SHIFT OF THE TERMS OF HYDROGEN-LIKE ATOMS

AND THE ANOMALOUS MAGNETIC MOMENT OF THE ELECTRON

Ya. A. Smorodinskii

1. SHIFT OF THE TERMS OF HYDROGEN AND HELIUM

The development of the technique of radio spectroscopy of atoms and molecules has led to a sharp increase in the accuracy of measurements of the fine and hyperfine structures in the spectra of atoms and molecules.

One of the most interesting problems that could be posed thanks to such an increase in accuracy was a further test of the relativistic wave equation for one electron (the Dirac equation).

The object of investigation chosen was the simplest system—the hydrogen atom. It is well known that, according to quantum mechanics, the energy of an electron in a hydrogen (or hydrogen-like) atom, with relativistic effects taken into account, is determined by the formula

\[ W = mc^2 \left[ 1 + \left( \frac{\alpha Z}{\,n - k + (k^2 - \alpha^2 Z^2)^{1/2}\,} \right)^2 \right]^{-1/2}, \tag{1.1} \]

where \(n\) is the principal quantum number, \(k = j + \frac{1}{2}\), \(j\) is the quantum number of the total orbital angular momentum (the internal quantum number), \(Z\) is the nuclear charge, and \(\alpha = e^2/\hbar c\).

If terms of order \(1/c^4\) are neglected (effects of such magnitude cannot be detected experimentally by present-day methods), then the energy (1.1) may be written in the form:

\[ W \simeq mc^2 - \frac{RZ^2}{n^2} - R \frac{\alpha^2 Z^4}{n^3} \left( \frac{1}{k} - \frac{3}{4n} \right). \tag{1.2} \]

Here \(R = \dfrac{me^4}{2\hbar^2}\) is the ionization potential of hydrogen in the nonrelativistic theory (13.5 eV). The first term in this formula represents the energy

rest, the second is the term energy in the nonrelativistic theory, and the third is the relativistic correction.

A remarkable feature of formula (1.1), and, of course, (1.2), is the independence of the level energy from the orbital quantum number. This circumstance leads to the fact that the terms of the hydrogen-like atom prove to be doubly expressed*) by the two possible values of \(l\) for a given \(j\):

\[ l=j\pm\frac{1}{2}. \]

A large number of experiments devoted to the study of the hydrogen spectrum had always confirmed both the very fact of the existence of the expression and the quantitative agreement with formula (1.2). Existing indications \(^{1,2}\) of possible discrepancies between experiment and theory were not entirely convincing, since the discrepancy lay within the limits of experimental errors.

For the first time, disagreement between theory and experiment was demonstrated in 1947 by Lamb and Retherford \(^{3}\), who investigated the structure of the expressed term \((2^{2}S_{1/2}-2^{2}P_{1/2})\) in the hydrogen atom.

In these experiments use was made of the fact that when the electron is on the \(2^{2}S_{1/2}\) level, the state of the atom is metastable. Indeed, the only level with a smaller principal quantum number is the \(1^{2}S_{1/2}\) level, but transition to this level with emission of a single quantum is forbidden, since the quantum number \(l\) does not change. A transition with emission of two quanta has a negligible probability: the lifetime corresponding to this transition is approximately \(1.4\cdot 10^{-1}\) sec., whereas the lifetime of an excited atom in a \(P\)-state is \(1.6\cdot 10^{-9}\) sec.

The investigation of transitions from the metastable state \(2^{2}S_{1/2}\) was carried out in the experiments of Lamb and Retherford.

A beam of hydrogen atoms was irradiated by an electron beam), after which the hydrogen atoms entered a detector, which was constructed so that it could register only atoms in an excited (metastable) state*).

