Revisiting the Lamb Shift

Transcription

1 Revisiting the Lamb Shift axiv: v1 [physics.gen-ph] 3 May 017 Abhishek Das, B.M. Bila Science Cente, Adash Naga, Hydeabad , India B.G. Sidhath, B.M. Bila Science Cente, Adash Naga, Hydeabad , India Abstact In this pape we endeavou to detemine the enegy levels of an atom by vitue of the modified Diac equation. It has been found that the enegy levels contain an exta tem in the expession which accounts fo the zittebewegung effects in the Compton scale. Applying ou pespective to the hydogen atom we have been able to find the Lamb shift fo the S1 P1 states. This esult substantiates that a slight modification of the Diac equation suffices to explain the phenomenon, whee the modification of the Diac equation aises due to the non-commutative natue of space-time. Besides, seveal othe unexplained phenomena can emege as a natual consequence of this modification. 1 Intoduction The spin- 1 natue of the electon can be natually accommodated by the Diac equation ([1]). This is fo the point electon a diffeentiable spacetime. Ove the past fifteen yeas, Sidhath had investigated this scenaio fom the point of view of fuzzy spacetime as in Quantum Gavity appoaches. In contadistinction to othe authos, Sidhath had deduced fundamentally athe than phenomenologically a modified enegy-momentum elation as E = p +m λ l p 4 The last tem on the ight h side aises owing to the non-commutative natue of spacetime. As discussed elsewhee ([], [3]), this leads to the following modification in the Diac equation fo the electon: (γ µ µ +m λlp )ψ = 0 (1) pabihtih3@gmail.com bilasc@gmail.com 1

2 whee λ is a small constant ([3]-[6]) aising due to the effects of non-commutative spacetime l (= h ) is the educed Compton wavelength. It may be mentioned that as shown mc peviously ([7]), λ α π 10 3, whee α is the fine stuctue constant. We now use this modification to obtain the Lamb Shift. As is well known, the Lamb shift was obseved by Lamb Rethefod ([8]) while caying out an expeiment using micowave techniques to stimulate adio-fequency tansitions between S 1/ P 1/ levels of the hydogen atom. Hans Bethe ([9]) was the fist peson to give a pecise explanation of this phenomenon elying on Diac s theoy adiative coections, thus laying the foundations of quantum electodynamics. The contibution of Bethe, Koll & Lamb Fench & Weisskopf ([10]) yielded the value of S 1/ - P 1/ splitting as E(S 1/ ) E(P 1/ ) 105.1MHz Moe pecise theoetical values of the Lamb shift wee given by Eickson ([11]) as ± MHz by Moh ([1]) as ± MHz Again, T.A. Welton ([13]) had given a somewhat qualitative desciption of the Lamb shift giving the fomula fo enegy diffeence ([14]) as E n = 8 Z 4 α 5 (ln 1 3π n 3 Zα )m δ l,0 which in case of hydogen atom (Z = 1) fo n = l = 0 gives E n 1000MHz Besides, fom diffeent pespectives, Peteman ([15]) Kashenboim ([16]) obtained diffeent values of the Lamb shift as ± 0.011MHz (1)MHz espectively. Now, it is well known that the phenomenon of Lamb shift is caused by fluctuations of the Zeo Point Field, as indeed is (1). Let us see if we can deduce it pecisely using (1). Ou appoach encompasses this modified Diac equation (1) the Hamiltonian concened with it. At the same time the pesent appoach is moe geneal as it deduces not only the Lamb shift, but othe effects also, as fo example the neutino, anti-neutino obseved symmety ([17]). Now, we follow Whitehead ([18]), Diac ([19]) othe authos ([0]) to fist deive a set of tansfomations which tun the Hamiltonian fo the system into a fom that depends only on the adial vaiables p. Then we solve the adial equations by conventional methods obtain the enegy levels coesponding to an atom. In the pocess

