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~ fd 3 X (E2 + B2) = I I in VS .NET
~ fd 3 X (E2 + B2) = I I PDF417 2d Barcode Decoder In VS .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. PDF417 Maker In .NET Framework Using Barcode creation for VS .NET Control to generate, create PDF 417 image in Visual Studio .NET applications. A= 1,2 k
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in rough agreement with the observed shift of 1057 MHz. The comparison of this term with the Darwin term in Eq. (282) shows a reduction factor of order IX In 1/1X. These approximations will be considerably improved in the sequel (see Chap. 7). Nuclear effects The nucleus has a finite size and its charge distribution is not concentrated at a point. For the proton this is represented by a form factor. This affects predominantly the swave states, since higher iwave functions vanish at the origin. An interesting consequence is the isotopic effect, predicted as early as 1932, where different isotopes would have slightly shifted levels. For light nuclei, the main contribution comes from the mass difference of the various isotopes: since the reduced mass of the electrons is 11m = lime + 1/mnu cl. For heavy nuclei, however, the finite size of the charge distribution is the leading effect. In a nonrelativistic approximation, we write the correction as where V(r) is the true potential and (Zelr) its Coulomb approximation.
THE DIRAC EQUATION
f + 4~: ] ~~it/l(OW fd3xr2~V
~ e it/l(OW
d x [V(r) where we have inserted ~r2 = 6, integrated by parts, and used the Poisson law: ~ V =  p, P d 3 x = Ze. According to (284), the effect is proportional to Z4. This effect might be used in practice for isotopic separation. A given isotope is excited to some level by a first laser ray and then ionized by a second laser ray. An electrostatic separation may then be achieved. A second consequence is that the critical value Zc ~ 137 beyond which the ground state becomes unstable is pushed to larger values Z ~ 175. A correct treatment should also include the recoil of the nucleus. This is a difficult problem, which may be tackled by a relativistic twobody equation (Chap. 10). An heuristic reasoning may, however, shed some light on this point. From the predictions of the Schrodinger and KleinGordon equations [(283) and (286)], we see that in both cases they can be interpreted by imposing the following condition on the velocity v: Twobody relativistic corrections
i!X .. . R = n = posItIve lllteger
Indeed, in the Schrodinger case, mv 2 m!X 2 E==2 2 2n whereas for the KleinGordon equation v = or
p2/E2, and the desired result is obtained by replacing n by n  bj as in (286). In a twobody problem, let us assume that the relevant velocity is the relative one, namely, the velocity of one of the particles measured in the rest frame of the other. Then E tot
= (p +
= m + M + (1 _ m~ M.
V 2 )1/2 which is symmetric in the interchange
Using again our empirical rule
we obtain
QUANTUM FIELD THEORY
IX mM 1X4 mM mM = m + M     +    [ 3=2n 2 m + M 8n 4 m + M (m + M)2 1X4 mM 1 6  n 3 m + M 2j + 1 + O(IX ) This expression reproduces fairly well the recoil effects obtained refined treatment.
a more
24 HOLE THEORY AND CHARGE CONJUGATION 241 Reinterpretation of Negative Energy Solutions
In spite of the successes of the Dirac equation, we must abandon our ostrich policy and face the interpretation of negative energy solutions. As explained previously, their presence is intolerable, since they make all positive energy states unstable in the final analysis. A solution was proposed by Dirac as early as 1930 in terms of a manyparticle theory. Although this shall not be the final standpoint, as it does not apply to scalar particles, for instance, it is instructive to retrace his reasoning. It provides an intuitive physical picture useful in practical instances, and permits fruitful analogies with different situations such as electrons in a metal. Its major assumption is that all the negative energy levels are filled up in the vacuum state. According to the Pauli exclusion principle, this prevents any electron from falling into these negative energy states, and thereby insures the stability of positive energy physical states. In turn, an electron of the negative energy sea may be excited to a positive energy state. It then leaves a hole in the sea. This hole in the negative energy, negatively charged states appears as a positive energy, positively charged particlethe positron. Besides the properties of the positron, its charge Ie I =  e and its rest mass me, this theory also predicts new observable phenomena: 1. The annihilation of an electronpositron pair. A (positive energy) electron falls into a hole in the negative energy sea with the emission of radiation. From energy momentum conservation at least two photons are emitted, unless a nucleus is present to absorb energy and momentum. 2. Conversely, an electronpositron pair may be created from the vacuum by an incident photon beam in the presence of a target to balance energy and momentum. This is the process mentioned above; a hole is created while the excited electron acquires a positive energy. Thus the theory predicts the existence of positrons which were in fact observed in 1932. Since positrons and electrons may annihilate, we must abandon the interpretation of the Dirac equation as a wave equation. Also, the reasons for