A constant magnetic field was placed in the path of the atomic beam, and, in a direction perpendicular to it, a high-frequency field; the magnitude of the constant field in which a resonant transition to the \(2^{2}P_{3/2}\) and \(2^{2}P_{1/2}\) levels occurs was measured (for several values of the frequency). These transitions were detected owing to the fact that

*) Except for the terms \(1^{2}S_{1/2}\), \(2^{2}P_{3/2}\), and other terms for which \(l\) has the maximum possible value for a given \(n\), and the angular momenta of spin and orbit are parallel to each other.

**) The intensity of the electron beam was sufficient to transfer approximately one hundred-millionth part of the hydrogen atoms into the metastable state.

***) The detector was a metallic target from which, owing to the excitation energy of the incident atoms, electrons were knocked out.

SHIFT OF TERMS OF HYDROGEN-LIKE ATOMS

atoms, having entered a new, no longer metastable state, emit quanta and pass into the normal state, after which they cease to be registered by the detector. Thus, when the detector measures the flux of excited atoms as a function of the magnitude of the external field, a minimum will be found at that value of the field at which the resonance transitions occur.

These resonance transitions occur when the energy of the quanta of the high-frequency field \(h\nu\) becomes equal to the difference of the energies of two levels. The expression for the energy difference has the form

\[ \Delta W_H = (\Delta W)_0 + a_{m_1 m_2}H, \tag{1.3} \]

where \((\Delta W)_0\) is the splitting of the term in the absence of a field, \(a_{m_1 m_2}\) is a coefficient depending on the projections, on the direction of the external field, of the magnetic moments of the atom in both states (characterized by the magnetic quantum numbers \(m_1\) and \(m_2\)), and \(H\) is the external field.

From (1.3) it is seen that \(\Delta W_H\) depends linearly on \(H\), and the graph of the dependence of the resonance-field frequency on \(H\) must be a straight line, whose intersection with the \(\nu\)-axis gives the required value of the splitting \((\Delta W)_0\).

The figure shows the arrangement of three terms of the hydrogen atom (without a field) with \(n=2\), obtained in the first experiments of Lamb and Retherford. Alongside, for comparison, is shown the arrangement of the same terms according to the Dirac equation. The distances between terms are given directly in frequencies (in Mc/s)*.

Arrangement of hydrogen terms (at \(n=2\)) according to Dirac’s theory (a) and as determined experimentally (b). The distances are given in megacycles.

The measured doublet splitting, according to the latest measurements, proved to be\(^4\) \(1062 \pm 5\) Mc/s**).

* \(1\) Mc/s \(= 0.33 \cdot 10^{-4}\ \text{cm}^{-1} = 4.1 \cdot 10^{-9}\ \text{eV}\). The term \(2P_{1/2}\) turned out to lie below the term \(2S_{1/2}\); therefore, strictly speaking, the transition \(2S_{1/2} \to 2P_{1/2}\) is possible even in the absence of a field. However, since the splitting is small, and the transition probability is proportional to the cube of the distance between the terms, this probability is very small (the lifetime is of the order of several years).

**) We note that in this last work the splitting was measured both for hydrogen and for deuterium. In both cases it turned out to be the same. In this measurement it was not the intensity of the beam that was measured; rather, the quantum emitted by the atom in its transition from the metastable state to the normal state was observed directly.

In addition to hydrogen, a displacement of the S term was also observed in the spectrum of singly ionized helium (He II).

Measurements by Mack and Austern\(^5\) gave, for the displacement of the term \(3^2S_{1/2}\), the value \(3390 \pm 420\) MHz.

The displacement of the term \(2^2S_{1/2}\) was measured by Fowles\(^6\) and, more accurately, by Skinner and Lamb\(^7\). The latter authors give for it the value \(14100 \pm 300\) MHz.

2. NONRELATIVISTIC THEORY

The causes of the discrepancy between theory and experiment were indicated by Weisskopf, Oppenheimer, and Schwinger (cf. \(^8\)) and consist in neglecting the effects of the interaction of the electron with the radiation field (“zero-point oscillations” in the vacuum)\(^*\).

