagonal matrix elements of the perturbation (34) vanish for the mutually-degenerate states considered below. Avoiding cumbersome calculations, the off-diagonal matrix elements are easily obtained for each specific case of the superposition (29)-(30). Emphasize that there is an important difference between the Lamb shifts and the motion- induced shifts. The Lamb shifts lead to the diagonal corrections to the energies separately for each of the degenerate states. The interactions (26) and (27) depend on the velocity of the atom’s motion, and result in the off-diagonal energy corrections between the mutually degenerate states. Therefore, it is of interest to analyze the joint effect of these perturbations.
8 5.1 The splitting of hydrogen 2s1/2 and 2p1/2 levels As well known, the radiation corrections for the hydrogen atom is
5 δε2s = 0.4065α m for the 2s1/2 level and δε = 0.0050α5m. 2p − for the 2p1/2 level. Here the two main contributions that are 1) the emission and absorption by the bound electron of virtual photons and 2) the vacuum polarization, are taken into account. The correction contributions of other effects to the Lamb shift will be cosidered in Section 6. For the considered states the superposition (29) with account for (30) takes the form:
0 1 1 1 1 2 1 F Y0,0 0 F Y1,0 F 1 Y 2 2 √3 2 2 3 1, 1 0 − 2 2 − F 1 Y − 0 2 0,0 2 1 q1 1 2 3 F 1 Y1,1 √ F 1 Y1,0 ψ = a 1 1 0 +a 1 +b 1 2 2 +b 1 3 2 2 G 1 Y 2 0 2 √ 1,0 2 G 1 Y 2 1 2 3 2 2 − 3 1, 1 q G 1 Y − 0 − − 2 2 − 2 0,0 2 0 q1 0 − 2 1 3 G 1 Y1,1 √ G 1 Y1,0 0 G 1 Y0,0 2 2 3 2 2 − 2 2 q (35) l l Here F 1 (r), G 1 (r) are the well-known real radial wave functions [16], 2 2 2 2
l F 1 3/2 1/2 2 2 (2λ) (m ε)Γ(2γ + nr + 1) γ 1 λr l = ± ± (2λr) − e− G 1 Zαm Zαm 2 Γ(2γ + 1) 4m κ n ! 2 h λ λ − r i Zαm κ F ( n , 2γ + 1, 2λr) F (1 n , 2γ + 1, 2λr) , (36) λ − − r ∓ − r l where F is the degenerate hypergeometric function, the upper signs refer to F 1 , the lower 2 2 l signs - to G 1 , the number nr = 1, the value κ = 1 for l = 0 and κ = 1 for l = 1. The other 2 2 − parameters in (36): Z = 1, γ = √1 α2, ε/m = (1+ γ)/ (1 + γ)2 + α2 and λ = √m2 ε2. − − Denoting the state with l = j 1 = 0 by the s index, and l = j + 1 = 1 by the p index, − 2 p 2 from (32) and (33) we obtain the off-diagonal elements of the perturbation (31):
mm mm 1/2 m vz W = W = ( 1) − I , (37) sp ps − c 1
1 1 1 1 2 , 2 2 , 2 vx ivy Wsp − = Wps − = − I , (38) c 1 1 1 1 1 2 , 2 2 , 2 vx + ivy Wsp− = Wps− = I . (39) c 1 Here α ∞ 0 1 1 0 1 I1 = rdr F2 1 G2 1 + G2 1 F2 1 . (40) 2 0 2 2 3 2 2 Z It remains for us to calculate the matrix elements of the perturbation (34). It is convenient to use the spherical coordinates in which
π 2π i ∞ α 2 ∂ vθ ∂ vφ ∂ + Q = + 2 r dr sin θdθ dφ vr + + ψ ψ (41) −2mc 0 r E 0 0 ∂r r ∂θ r sin θ ∂φ Z Z Z h i and the velocity components are given by:
vr = vxsinθcosφ + vysinθsinφ + vzcosθ
v = v cosθcosφ + v cosθsinφ v sinθ θ x y − z
9 v = v sinφ + v cosφ. φ − x y mm1 mm1 For the superposition (35) all the diagonal matrix elements Qss and Qpp vanish as a result of the integration over the solid angle in (41). Calculating the off-diagonal matrix elements between the s and p states, one should pay attention to the following. In (35) the 0,1 0,1 functions G 1 are small in comparison with F 1 since they have the additional multiplier α 2 2 2 2 arising from the factor (m ε)1/2 (see (36)). Hence, taking into account the product (v ψ+)ψ − ∇ in the operator (34), the contributions to the off-diagonal matrix elements from the functions 0,1 2 0,1 G 1 will be in α times smaller than that from the functions F 1 . Formally, instead of the 2 2 2 2 0,1 functions G 1 , we can simply use zeros in the superposition (35). As a result, we find: 2 2
