20: !1E Bl = - (2q> coth 2q> - 1) In n K in .NET framework

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Our previous result for B was [Eq. (7-88)]
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20: { 2!1E (2q> coth 2q> - 1) In n 11
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1 1 - f32 - lcosh 2q> f3 sin (8/2)
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For bound electrons the velocity is small ([3 ~ Z()(). It is therefore appropriate to match Bl and B in the limiting case where [3 -+ 0, q> ~ [3 sin (0/2). Under these circumstances the integral appearing in the expression for B is approximately equal to
We may then identify Band Bl provided that
ln~= -~
2K 6
(7-110)
The Lamb-shift calculation only involves one virtual photon line. Furthermore, the nucleus dictates a privileged frame for which the splitting k ~ K is legitimate. We decompose accordingly (jE(1) into two parts, (jE> and (jE<. For (jE> we use the result (7-107) with the substitution implied by (7-110). The remaining piece, (jE<, requires a new calculation. Our expression is more correctly
(jE(l) = (jE<
+ (jE>
(7-111)
(Z()()4 (jE = -4 m()(-- ( In - m 3
5 3 1) + - - - - - (j[ 0 2K 6 8 5 '
This will be meaningful if the dependence on K cancels between (jE< and (jE>. To obtain (jE< we return to the original procedure of radiative corrections with the proviso that the momenta occurring in the virtual radiation field are limited by K and hence can be treated by second-order nonrelativistic perturbation theory starting from a Schrodinger equation of the form (7-112) Here l/J is a wave function for both the electron and the radiation field (A~, Aq) which describes in the unperturbed state a bound electron and the electromagnetic vacuum. The quantum part effectively reduces to Aq (X ) A~(x)
Ikl<K
k 2I kd k l(2)3 [ " 8. )a('<) (k) eik x ~
),=1,2
+ hc]
(7-113)
We treat the interaction with the radiation field to lowest nonvanishing order and find two contributions. The first one is a "seagull" contribution arising from the quadratic term in A q It may be reabsorbed in E as a contribution to mass renormalization, since it affects indiscriminately all levels. The second one has the form
RADIATIVE CORRECTIONS
,k x x fd 3X Il/I~(x)[(l/i)Vos).(k) e ' + eik'xs).(k) (l/i)V]l/In'(xW ' (7-114) En - En' - k where En and En' stand for the unperturbed levels, En ~ m - mZ 2rx 2/2n 2 The range of the x integration is effectively limited by the Bohr radius. Accordingly, K Ix I ;:S K/mZrx. From our hypothesis on K this may be considered as very small, justifying the dipole approximation where e ikox is replaced by unity. Define vop = (l/im)Vand note that
2k(2n)
I--2 ). (2m)
).J;,21<nlvopos).(k)ln')12 and
k~~b)<nlv~pln')<n'lv~pln)
b dn (bab _ kak ) = ~ b k2 3 ab f 4n
We therefore conclude that
bE:= 2rx 3n I n'
dkk l<nlvop ln')1 2 En - En' - k
(7-115)
Formula (7-115) is not yet the correct one since it does not take into account mass renormalization in the sense that we are using in vop the expression of the physical mass. The contribution of very soft photons (k < K) to the self-mass has yet to be subtracted from it. In other words, a counterterm of the form -(bm/2m2)pop2 = -(bm/2)vo/ had to be inserted in the hamiltonian with bm adjusted in such a way that bE< vanishes for a free electron. Since <nl vo/ In) = Ln' I<n Ivop In') 12 and, owing to the fact that K 1 En - En' I, it is clear that the correct expression for bE< is
(7-116) At that stage we can only do a numerical evaluation. Bethe, who was the first to determine this nonrelativistic contribution, has introduced the following logarithmic average. For a state In) pertaining to an s wave define <En) through
In < ) = En
In,l<nlvopln')12(En,-En)lnIEn,-Enl
In' I <n Ivop In) I (En' -
(7-117)
using an arbitrary, but fixed, energy scale. This definition becomes meaningless for higher waves (I "# 0) where the denominator vanishes, as we shall soon realize. Therefore, in this case bE< becomes independent of K, a fortunate circumstance.
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