If we consider the Dirac equation for a free electron, then, strictly speaking, we must include in this equation terms that take into account the interaction of the electron with the radiation field of the vacuum.

In the language of the quantum theory of radiation, the electron can virtually emit and absorb quanta of any frequency. Taking into account processes\(^ {**}\) of this kind, as is known, leads to divergent integrals in the expression for the energy of the electron, and therefore the corresponding terms were discarded. In ordinary problems of the quantum theory of radiation it was considered self-evident that it makes sense to carry out calculations only up to such approximations in which infinite integrals do not appear. Such a formulation of the problem is in fact equivalent to neglecting the effects of interaction with the vacuum field. It is possible, however, to take this interaction into account, at least in the first approximation, if one reasons as follows\(^8\).

Let us consider an electron in a Coulomb field. For simplicity of reasoning we shall regard the problem as nonrelativistic. If there were no interaction with the radiation field, then for the energy of the level we would obtain the usual value \(R \dfrac{1}{n^2}\). If now one takes into account the possibility of emission and absorption of a virtual quantum, then a divergent expression proportional to \(\alpha\) will be added to this expression. It is possible, however, to give this expression a physical meaning if one pays attention to the fact that an analogous divergent expression

\(^*\) Similar ideas were independently developed by Tomonaga and his collaborators.

\(^ {**}\) Since the interaction of the electron with the field is proportional (in dimensionless units) to \(e/(\hbar c)^{1/2}\), such a second-order process (emission and subsequent absorption of a virtual quantum) gives terms in the Hamiltonian proportional to \(e^2/\hbar c\).

will result if, in the second approximation, we calculate the energy of a free electron.

This last result can have a physical meaning only in the event that we assume that the divergence in the existing theory is connected with an incorrect accounting of the interaction with quanta of very high energy, and that some future theory will give a finite result for this expression.

If this is so, then we must consider that the parameter \(\mu\) (mechanical mass) entering into the wave equation
\[ H\psi=\frac{\hbar^2}{2\mu}\Delta\psi+U\psi=E\psi \]
is not the full mass of the electron, but only a part of it. The full mass of the electron is obtained only after adding to it terms arising from the accounting of the interaction of the electron with the vacuum field (the electromagnetic mass). And since in experiment only the full mass can be observed, the very separation of it into two parts is meaningless. Therefore, in calculating all other effects it is precisely the full mass that must appear, and not only its finite part—the mechanical mass. In this case one may hope that the infinite parts arising in the calculation of these effects are nothing other than the result of the appearance of a formally infinite electromagnetic mass.

In the case of an electron in a Coulomb field one may try to identify the divergent part of the energy with the electromagnetic mass, and to regard as the correction to the term value the difference between the interaction energies of the bound and the free electrons with the radiation field. In doing so we must ignore the fact that both expressions are infinite, and be interested only in whether their difference is finite. As Bethe showed\(^{8}\), in this way one indeed obtains a finite expression, in agreement with experiment.

Bethe’s conclusion is as follows. The ordinary theory of radiation shows that the theory of the interaction of the electron with the electromagnetic vacuum field, which is in the state “0,” is determined by the divergent integral
\[ W=-\frac{2e^2}{3\hbar c^3m^2}\int_0^K k\,dk\cdot\sum_n \frac{|(0|p|n)|^2}{E_n-E_0+k}, \tag{2.1} \]
where \((0|p|n)\) is the matrix element of the momentum, and \(E_0\) and \(E_n\) are the energies in the initial and “intermediate” states; the summation is carried out over all intermediate states.

The upper limit of integration is denoted by \(K\). Formally \(K\to\infty\), but for large wave vectors the nonrelativistic theory is no longer applicable; therefore the question of the upper limit will be considered separately.

We must consider the difference of the expressions (2.1), calculated for free and bound electrons.