1 1 1 1 2 2 2 2 vz Q−sp − = Qsp = I , (42) − c 2
1 1 1 1 2 2 2 2 ∗ vx ivy Qsp− = Q−sp = − I , (43) c 2 1 1 1 1 2 2 2 2 vz Qps = Q−ps − = I (44) − c 3 and 1 1 1 1 2 2 2 2 ∗ vx ivy Qps− = Q−ps = − I , (45) − c 3 where 1 ∞ 2 α 1 d 0 I2 = r dr + 2 F2 1 F2 1 (46) 6m 0 r E 2 dr 2 Z and 1 ∞ 2 α 0 d 1 2 0 1 I3 = r dr + 2 F2 1 F2 1 + F2 1 F2 1 ] (47) 6m 0 r E 2 dr 2 r 2 2 Z h Calculating the integrals (40), (46) and (47) we take γ = 1 instead of γ = √1 α2, and 2 − ε/m = 1 α m that means neglecting the terms αn with n 4. Then from (36) the radial − 8 ∝ ≥ wave functions are given by:
3/2 0 αm αmr F 1 = e− 2 (2 αmr), (48) 2 2 2 − 3/2 0 αm αmr αmr G 1 = α e− 2 (1 ), (49) 2 2 − 2 − 4 1 1 αm 3/2 αmr F 1 = e− 2 αmr, (50) 2 2 −√3 2 and 1 √3α αm 3/2 αmr αmr G 1 = e− 2 (1 ). (51) 2 2 − 2 2 − 6 Using (48)-(51), we obtain:
√3 √3 √3 I = α3m,I =+ α3m,I = α3m. (52) 1 −144 2 144 3 −144 Note that the use of the exact values of γ and ε leads to corrections of the values (52) that are proportional to α5. Taking into account of (25), (52) and v = v , we sum the off-diagonal matrix elements z √3 (37)-(39) and (42)-(45):
1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ sp = Wsp + Qsp = (I I )= α m E c 1 − 2 −72 c
10 1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ sp− − = Wsp− − + Q−sp − = (I I )=+ α m E − c 1 − 2 72 c 1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ ps = Wps + Qps = (I + I )= α m E c 1 3 −72 c 1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ ps− − = Wps− − + Q−ps − = (I + I )=+ α m E − c 1 3 72 c All the other matrix elements of the operator Wˆ + Qˆ are zeros. Thus, we can carry out the following consideration. There are the two different perturba- tions for the initially degenerate states 2s1/2 and 2p1/2. Interaction between vacuum energy fluctuations and the hydrogen electron leads to the diagonal corrections for each energy levels. These corrections named the Lamb shifts, result in the energy difference of the states 2s1/2 and 2p1/2. The second perturbation which is caused by the interaction between the hydrogen atom momentum and the hydrogen electron, leads to the off-diagonal matrix element between these states. Then, using (25), the secular equation for the energies of these states has the form:
1 1 2 2 g δε2s 0 δ sp 0 E − E 1 1 2 2 0 g δε2s 0 δ sp− − det 1 1 E − E = 0. (53) 2 2 δ ps 0 g δε2p 0 E 1 1 E − 2 2 0 δ ps− − 0 δε E Eg − 2p Solving (53), we find:
2 2 δε2s + δε2p δε2s δε2p 1 6 v 2 g = − + α m . E ± 2 ± s 2 722 c2 Accordingly, the photon energy corresponded to the the transition between the 2s1/2 and 2p1/2 states is given by: 2 2 1 6 v 2 hν = (hνL) + α m . (54) s 64 c2
Here hνL is the photon energy equal to the splitting the 2s1/2 and 2p1/2 states due to the radiative shifts. Lamb and Rutherford found that hνL = 1060 MHz [17]. The second term under the root in (54) presents the splitting effect due to the interaction between the hydrogen atom momentum and the hydrogen electron. The main result is that, according to (54), this photon energy of the transition 2s 2p 1/2 → 1/2 is predicted to depend on the hydrogen atom velocity. The necessary discussions and evaluations will be postponed to Section 6.