3 it has been agued that the Lamb shift is connected explicitly with the modification tem of the Snyde-Sidhath Hamiltonian ([4],[5],[6]) fom ou appoach the value of Lamb shift has also been obtained without elying on the featues of quantum electodynamics. The Modified Diac Equation The Hamiltonian of the modified Diac equation ([]) fo electomagnetic coupling can be witten as whee H = eφ cα 1 σ.( p e A c )+α 3mc α 3 λlc h ( σ. p) () ( ) 0 I α 1 = = I ( ) I 0 α 3 = = 0 I α 3 λlc h ( σ. p) is a modification tem due to the Snyde-Sidhath Hamiltonian. Also, the σ s ae the extended Pauli matices. Consideing cgs units we can wite fo the Coulomb potential also eφ = ze A = 0 Theefoe () can be witten as H = ze cα 1 σ. p+α 3 mc λlc α 3 h ( σ. p) (3) It is ou objective to expess (3) only in tems of the adial vaiables p. We do this by looking fo quantities that commute with the tems of the Hamiltonian, as it has been conventionally done ([18]-[0]). Now, using the following identities 3

4 ( σ. L)( σ. p) = ( L. p)+i σ( L p) = i σ.( L p) ( σ. p)( σ. L) = ( p. L)+i σ( p L) = i σ.( p L) we would obtain ( σ. L+ h)( σ. p)+( σ. p)( σ. L+ h) = 0 (4) which is an anti-commutation elation. Nonetheless, it is easy to show by vitue of equation (8) that α 3 α 1 ( σ. L+ h) will commute with cα 1 ( σ. p). Again, let us investigate the following identities ( σ. L)( σ. p) = [( L. p)+i σ( L p)]( σ. p) ( σ. p) ( σ. L) = ( σ. p)[( p. L)+i σ( p L)] Fom these two equation we would obtain ( σ. L)( σ. p) +( σ. p) ( σ. L) = 0 (5) which is also an anti-commutation elation. Futhe investigation shows that in this case α 3 α 1 ( σ. L) commutes with α 3 ( σ. p). Also, α 3 α 1 ( σ. L + h) commutes with α 3 ( σ. p) α 3 α 1 ( σ. L) commutes with cα 1 ( σ. p). Again, using the following linea opeato as done by vaious othe authos ([18], [19]) the elation ǫ 1 = α 1 ( σ. x) (6) ( σ. x)( σ. p) = p +i h(α 3 j 1) we have α 1 ( σ. p) = ǫ 1 p ǫ 1i h + ǫ 1i hα 3 j Hee, it is known that ǫ 1 has the popety ǫ 1 = 1, 4

5 J = L+ 1 hσ (j h) = J h Similaly, we can define anothe linea opeato ǫ = α 3 σ( σ. x) = σ( σ. x) (7) fom whence, it can be shown that ǫ has the popety ǫ = 1 Now, consideing the modification tem in the Hamiltonian (3) we will obtain the elation ( σ. x)( σ. p) = σp hσp fom whence, we can deduce σ( σ. x)( σ. p) = p hp With the use of (7) we get α 3 ( σ. p) = ǫ p ǫ hp Theefoe, the Hamiltonian of the modified Diac equation finally can be witten as H = ze cǫ 1(p i h )+ ciǫ 1 hα 3 j +α 3 mc λlc h [ǫ p ǫ hp ] (8) Now, in ([18]) it has been consideed that α 3 = ( ) ǫ 1 = ( ) 0 i i 0 5

