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acting with an external field. This leads to a modified Dirac equation of the form . {[ rx ( m 3 [ Iii - m - yP 1 - - - In -J1. - - - - D 3nm 2 8 5
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rx - VP 0 (J 2n 2m
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v} eAP(x) ] ljJ(x) = 0
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(7-103)
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including vacuum polarization and vertex corrections evaluated for a slowly variable external field. The differential operators inside the curly brackets operate on the Coulomb potential AP(x) only. The quantity ljJ is to be interpreted as a matrix element of the field operator between the vacuum and a one-electron state in the presence of the external potential. According to the perturbative rules, the added term can only be taken into account to first order. Otherwise our calculation would be logically inconsistent. The unperturbed states are solutions of the Dirac equation
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(iii - m - eJ)ljJ(x) = 0
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Ze 0 AP(x) = - 4nlxl gP
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(7-104)
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which have been discussed in Sec. 2-3. Consequently, the level shift is given by the expression
where ljJ stands for a solution pertaining to the quantum numbers (n, [,j) of Eq. (7-104). As a first approximation we may even use the nonrelativistic value for ljJ in Eq. (7-105) in such a way that
I ljJ~r(o) 12 = _1_ = (Zrxm na~ nn
This yields
t d 3x ljJn,I,Ax) [ 0 ] ljJn,l,j(X) ~A (x)
4m (Zrx) 4 Dl,O 7
Define now the separate contributions of the two terms in (7-105) as
= DE(l) + DE(2)
(7-106)
It follows that
(1) erx DEn,l,j = 3nm 2
t m d 3 x ljJn,I,Ax) (In -; -"8 -"5
4mrx D - - (Zrx) 4 ( In -m - - - 10 , 3nn 3 J1. 8 5
3 1) [ 3 1)
~A 0 (x)] ljJn,l,j(X)
(7-107)
QUANTUM FIELD THEORY
The relativistic wave functions are singular at the origin. A more rigorous treatment would require avoiding an expansion of the effective interaction in powers of q21m 2. Since only mean values of the potential enter the expression (7-105), the above result remains accurate to order ()((Z()()4. We turn now to the second term C5E(2), which we intend to compute in the same spirit. Some limited use of the Dirac equation is necessary since y matrices relate small and large components. To this end we express C5E~:I,j as
(2) ie()( C5En,l,j = 4nm
d x ljI n,l,Ax)y" E(X)ljIn,l,Ax)
(7-108)
which represents the effect of the anomalous magnetic moment inducing an electric dipole moment for a moving electron. The large (ep) and small (X) components of ljI are approximately related by
X(x)
= - i 2m ep(x)
O'"V
ignoring the effects of the Coulomb potential. Since eE(x) =
Z()((x/l X 1 3 ) and
y"X ljI t IX13 ljI
O'"X O'"X 1 (O'"xO'"V ep t IX13 X - Xt IX13 ep ~ 2im ep t IX1 3 ep
+ hc
we find after an integration by parts that
to obtain
d x ljIt(x)
I~I~ ljI(x) ~ 2~
d x ept [0'" V,
iX"I~]ep
As in the case of the hyperfine interaction, some care is required when handling 3 such a singular quantity as xii x 13. We recall that xii x 1 = - VIII x Iand ~1/1 x 1= 3 -4nC5 (x). We can take for ep a Pauli spinor and use the expression of the nonrelativistic angular momentum operator L = (1/i)x x V together with the identity [O'aAm O'bBb]
C5ab(AaBb - BbAa)
+ iSabcO'c(AaBb + BbAa) + 4 L"S] epn,l,j(X) I X 13
C5En,t,j - 8nm2
(2) _
Z()(2
d x epn,l,j(X) 4nC5 (x)
We know the value of the wave function at the origin. The matrix element of L"S is equal to (1 - C5 l ,o)UU + 1) - l(l + 1) - ~]/2. Finally, the mean value of III x 13 for states of angular momenta equal or larger to one is
+ 1)(21 + l)n3
(Zm()()
Putting everything together we find
(2) _
C5E n,t,j -
()((Z()()4
1 2nn 3 m 21 + 1 Cj,l
RADIATIVE CORRECTIONS
c. J,I -
(1 _ b ) j(j
+ 1) -
1(1 + 1) - i + 1)
(7-109)
ifj=l+i
if j = I -
I~ 1
For states with nonvanishing orbital momentum, bE(l) vanishes, while bE(2) gives a small contribution of order 0:(Zo:)4 originating from the electron anomaly. For s states our calculation of bE still contains the fictitious mass 11 and diverges as 11 --+ 0, a signal that we have omitted some essential contribution. As the qualitative argument indicated, an effective infrared cutoff was expected of the order of the Bohr radius (Zmo:) - 1 instead of the arbitrary value 11- 1. The mistake can be traced to the use of the effective interaction (7-77) for arbitrary large wavelengths. For wavelengths comparable to or larger than the Bohr radius the radiative corrections should in turn take into account the Coulomb interaction, i.e., the fact that they apply not to a free but to a bound electron. In order to correct for this effect we use the observation that the ratio of the binding energy to the rest mass energy is very small, of order iZ 2 0: 2 . Let us introduce a cutoff K on the three-momentum of virtual photons such that mZ 2 0: 2 K m. For k < K it will be possible to treat the electrons in a nonrelativistic approximation and we shall be able to take the nuclear potential into account. For k > K we will neglect the effects of binding and use the previous results, except for the fact that we need an accurate relation between K and 11. To do this we return to Sec. 7-2-3, where we computed the rate of soft bremsstrahlung by integrating the quantity 2 2 3 0: d k [ 2p'p' m m ] B = 4n2 ()J~/!"E 2w k pk' pi - (k' p - (k' p')2
The latter was evaluated for 11 !1E but 11 =I- O. Let us repeat the calculation setting 11 = 0 and using instead the lower cutoff K(K !1E) on photon threemomenta. Denote such a quantity as B 1 . The comparison of B with Bl will offer a means of relating K and 11. Keeping the definition of the hyperbolic angle q> such that p' pi = m 2 cosh 2q> we find
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