For a free electron the momentum, as is known, has only diagonal elements (the wave function is a plane wave with a specified wave vector); therefore, for it*)

\[ W_{\mathrm{free}}=-\frac{2e^2}{3\pi\hbar c^3m^2}\int_0^K(0|p^2|0)\,dk. \tag{2.2} \]

For a bound electron we write (2.1) in the form

\[ W_{\mathrm{bound}}=-\frac{2e^2}{3\pi\hbar c^3m^2} \left[ \int_0^K dk\,(0|p^2|0)+ \right. \]

\[ \left. +\int_0^K dk\sum |(0|p|n)|^2\cdot \left(\frac{k}{E_n-E_0+k}-1\right) \right]. \tag{2.4} \]

Choosing the momenta of the free and bound electrons to be the same and forming the difference \(W_{\mathrm{bound}}-W_{\mathrm{free}}\), we obtain:

\[ W'=W_{\mathrm{bound}}-W_{\mathrm{free}} =\frac{2e^2}{3\pi\hbar c^3m^2} \int_0^K \sum_n \frac{|(0|p|n)|^2(E_n-E_0)}{E_n-E_0+k}\,dk. \tag{2.5} \]

Let \(K\) be much larger than all possible energy differences \(E_0-E_n\); then, integrating over \(k\), we obtain:

\[ W'=\frac{2e^2}{3\pi\hbar c^3m^2} \sum_n |(0|p|n)|^2(E_n-E_0)\ln\frac{K}{|E_0-E_n|}. \tag{2.6} \]

It is evident from this expression that the upper limit of the integral enters only under the logarithm. If one assumes that, when relativistic effects are taken into account, the integral proves to be convergent (which, as will be seen from what follows, is indeed the case), and that therefore values of \(k\) exceeding \(mc^2\) in fact have little effect on the integral, then, owing to the comparatively small sensitivity of (2.6) to the exact value of the cutoff boundary, one may put

\[ K\simeq mc^2. \tag{2.7} \]

For this value of \(K\), the argument of the logarithm turns out to be very large, and one may, replacing \(\ln|E_0-E_n|\) by the mean value \([\ln|E-E_0|]_{\mathrm{av}}=\ln(\Delta E)_{\mathrm{av}}\), take the logarithm outside the summation sign.

*) According to the rules for multiplying (Hermitian) matrices,

\[ \sum_n(0|p|n)\overline{(0|p|n)} =\sum_n(0|p|n)(n|p|0)=(0|p^2|0). \tag{2.3} \]

Then, in order to calculate (2.6), it remains only to find the value of the sum

\[ A=\sum |(0|p|n)|^2(E_n-E_0). \tag{2.8} \]

In order to compute this sum, let us note that the matrix element \((0|p|n)\) contains the factor

\[ \exp\left[-\frac{i}{\hbar}(E_0-E_n)t\right]. \]

Therefore

\[ (0|p|n)(E_n-E_0)=\frac{\hbar}{i}\frac{d}{dt}(0|p|n)=\frac{\hbar}{i}\left(0\left|\frac{dp}{dt}\right|n\right), \tag{2.9} \]

according to the definition of the derivative of a matrix. The sum (2.8) is then transformed into the form:

\[ A=\frac{\hbar}{2i}\left(0\left|\frac{dp^2}{dt}\right|0\right). \tag{2.10} \]

According to the rule for differentiating operators with respect to time,

\[ \frac{d}{dt}p^2=\frac{i}{\hbar}(Hp^2-p^2H), \tag{2.11} \]

where

\[ H=\frac{p^2}{2m}-eV \tag{2.12} \]

is the Hamiltonian. Hence

\[ -\frac{d}{dt}p^2=-\frac{ie}{\hbar}(Vp^2-p^2V)=ie\hbar(V\Delta-\Delta V), \tag{2.13} \]

where \(\Delta\) is the Laplace operator.