5.2 The splitting of hydrogen 3s1/2 and 3p1/2 levels The Lamb shift for these states is ν = 314.894 0.009MHz [23]. L ± For the considered states the superposition (29) with account for (30) takes the form:
0 1 1 1 1 2 1 F Y0,0 0 F Y1,0 1 3 2 √3 3 2 3 F Y1, 1 0 − 3 2 − F 1 Y − 0 3 0,0 2 1 q1 1 2 3 F 1 Y1,1 √ F 1 Y1,0 ψ = a 1 1 0 +a 1 +b 1 3 2 +b 1 3 3 2 G 1 Y 2 0 2 √ 1,0 2 G 1 Y 2 1 2 3 3 2 − 3 1, 1 q G 1 Y − 0 − − 3 2 − 3 0,0 2 0 q1 0 − 2 1 3 G 1 Y1,1 √ G 1 Y1,0 0 G 1 Y0,0 3 2 3 3 2 − 3 2 q (55) Here, using (36) the radial wave functions are given by:
3/2 αmr 0 αm 2 2 2 2 2 F 1 = 2 e− 3 (1 αmr + α m r ), (56) 3 2 3 − 3 27 11 0 α αm 3/2 αmr 10 2 2 2 2 G 1 = e− 3 (3 αmr + α m r ), (57) 3 2 − 3 3 − 9 27 3/2 1 4√2 αm αmr 1 F 1 = e− 3 αmr(1 αmr), (58) 3 2 − 9 3 − 6 and 3/2 1 2√2α αm αmr 1 1 2 2 2 G 1 = √3 e− 3 (1 αmr + α m r ). (59) 3 2 − 3 3 − 3 54 Making the obvious replacement of the wave functions in (40), (46), and (47) by the wave functions (56) - (59) and calculating the corresponding integrals, we obtain:
√2 √2 √2 I = α3m,I =+ α3m,I = α3m. (60) 1 −182 2 182 3 −182 Then the secular equation for the energies of these states takes the form (53) with the replace- ments δε δε and δε δε . Respectively, considering (37)-(39) and (42)-(45) but now 2s → 3s 2p → 3p with (60), in (53) the non-zero matrix elements of the interaction (25) should be replaced by:
1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ sp = Wsp + Qsp = (I1 I2)= α m E c − −21/239/2 c
1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ sp− − = Wsp− − + Q−sp − = (I I )=+ α m E − c 1 − 2 21/239/2 c 1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ ps = Wps + Qps = (I + I )= α m E c 1 3 −21/239/2 c 1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ ps− − = Wps− − + Qps− − = (I + I )=+ α m. E − c 1 3 21/239/2 c Such modified secular equation yields the result for the photon energy of he transition between the 3s1/2 and 3p1/2 states:
2 2 2 6 v 2 hν = (hνL) + α m , (61) s 39 c2 where the Lamb shift for these states is ν = 314.894 0.009MHz. L ± Discussion of this result is deferred to Section 6.