6 In ou case, we conside the matix ǫ = ( ) 0 i i 0 confoming with the popety ǫ = 1 whee the matix ǫ has been put by h in contadistinction to ([18]). Theefoe, the modified Diac equation fo stationay states would be Hψ = Eψ i.e. ( ze +mc icp c h cj h + iλclp h iλcl p icp + c h cj h iλclp +iλcl p h ze mc ) (ψ1 ) ψ = Λ ( ) ψ1 () ψ () 3 Enegy levels fom the Modified Diac Equation Now, this epesentation is diffeent fom that of Whitehead ([18]) othes ([19], [0]) since thee ae two exta tems which oiginate fom the modification tem in Diac equation (). Now, educing the system to coupled diffeential equations, we would solve them by substituting an unknown function in the fom of a infinite seies, i.e. by the method of powe seies. Rewiting (H ΛI) = 0 as a system of coupled equations we obtain ( Λ ze +mc )ψ 1 c h( d d 1 + j d iλl d λl1 d d )ψ = 0 (9) ( Λ ze mc )ψ +c h( d d 1 + j d iλl d λl1 d d )ψ 1 = 0 (10) Now, fo the sake of simplicity we neglect the tems involving second ode deivative obtain ( Λ ze +mc )ψ 1 c h( d d 1 + j λl1 d d )ψ = 0 (11) ( Λ ze mc )ψ +c h( d d 1 + j λl1 d d )ψ 1 = 0 (1) Substituting α = e hc (fine-stuctue constant), a 1 = h mc Λ c ( 1 a 1 zα )ψ 1 +( d d j 1 6 +λl 1 a = h mc+ Λ c we get d d )ψ = 0 (13)

7 ( 1 a + zα )ψ +( d d + j +1 +λl 1 Now, as it is conventional, we assume solutions of the type ψ 1 () = 1 e a x() d d )ψ 1 = 0 (14) ψ () = 1 e a y() whee, a = a 1 a = (14) h m c Λ c. Using the afoementioned solutions we obtain fom (13) ( 1 zα a 1 )x()+[ d d 1 a j + λl ( d d 1 a 1 )]y() (15) ( 1 + zα a )y()+[ d d 1 a + j + λl ( d d 1 a 1 )]x() (16) Now, we exp the unknown functions x() y() as seies which will then be substituted into the given system of equations. We have x() = t x t t y() = t y t t By vitue of the powe seies method we know that in ode fo the equation to be zeo as equied, each tem in the esulting seies must sepaately be zeo. Theefoe, afte aanging we have the coefficients of the s tems as x t a 1 y t a zαx t+1 +(t+1 j λl a )y t+1 λl(t 1)y t = 0 (17) y t a x t a +zαy t+1 +(t+1+j λl a )x t+1 λl(t 1)y t = 0 (18) Now, multiplying equation (17) by a equation (18) by a adding them we get x t ( a a 1 a a ) zαax t+1+zαa y t+1 +(t j λl a )ay t+(t+j λl a )x ta +λl(t 1)x t +λl(t 1)y t = 0 (19) 7

8 This can be witten as x t [ zαa+(t+j λl a )a +λl(t 1)]+y t [zαa +(t j λl )a+λl(t 1)] = 0 (0) a The functions x() y() must go to zeo at = 0, because the ψ() functions would othewise divege thee due to the 1 tem which entails that thee is some smallest t below which the seies does not continue. Let this be t s which will have the following popety accoding to ([18]): x ts 1 = y ts 1 = 0 Applying this to equations (17) (18) we have zαx ts (t s j λl a )y t s = 0 (1) zαy ts +(t s +j λl a )x t s = 0 () Fom these two equations we get the value of t s as t s = λl a + j z α (3) In equations (17), (18), (0) hencefoth we choose to neglect the tem λl(t 1), since the educed Compton length l (= h mc ) is extemely small λ Again, it can be shown that the seies must teminate if the enegy eigenvalue Λ is to be less than mc ([19]). This implies that if the seies teminates at t 1 such that x t1 +1 = y t1 +1 = 0 then using equations (17), (18) (0) we would obtain 1 a (t 1 λl a ) = 1 [ 1 a 1 1 a ]zα (4) Now, equations (3) (4) epesent the lowe uppe bounds of the seies espectively. We shall find late that λl ( a 10 5 ) is consideably small. But, we have not neglected it in equation (3) emembeing that it is the lowe bound. Hee, it is obvious that t 1 λl since t a 1 is the uppe bound of the seies hence we wite t 1 a = 1 [ 1 a 1 1 a ]zα (5) Using the values of a, a 1 a we would get Λ = mc [1+ z α ] 1 (6) 8 t 1

