Numerical Value of the Lamb Shift

Bethe, Hans A.; Brown, Laurie M.; Stehn, J. R.

doi:10.1103/physrev.77.370

Cited by 122 publications

(46 citation statements)

References 10 publications

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“…In the context of the spectral representation, we recall that in Table II of Ref. [17] and Table I of Ref. [18] one may even find results for the particular contributions of the discrete spectrum and of the continuum to the Bethe logarithm of selected low-lying states.…”

Section: Introductionmentioning

confidence: 99%

“…While a complete account of all previous work on the Bethe logarithm would result in an excessively long list of references, it might be instructive and appropriate to recall a few previous investigations on this subject [17,18,19,20,21,22,23,24,25,26,27]. Recently, the Bethe logarithm has been re-evaluated, for selected hydrogenic states, in the context of lower-order terms acting as preparatory calculations for higher-order relativistic corrections to the self-energy [28,29,30].…”

Section: Introductionmentioning

confidence: 99%

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Calculation of hydrogenic Bethe logarithms for Rydberg states

Mohr

2005

Phys. Rev. A

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We describe the calculation of hydrogenic (one-loop) Bethe logarithms for all states with principal quantum numbers n ≤ 200. While, in principle, the calculation of the Bethe logarithm is a rather easy computational problem involving only the nonrelativistic (Schrödinger) theory of the hydrogen atom, certain calculational difficulties affect highly excited states, and in particular states for which the principal quantum number is much larger than the orbital angular momentum quantum number. Two evaluation methods are contrasted. One of these is based on the calculation of the principal value of a specific integral over a virtual photon energy. The other method relies directly on the spectral representation of the Schrödinger-Coulomb propagator. Selected numerical results are presented. The full set of values is available at arXiv.org/quant-ph/0504002.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Calculation of hydrogenic Bethe logarithms for Rydberg states

Mohr

2005

Phys. Rev. A

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“…We evaluate the integral over y in in the following way: The integral over t in (9-6) is evaluated as follows: The results of the numerical integrations are given in Table 9-1-In that table, where three values are given for a single point, the middle value is the result obtained with the above described method of integration; the upper value is the result of evaluating the inte grals with a number of integration points in each integral which is one less than the number of integration points specified for that integral in the above method; the lower vaiue is the result obtaired with one extra integration point in each integral. The single vaiues in that (2) 0.01*6 (1) 0.1277 (5) 0.2133 (7) 0.301*5 (8) 0.1*021*{1*) 0.5107(1*) 0.6341 (6) 0.7797 (7) 0.9638 (8) 1.2171 (8) in the sum over K is effective.…”

Section: -Rmentioning

confidence: 99%

“…We now consider the integral r S1(y,t,r) = / dr S(r,y,t,r) - (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) ^y.t.r) = / dx i s (1 -fiy,t,r) Jo N = 8 (5-25) where N is the number of integration points used to evaluate the integral, gf is given by dx e -;jUA e"* m 5 X l(f J "> e" y . (5.27) We arrived at the above prescription in the following way.…”

Section: {K -So)mentioning

confidence: 99%

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Radiative corrections in hydrogen-like systems

Mohr

1973

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The lowest order term was subsequently calculated relativlstiually, and evaluation of successively higher order terms followed that. To display th?results of these calculations, we express the level shift in the formwhere F(aa) = k k0 * A kl en(za)" 2 + A^za) + A 6n {2afIn Table 1 for all Zot and agrees, by construction, with the small la expan-12 sion.The results of these calculations appear in Fig. 10.1.-it- VW --frThe bar over the wave function denotes the adjoint: 0 ^x) = 0 <(x)rTo keep the terms in the energy shift separately finite, we use the 19 20 covariant regulator method, where in momentum space the following replacement is made for the photon propagator:The limit is to be taken after the integrals in Eq. In order to do the integration over t" -t^ first, we place in the integrand a convergence factor exp(-6Jt -t |), where 6 temporarily satisfies the conditionWe shall eventually let & tend to zero. We now integrate over(2.12) Equation (2.10) can be written -6m Id^tfx)^^) (P.13)whereThe integration over k in T p is done next. It is convenient to integrate over k. firstThen we perform the integration over k in I_ and obtainThe branches of the square roots are determined by the conditionsThe .'-n. •M^'ljl^nlIn order to facilitate the evaluation of (2."9), we change the contour of integration C" to a new contour, and divide the integral The total energy shift is -12-For these contributions, we are interested in the limit as e -» 0+, as 2, and z~ approach zero from below and above the real axis respectively, and as R -*». This limit will be considered first for AEj. and then for /sE".-13-The low-energy part of the energy shift is * |x -x j texp(-bjj 2 -xj) -expC-b'lx^ -xj)) . {3.1)Tne contour CT is shown in Pig. (3.'*)Henceforth we assume tha'. A > E . We then have n X

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Negative refraction and quantum vacuum effects in gyroelectric chiral medium and anisotropic magnetoelectric material

Shen¹,

Norgren²,

2006

Ann. Phys.

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Numerical Value of the Lamb Shift

Cited by 122 publications

References 10 publications

Calculation of hydrogenic Bethe logarithms for Rydberg states

Calculation of hydrogenic Bethe logarithms for Rydberg states

Radiative corrections in hydrogen-like systems

Negative refraction and quantum vacuum effects in gyroelectric chiral medium and anisotropic magnetoelectric material

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