5.3 The energy separation in He+, n=2 As was found in [24], the energy separation between 2s and 2p is equal to ν = 14040.2 1/2 1/2 L ± 1, 8MHz in He+. The superposition of these states has the same form (35). Considering Z = 2, the radial wave functions (36) are rewritten as:
3/2 0 (2αm) αmr F 1 = e− (1 αmr), (62) 2 2 √2 −
3/2 0 (2αm) αmr αmr G 1 = α e− (1 ), (63) 2 2 − √2 − 2 3/2 1 (2αm) αmr F 1 = e− αmr (64) 2 2 − √6 and 3/2 1 (2αm) αmr αmr G 1 = α√6e− (1 ) (65) 2 2 − 23/2 − 3
12 Substituting (62)-(65) into (40), (46) and (47), we obtain:
√3 √3 √3 I = α3m,I =+ α3m,I = α3m. (66) 1 − 36 2 36 3 − 36 Then the secular equation for these states in He+ has the same form as (35) but now, using (37)-(39), (42)-(45) and (66), the nonzero elements of the matrix are given by:
1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ sp = Wsp + Qsp = (I I )= α m E c 1 − 2 −18 c
1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ sp− − = Wsp− − + Q−sp − = (I I )=+ α m E − c 1 − 2 18 c 1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ ps = Wps + Qps = (I + I )= α m E c 1 3 −18 c 1 1 1 1 1 1 2 2 2 2 2 2 vz 1 v 3 δ ps− − = Wps− − + Qps− − = (I + I )=+ α m. E − c 1 3 18 c Solving (53) with the matrix elements presented above, we find the photon energy corre- + sponded to the transition between the 2s1/2 and 2p1/2 states for the He ion:
2 2 1 6 v 2 hν = (hνL) + α m . (67) s 34 c2
Here νL = 14040MHz is the separation of the energy levels 2s1/2 and 2p1/2 due to the radiative shifts. The second term under the root presents the splitting effect due to the interaction between the helium ion momentum and the electron of He+.
6 Discussion and Conclusion
We have established above that the remaining degeneracy of the energy levels of hydrogen-like atoms is removed not only by the well-known radiative relativistic corrections, but also by the interaction of the atomic electron with the atom momentum. The joint effect of these two different types of the interactions is investigated. The total energy separation of the initially degenerate states for the hydrogen atom, (54) and (61), and for the He+ ion, (67), was obtained. While the radiative corrections are proportional to α5m, the shifts caused by the atom motion, 3 v are proportional to α c m. Therefore, the crossover from the Lamb shifts to the motion-induced shifts is possible. Lamb and Rutherford [17] found experimentally that the energy separation of the hydrogen states 2s1/2 and 2p1/2 is equal to 1060 MHz. In these experiments the velocities of hydrogen atoms were about 8 km/s relative to the observer on the Earth. The current theoretical value of the Lamb shift is ν = 1058.911 0.012 MHz. L ± For the states considered, the expression (54) can be rewritten as:
v2 ν = ν2 + ν2 , (68) s L H2 c2 where α3 mc2 ν = = 1.326THz. (69) H2 36 h Both these terms under the root in (68) are equal to each other at v∗ = 0.799 10 3 that c × − means the hydrogen velocity v = 240 km per second. Usually, stars in the Universe consist mostly of hydrogen and helium.∗ In addition, stars may have significantly higher velocities than
13 v = 240km/s. Therefore, for such fast stars the photon energy corresponding to the radiative ∗ transition between hydrogen states 2s1/2 and 2p1/2 could depend on the star velocity. For the hydrogen states 3s and 3p the Lamb shift is equal to ν = 314.894 0.009MHz. 1/2 1/2 L ± The corresponding expression (61) gives us:
v2 ν = ν2 + ν2 , (70) s L H3 c2 where √2α3 mc2 νH3 = = 0.481THz. (71) 81√3 h Both these terms under the root in (70) are equal to each other at v∗ = 0.655 10 3 that is at c × − the hydrogen velocity v = 197 km per second. ∗ + Finally, we made the same estimation for the He ion with νL = 14040MHz for the separa- tion of the 2s1/2 and 2p1/2 states. The equation (67) takes the form:
v2 ν = ν2 + ν2 , (72) s L He2 c2 where α3 mc2 ν = = 5.304THz. (73) He2 9 h Then, from (72) we obtain the hydrogen atom velocity at which the radiative shift is equal to the energy-level shift induced by the He+ motion: v = 794 km per second. ∗ Note that in (68), (70) and (72) the velocity v is the absolute velocity of the hydrogen and hydrogen-like atoms that should be considered as a part of objects moving in the Universe. Let hydrogen atom be at rest relative to the observer on the Earth. The Earth has the rotation velocity 0.5km/s, the orbital velocity V 30km/s of motion around the Sun. The Sun ≃ orb ≃ itself is moving with respect of the average of the stars in its vicinity. The Sun velocity with respect to the observer’s local standard of rest is about VLSR = 16.5 km/s [25]. We carry out estimations for the most studied transition 2s 2p of the hydrogen 1/2 → 1/2 atoms. Assume that the orbital velocity Vorb is perpendicular to the Sun velocity VLSR. We use the experimental result obtained by Lamb and Rutherford [17] that the energy separation between the 2s1/2 and 2p1/2 hydrogen states is equal to νexp = 1060 MHz. In their experimental setup the velocities of hydrogen atoms were about 8 km/s relative to the observer on the Earth. But unfortunately we can not take it into account, since the direction of this velocity is not known. Then, using Eq. (68), the theoretical value of the energy separation of the 2s1/2 and 2p1/2 in the resting hydrogen atom could be:
ν = ν2 δν2, (74) L exp − q where δν is the atom-motion-induced correction,:
2 2 Vorb + VLSR δν = ν . (75) H2 q c Using (69), we obtain ν = ν (1 0.01019) = 1049.14MHz. The latter is very close to the L exp − value [16] E E = 0.41α5mc2 = 1050MHz (76) 201/2 − 211/2 that was obtained with account for the two main radiative effects that are of order of α5m: 1) the emission and absorption by the bound electron of virtual photons, which leads to the change in the effective mass and the appearance of the anomalous magnetic moment of the electron; 2)
14 the possibility of virtual creation and annihilation of electron-positron pairs in vacuum or, in other words, the vacuum polarization, which distorts the Coulomb potential of the nucleus at distances of the order of the Compton electron wavelength. Both these contributions result in 1050,556 MHz according to data for 1996. However, Eq. (76) do not include other more subtle corrections of order of α6m [23, 26] which are, in total, 7.355MHz. ≃ In order to test the theory developed, the following experiment can be proposed. Let there be an atomic hydrogen beam with the atom velocity equal to the orbital velocity of the Earth, vb = Vorb = 30km/s. This velocity is 3.75 times more than that in the experiments [17]. In the first experimental geometry the beam velocity and the orbital velocity of the Earth have the same direction. In this case the frequency which corresponds to the energy separation of the 2s1/2 and 2p1/2, is denoted by ν+. In the second one, when the beam velocity and the orbital velocity of the Earth are opposite directed, the frequency is ν . Then from (68) we find, with − good accuracy, the frequency difference
2 2 νH2 4Vorb δν = ν+ ν = 2 (77) − − 2νL c Using (69), Eq. (77) gives us δν = 33.18MHz which can be measured experimentally.
Appendices
Appendix A Test of the equations obtained
The integral equation (20) and the differential equation (21) can be reduced to the Dirac equa- tion for the hydrogen atom. To do this, we should put g=0, q = 0, ω = M , and → ∞ 1 0. The notation (23) is replaced as E = M + m + , and the equation (19) is rewritten 2E → E as ∆ = 4M 2((m + )2 ε2). Accordingly, E − E2 ε2 + ω2 1 0 Mβ + α qˆ + − 2M p p 2E → 0 0 !p and E2 + ε2 ω2 mβ + α pˆ + − mβ + α pˆ + (m + ). e e 2E → e e E As a result, Eq. (20) is reduced to the form:
′ α if(r r ) (m + )+ mβe + epˆ ψ(r)= dr′ e − E V (r′)ψ(r′), (m + )2 ε2 Z Xf E − and the corresponding differential equation is
(m + )2 ε2 ψ(r)= (m + )+ mβ + α pˆ V (r)ψ(r), (A1) E − E e e e2 where is the energy of the discrete levels, V (r) = − . In the same limits Eq. (A1) can be E r deduced from the differential equation (21). Eq. (A1) is the Dirac equation, (α pˆ + mβ + V (r))ψ = (m + )ψ, which describes the e e E motion of the electron in the external Coulomb field. Now consider the non-relativistic limit of (21). We assume that the ratio of atom velocity to the speed of light v g < α, and take into account that α2m,
15 2 2 2 2 From Eq. (23) we have E = Eg + 2(M + m) , where Eg = (M + m) + g . Substituting 4 E the latter in Eq. (19), the terms with Eg disappear, and in the firstp order in the binding energy we get: ∆ = 8mM(M + m) 4(M + m)2ˆf 2. E− In the first order over α2 the left-hand side of Eq. (21) is reduced to:
1 0 1 0 e2 8(M + m)mM − ψ(r). 0 0 ! 0 0 ! r p e Thus in the considered case Eq. (21) takes the form of the Schro¨dinger equation:
ˆf 2 e2 φ = φ, 2µ − r E where µ = mM/(M + m) is the reduced mass, and φ is the upper spinor of the bispinor ψ(r). In the same limits the Schr¨odinger equation can be deduced from the integral equation (20) as well.