9 consideing only the positive values. Now, the two teminal points of the seies indices t s t 1 ae sepaated by an intege numbe of steps. If we call this intege N, then we can wite t 1 = N +t s fom which we get the modified enegy levels as E N,j = mc [1+ z α ] = mc N+ λl a + j z α [1 z α (n+ λl a ) z4α4 (n+ λl a )3 (j +1) + ] (7) whee, fom ([18], [19]) we wite j = j + 1 N = n j = n j 1, n being the pincipal quantum numbe j being the total angula momentum quantum numbe. 4 The Lamb Shift Now, let us assume that fo the S1 (without modification) as state the enegy is given by the nomal elation E(S1) = that of the P1 mc [1+ z α ] = mc [1 z α n z4 α 4 n 3 (j +1) + ] N+ j z α state is given by equation (7). The ationale fo this assumption is state will have a geate enegy than the S1 highe states. that it is only feasible to assume that the P1 state. Also, wepesumethatthemodificationcomesintoplayfothep1 Now, accoding to ou intuition thee will be a cetain enegy diffeence between these two states. Weinfe thattheenegy diffeence isgiven by theelation(z = 1fohydogen atom) [E(S 1 ) E(P1 )]+[E(S1 ) E(P1)] 0+ mc [ α n + α (n+ λl )] a whee the fist tem on the left h side gives the contibution 0 consideing the nomal enegy levels fo both S 1 P1 states the second tem gives the contibution { mc + [ α ]} consideing the n (n+ λl a ) P1 state to have acquied the modified enegy levels. Thus, the aveage enegy diffeence is given by Now, a is given by E(S1) E(P1) mc 4 [ α n + α (n+ λl (8) )] a a = h m c Λ c 9

10 Again, taking the electon mass m = ev the nomal enegy (without modification) c of the P1 state would be appoximately given by eV Theefoe, λl a λl hc c 1318eV which gives λl a = π whee c cm/s l = h πmc = cm π λ h ev = s Theefoe λl a isoftheode10 6. Now, letuslookatequation(8)findtheappoximate value. It can be witten as which would give us E(S1) E(P1) mc 4 [ α { 1 n 1 (n+ λl)}] a E(S1) E(P1) mc + λ l a a 4 [ α { nλl n (n+ λl)}] a Neglecting λl with espect to n (= ) neglecting λ l with espect to n λl a a a finally E(S1) E(P1) mc α λl n 3 a we obtain (9) 10

11 Now, l = h mc, λ 10 3 λl a λl hc c 1318eV which gives E(S1) E(P1) λα n 1318eV ev (30) Altenatively, fom (30) we can deduce the enegy shift as E(S1) E(P1) 1056MHz (31) which is vey nealy equal to the Lamb shift. Now, this justifies ou assumption that the P1 state has the enegy level given by equation (7) wheeas fo the S1 state it is given by the nomal enegy levels fom the Diac equation, fo although we consideed the P1 acquies state to have highe enegy the Compton-scale effects come into play the S1 highe enegy. Theefoe we can ague that the effects of the modification tem in the Snyde-Sidhath Hamiltonian is the eason of the Lamb shift. Moe intuitively, we can infe that this shift aises due to the inteaction of the electons with the zittebewegung fluctuations of the quantized adiation field, a phenomenon that can be attibuted to the Compton scale the non-commutative natue of space-time. Also, we may deive the enegy diffeence (although negligible) between the 3P 3 states. Following the same methodology as above we would obtain which will yield E(3P3) E(3D3) 3mc α 4 λl 8n 4 a E(3P3 which is a vey small diffeence of enegy. Thus, the 3P3 3D3 (3) ) E(3D3) MHz (33) states have nealy equal 3D3 enegy this diffeence is negligible. Thus, we can see that ou appoach is consistent with the spectum of hydogen atom. 5 Conclusions It is vey inteesting that we obtain the obseved Lamb shift meely by esoting to the modified Diac equation in lieu of the conventional Diac equation. All of this accounts fo the lucid fact that in the Compton scale thee exists some exta effects due to the noncommutative natue of space-time due to the fluctuations of the field. In such cases, the modified Diac equation the enegy levels deived fom it would be necessay to explain atomic sub-atomic phenomena. Of couse, it is known that quantum electodynamics can explain such phenomena, but ou appoach is simple moe geneal in the sense that it applies to othe phenomena as well. It may be mentioned that the above consideations lead to the conclusion that thee is a mysteious cosmic adio wave backgound, that has been ecently obseved ([1]) by NASA s ARCADE expeiments, a mystey that was hitheto inexplicable by conventional theoies. 11