Multiplying (2.13) on the right and on the left by the electron wave function \(\psi\) and integrating over all space to obtain the matrix element, we get:

\[ A=\frac{1}{2}e\hbar^2\int \psi [V\Delta\psi-\Delta(V\psi)]\,d\tau = \]

\[ =-\frac{1}{2}e\hbar^2\int \psi^2\Delta V\,d\tau =2\pi\hbar^2 e^2 Z|\psi(0)|^2, \tag{2.14} \]

since, according to Poisson’s equation, \(\Delta V=-4\pi eZ\delta(\mathbf r)\), where \(Z\) is the charge of the nucleus; \(\psi(0)\) is the value of the electron wave function at the origin of the coordinates.

For the shift of the level of a hydrogen-like atom we obtain, according to (2.6):

\[ W'=\frac{4Ze^3}{3}\left(\frac{e^2}{\hbar c}\right)\left(\frac{\hbar}{mc}\right)^2 \ln\frac{mc^2}{(\Delta E)_{\mathrm{cp}}}\,|\psi(0)|^2. \tag{2.15} \]

As is known, in the nonrelativistic theory \(\psi(0)\) differs from zero

only for \(S\)-terms. Therefore only these terms should exhibit a displacement of levels. Taking relativistic effects into account shows that for other terms there will exist only a very small displacement connected with the presence of an anomalous magnetic moment of the electron (see § 4).

Since the displacement of all the other terms is considerably smaller than the displacement of the \(S\)-term, these effects may legitimately be considered within the framework of nonrelativistic theory. They will therefore be considered below. For a hydrogen-like atom

\[ |\psi(0)|^2=\frac{1}{\pi}\left(\frac{Z}{na}\right)^3, \tag{2.16} \]

where

\[ a=\hbar^2/me^2 . \]

Substituting in (2.15), we obtain:

\[ W'=\frac{4}{3\pi}\frac{me^{10}}{c^3\hbar^5}\frac{Z^4}{n^3}\ln\frac{mc^2}{(\Delta E)_{\mathrm{cp}}}, \tag{2.17} \]

or, introducing the ionization potential of hydrogen \(R=me^4/2\hbar^2\), we finally obtain:

\[ W'=-\frac{8}{3\pi}\left(\frac{e^2}{\hbar c}\right)^3 R\frac{Z^4}{n^3}\ln\frac{mc^2}{(\Delta E)_{\mathrm{cp}}}. \tag{2.18} \]

For the hydrogen term \(2^2S_{1/2}\), \(Z=1\), \(n=2\), and, according to Bethe’s calculations, \((\Delta E)_{\mathrm{cp}}=16.7R^*)\) (\(R=13.5\ \mathrm{eV}\)). As a result we obtain:

\[ W'=1040\ \mathrm{Mc}, \]

which is in excellent agreement with experiment \((1062\pm5\ \mathrm{Mc})^{**})\).

For the \(2^2S_{1/2}\) term of He the theoretical value should be approximately 13 times larger (\(Z^4=16\) and \((\Delta E)_{\mathrm{cp}}\) changes somewhat). As we have seen, experiment gives \(14100\pm300\), which also agrees well with theory. Finally, the last of the published values—the displacement of the term \(3^2S_{1/2}\) He II, equal to \(3390\pm420\ \mathrm{Mc}\), is also in agreement with theory, which gives for it the value \(3900\ \mathrm{Mc}\).

Before proceeding to a discussion of the result obtained, we shall show a very transparent way of obtaining formula (2.18), indicating—

*) Bethe later (Pocono Manor Conference, March 1948) gave the corrected value \(17.8R\); however, in view of the inaccuracy of the refinement formula such a correction is superfluous here (cf. § 4).

**) In 1938 Blokhintsev, in a report at a theoretical seminar of the Physics Institute of the Academy of Sciences of the USSR, pointed out that interaction with the radiation field can lead to a displacement of the level. However, in deriving the formula he made an averaging over high frequencies only, which meant that he took into account only transitions to nearby levels; this, of course, could not give an unambiguous result. Consequently, the formula obtained by Blokhintsev differs from (2.18) by an extraneous factor \(1/3\) and by the fact that instead of \((\Delta E)_{\mathrm{cp}}=16.7R\) he has \(68R\).

given by Welton\(^{10}\) and not explicitly connected with the subtraction of divergent integrals.