Appendix B Derivation of the equations (25), (26) and (27)
Analyzing the equation (21), we must take into account the following. The atom momentum g (M + m)v with, according to (24), v/c << α. Considering α2m, δ α3v/cm and ≃ E ∝ Eg ∝ f αm, all terms that do not exceed α6 and α5v/c, can be neglected. So, the terms δ 2, δ , ∝ Eg E Eg 2f 2, 2g2, 2fg, and 3 are omitted. E E E E Eq. (21) is presented in the form:
1 ∆ˆ α 1 α 1+ ψ = R R ψ. (B1) 4mM 2E 2rE 4mM e p − r Here ∆ˆ is given by Eq. (19). Considering (22), (23) and (24), the operator (19) is rewritten as:
mMg2 ∆ˆ = 4 2 (M + m)2 + mM + 8 mM(M + m)+ (M + m)ˆf 2 + (M m)gˆf E E M + m − − h i h i 2 +8δ mM(M + m) 4(M + m)2ˆf 2 4g2ˆf 2 + 4 gˆf . (B2) Eg − − 1 According to (B2) and (23), the expansion of (2E)− in the left-hand side of (B1) should be restricted by 1 1 + δ g2 = 1 E Eg (B3) 2E 2(M + m) − M + m − 2(M + m)2 Combining Eqs. (B2) and (B3), the left-hand side of Eq. (B1) is reduced to:
1 ∆ˆ α (M + m)2 mM g2 ˆf 2 (M m)gˆf 1+ ψ = 2 − + 1+ + − 4mM 2E 2rE E 2mM(M + m) E 2(M + m)2 − 2mM mM(M + m) h M + m 2(gˆf )2 g2ˆf 2 α +δ ˆf 2 + − 1+ ψ. (B4) Eg − 2mM 4mM(M + m) 2r(M + m) i Now we turn to the right side of Eq. (21). In the representation of this equation in the form (B1) the factor Re is: E2 + ε2 ω2 R = mβ + α p + − . e e e 2E
16 Using (22), (23) and (24), Re is rewritten as: m 1 2m R = mβ + α f + g + 2m(M + m)+ g2 + 2(M + m)( + δ )+ 2 + 2gˆf , e e e M + m 2E M + m E Eg E h (B5)i where the factor (2E) 1 is given by (B3). The potential energy that is α2, is present on the − ∝ right side of the equation (B1). Consequently in the approximation presented above, from (B5) we have: m M gˆf R = m(β + I)+ α f + g + + . (B6) e e e M + m M + mE M + m Here I is the 4 4 unit matrix. × The another factor in the right-hand side of (B1) is given by:
E2 ε2 + ω2 R = Mβ + α q + − p p p 2E The similar approach gives us:
M m gˆf R = M(β + I)+ α f + g + . (B7) p p p − M + m M + mE − M + m In Eqs. (B6) and (B7) there are two α matrices in the standard representation: αe acts on the electron bispinor and αp acts on the proton bispinor. The operator of the particle’s velocity can be found as [H,ˆ r]. Using the Dirac Hamiltonian for the free particle as H, we find [H,ˆ r] = cα. In the bound states the momenta of the electron and proton are proportional to f αmc, where α is the fine structure constant. Hence the velocities of the electron and proton ∝ are proportional to < α > α and < α > m α, respectively. | e |∝ | p |∝ M Combining Eqs. (B6) and (B7), the right-hand side of Eq. (B1) is reduced to:
1 1 M m R R V ψ = m(β + I)+ α f + M(β + I) α f + V ψ 4mM e p 4mM e e M + mE p − p M + mE h ih i 1 1 0 α g gˆf 1 1 0 α g gˆf + e + V ψ + p V ψ. (B8) 2 0 0 ! M + m m(M + m) 2 0 0 ! M + m − M(M + m) p h i e h i Here V = α . − r According to Eq. (23), the mass of the hydrogen atom is M + m + + δ , where + δ E Eg E Eg is the binding energy of the atom, and >> δ . At rest (g = 0), the atomic energy levels are E Eg determined by . The energy levels shifts caused by the g-dependent perturbations, are given E δ . Considering δ as a small correction caused by the motion of the atom, Eq. (B1) with Eg Eg acccount for (B4) and (B8) is divided into two equations: the first equation defines and the E second one - δ . As a result, the equation for takes the form: Eg E (M + m)2 mM ˆf 2 M + m α 2 − + 1 ˆf 2 1+ ψ = E 2mM(M + m) E − 2mM − 2mM 2r(M + m) h i 1 M m m(β + I)+ α f + M(β + I) α f + V ψ (B9) 4mM e e M + mE p − p M + mE h ih i One can see from Eq. (B9) that the binding energy does not depend on the hydrogen E momentum. The second equation for δ is given by: Eg g2 (M m)gˆf 2(gˆf)2 g2ˆf 2 δ + + − + − ψ = Eg E 2(M + m)2 mM(M + m) 4mM(M + m) h i
17 1 1 0 α g gˆf 1 1 0 α g gˆf e + V ψ + p V ψ. (B10) 2 0 0 ! M + m m(M + m) 2 0 0 ! M + m − M(M + m) p h i e h i Now we analyze Eq. (B9). It is reduced to the form:
m µ ˆf 2 α 2 1 + 2µ 1 ˆf 2 1+ ψ = µ E − M + m E − 2mM − 2r(M + m) h i
µ 1 0 αpf m m + mβe + αef + + E V ψ (B11) mE 0 0 ! − 2M 2M(M + m) h ih p i 2 µ 2 µ 4 m In the left-hand side of (B11) the terms M+m and f 2mM are proportional to α M . In the E E 2 αpf m 2 m left-hand side of the equation the terms and E α 2 . Omitting these four terms, 2M 2M(M+m) ∝ M (B11) is rewritten as:
m V µ (µ + )2 2 1+ ψ = m + mβ + α f + V ψ(r; g = 0), (B12) µ E −Eµ 2(M + m) e e mE h i h i
2 2 where µ = Mm/(M + m) is the reduced mass, and εµ = µ + ˆf . It is interesting to compare Eq. (B12), in which the finiteq mass of the proton is taken into account, with Eq. (A1), which is the Dirac equation for the electron in the external Coulomb field. It is seen that in the limit M the equation (B12) goes to Eq. (A1). →∞ Finally we consider Eq. (B10). In the Introduction we showed that δ g must be of order of 3 v E α c . In the left-hand of the equation the first term in parentheses and the last term in square 2 v2 brackets are proportional to α c2 . Taking into account (24), both these terms can be neglected. The remaining term in equation,
(M m)gˆf v − α3 m. E mM(M + m) ∝ c
Analyzing the right-hand side of (B10), it should be considered that in the bound states the velocities of the electron and proton are proportional to < α > α and < α > m α, | e |∝ | p |∝ M respectively. Therefore, the second term is small in comparison with the first one. Taking into account that
δ =< ψ Wˆ + Qˆ ψ > (B13) Eg | | where the perturbation operators are
α σ ˆ ev α 0 v e W = V = σ (B14) 2c −2cr v e 0 ! and vˆf v α Qˆ = V 2 = i ∇ + 2 . (B15) 2mc − E 2mc r E In Eqs. (B13)- (B15) the wave functions ψ are given. They are determined by (B12) or (A1), since M = 1836m and the proton mass corrections to the wave functions and the eigenvalues are small. Therefore, Eq. (B13) should be regarded as the equation for the atom-motion-induced corrections δ to the hydrogen energy levels for the known wave functions. Eg
18 References
[1] L.D. Landau and E.M. Lifshitz. Quantum mechanics (Non-relativistic theory), Bristol, Pergamon Press, 1965.