12 6 Discussions We would like to stess an impotant similaity with ou appoach that of C. Coda et al. ([]-[5]) whee it has been shown that the subsequent emissions of Hawking quanta nea the hoizon of a black hole can be intepeted as the quantum jumps among the quantum levels of a black hole. The fundamental consequence is that the black holes seem eally to be the gavitational atoms of quantum gavity. Refeences [1] J.J. Sakuai, Advanced Quantum Mechanics, Peason Education, Sixth Impession 009. [] B.G. Sidhath, Noncommutative Spacetime, Mass Geneation Othe Effects, Int.J.Mod.Phys.E, 19 (1), 010, pp , [3] B.G. Sidhath, Non-Commutative Geomety Symmety Beaking in Gauge Theoy, Int.J.Mod.Phys.E,14(),005, pp.15ff. [4] Lukasz Andzej Glinka, ÆTHEREAL MULTIVERSE, axiv: v [physics.genph], 8 Ma 011. [5] S. Sahoo, Mass of Neutino, Indian Jounal of Pue Applied Physics, Vol. 48, pp , Octobe 010. [6] A. Raoelina R. Chistian, A Study of the Diac-Sidhath Equation, EJTP 8, No.5, , 011. [7] B.G. Sidhath, Abhishek Das Aka Dev Roy, New Advances in Physics 9(1), 015. [8] Willis E. Lamb Robet C. Rethefod, Fine Stuctue of the Hydogen Atom by a Micowave Method, Phys. Rev., Vol. 7(3), pp , [9] Hans A. Bethe, The Electomagnetic Shift of Enegy Levels, Phys. Rev., Vol. 7(4), pp , [10] C. Itzykson J.B. Zube, Quantum Field Theoy, pp. 365, McGaw-Hill, [11] Glen W. Eickson, Physical Review Lettes, vol. 7, pp. 780, [1] Pete J. Moh, Physical Review Lettes, vol. 34, pp. 1050, [13] T.A. Welton, Some Obsevable Effects of the Quantum Mechanical Fluctuations of the Electomagnetic Field, Phys. Rev., vol. 74, 1157, [14] J.D. Bjoken S.D. Dell, Relativistic Quantum Mechanics, Mc-Gaw Hill, New Yok, 1964, pp.39. 1

13 [15] A.Peteman, A New Value fo the Lamb Shift, Physics Lettes B, Vol. 38, Issue 5, pp , 6 Mach 197. [16] Savely G. Kashenboim, The Lamb Shift of Hydogen Low-Enegy Tests of QED, axiv:hep-ph/ v1, Nov [17] B.G. Sidhath, The Modified Diac Equation, EJTP, Vol. 7, No. 4, July 010. [18] Alfed Whitehead, A Relativistic Electon in a Coulomb Potential, Physics 518, Fall 009. [19] P.A.M. Diac, The Pinciples Of Quantum Mechanics, 4th ed. Oxfod Univesity Pess, 198. [0] Walte Geine, Relativistic Quantum Mechanics: Wave Equations, nd ed. Spinge, [1] B.G. Sidhath, The Dak Enegy Signatue, to appea in Intenational Jounal of Moden Physics E. [] C. Coda, Effective Tempeatue, Hawking Radiation Quasi-Nomal Modes, IJMPD 1, 1403 (01). [3] C. Coda, Black Hole Quantum Spectum, EPJC 73, 665 (013). [4] C. Coda et al., Effective State, Hawking Radiation Quasi-Nomal Modes fo Ke Black Holes, JHEP 1306, 008 (008). [5] C. Coda, Time-Independent Schodinge Equation fo Black Hole Evapoation: No Infomation Loss, Annals of Physics, 353, pp. 71-8, (015). 13