[2] E.E. Salpeter, H.A. Bethe. A Relativistic Equation for Bound-State Problems. Phys. Rev. 84, 1232 (1951).
[3] E. E. Salpeter. Mass corrections to the fine structure of hydrogen-like atoms. Phys. Rev. 87, 328 (1952).
[4] R. Karplus and A. Klein. Electrodynamic Displacement of Atomic Energy Levels. III. The Hyperfine Structure of Positronium. Phys. Rev. 87, 848 (1952)
[5] Th. Fulton and Paul C. Martin. Two-Body System in Quantum Electrodynamics. Energy Levels of Positronium Phys. Rev. 95, 811 (1954).
[6] P. K. Kabir and E. E. Salpeter. Radiative Corrections to the Ground-State Energy of the Helium Atom Phys. Rev. 108, 1256 (1957).
[7] H. Araki. Quantum-Electrodynamical Corrections to Energy-Levels of Helium. Progress of Theoretical Physics, 17, 619642 (1957).
[8] N. Nakanishi. A General Survey of the Theory of the Bethe-Salpeter Equation. Progress of Theoretical Physics Supplement, 43, 181 (1969).
[9] M. Douglas, N.M. Kroll. Quantum electrodynamical corrections to the fine structure of helium. Annals of Physics 82, 89-155 (1974).
[10] F. Gross. Relativistic few-body problem. I. Two-body equations. Phys. Rev. C 26, 2203 (1982).
[11] R. Machleidt, K. Holinde, Ch. Elster. The bonn meson-exchange model for the nucleonnu- cleon interaction. Physics Reports 149, 1-89 (1987).
[12] W. Lucha, F.F. Schberl, Dieter Gromes. Bound states of quarks. Physics Reports 200, 127-240 (1991).
[13] P.J. Mohr, G.Plunien, G. Soff. QED corrections in heavy atoms. Physics Reports 293, 227-369 (1998).
[14] M.I. Eides, H. Grotcha, V.A. Shelyuto. Theory of light hydrogen-like atoms. Physics Re- ports 342, 63-261 (2001).
[15] C. Gutierrez, V. Gigante, T. Frederico, G. Salm, M. Viviani, Lauro Tomio. Bethe-Salpeter bound-state structure in Minkowski space. Physics Letters B 759, 131-137 (2016).
[16] V B Berestetskii L. P. Pitaevskii E.M. Lifshitz. Quantum Electrodynamics. 2nd Edition. Volume 4. Butterworth- Heinemann, 2012.
[17] W.E. Lamb Jr, R.C. Retherford. Fine Structure of the Hydrogen Atom by a Microwave Method. Phys. Rev. 72, 241 (1947).
[18] H.A. Bethe. The Electromagnetic Shift of Energy Levels. Phys. Rev. 72, 339341 (1947).
[19] M.I. Eides, Howard Grotch, V.A. Shelyuto. Theory of light hydrogenlike atoms. Physics Reports 342, 63-261 (2001).
[20] R. Pohl, A. Antognini, F. Kottmann. The size of the proton. Nature 466, 213216 (2010).
19 [21] S.G. Karshenboim. Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants Physics Reports 422, 1-63 (2005).
[22] R. P. Feynman. Space-Time Approach to Quantum Electrodynamics. Phys. Rev. 76, 769 (1949).
[23] S.J. dsk, S.D. Dll. The Present Status of Quantum Electrodynamics. Annual Beview of Nuclear Science, ed. by E. Segre, v. 20, Palo Alto, California, 1970, p. 147.
[24] E. Lipworth and R. Novick. Fine Structure of Singly Ionized Helium. Phys. Rev. 108, 1434 (1957).
[25] J. Binney and S. Tremaine. Galactic Dynamics, Princeton, NJ, Princeton University Press, 1987, 747 p.
[26] E.E. Salpeter. The Lamb Shift for Hydrogen and Deuterium. Phys. Rev. 89, 92 (1953).
20