A characteristic feature of the radiation field of the vacuum is zero-point oscillations. As is known, the state of this field having minimal energy is such a state in which each normal oscillation has energy \(\dfrac{1}{2}h\nu=\dfrac{1}{2}\hbar ck\) (\(k\) is the wave vector). As a result of the interaction of such oscillations with the electron, the coordinate of the electron does not remain constant, but fluctuates about some equilibrium position. If now some field \(V(r)\) also acts on the electron, then the fluctuation of the coordinate leads to the interaction of the electron with the field being averaged over a certain small region. As a result, the interaction energy is also changed.

It is not difficult to obtain the magnitude of the change in energy. Let the position of the electron correspond to the coordinate \(\mathbf q\). Denote the magnitude of the displacement by \(\Delta\mathbf q\), and expand the potential \(V(\mathbf q+\Delta\mathbf q)\) in powers of \(\Delta\mathbf q\):

\[ V(\mathbf q+\Delta\mathbf q)=\left\{1+\Delta\mathbf q\cdot\nabla+\frac{1}{2}(\Delta\mathbf q\cdot\nabla)^2+\cdots\right\}V(\mathbf q). \tag{2.19} \]

Averaging this expression over all values of \(\Delta\mathbf q\) and noting that

\[ (\Delta\mathbf q)_{\mathrm{av}}=0,\qquad (\Delta\mathbf q\cdot\nabla)^2_{\mathrm{av}}=\frac{1}{3}(\Delta q)^2_{\mathrm{av}}\Delta, \]

we obtain, retaining only terms of second order with respect to \(\Delta q\) (which, as we shall see, corresponds to Bethe’s approximation),

\[ [V(\mathbf q+\Delta\mathbf q)]_{\mathrm{av}} = \left\{1+\frac{1}{6}(\Delta q)^2_{\mathrm{av}}\Delta+\cdots\right\}V(\mathbf q). \tag{2.20} \]

Thus the correction to the interaction potential proves to be equal to

\[ \delta V=[V(\mathbf q+\Delta\mathbf q)]_{\mathrm{av}}-V(\mathbf q) = \frac{1}{6}(\Delta q)^2_{\mathrm{av}}\cdot\Delta V. \tag{2.21} \]

In the case of the hydrogen atom,

\[ \Delta V(\mathbf r)=-4\pi e\delta(\mathbf r). \tag{2.22} \]

In order to obtain the level shift, one must substitute the value (2.22) into (2.21), multiply it by the electron charge \((-e)\) and by the square of the modulus of the wave function \(|\psi(\mathbf q)|^2\), and integrate over all space.

As a result we obtain:

\[ W'=\frac{2\pi}{3}\,e^2|\psi(0)|^2(\Delta q)^2_{\mathrm{av}}. \tag{2.23} \]

It remains to calculate the quadratic fluctuation \((\Delta q)^2_{\mathrm{av}}\). Since we consider the velocity of the electron to be small (nonrelativistic), the action of

the magnetic field may be neglected, and therefore the equation of motion of the electron has the form

\[ m\ddot{\mathbf q}=e\mathbf E, \tag{2.24} \]

where \(\mathbf E\) is the field associated with the zero-point oscillations of the vacuum.

Expanding \(\mathbf E\) in a Fourier integral:

\[ E=\int E_\omega \cos \omega t\, d\omega, \tag{2.25} \]

we can write the solution of (2.24) in the form (putting \(\mathbf q=\dot{\mathbf q}=0\) at \(t=0\) and therefore denoting this solution by \(\Delta q\)):

\[ \Delta q=-\frac{e}{m}\int \frac{E_\omega}{\omega^2}\cos \omega t\, d\omega. \tag{2.26} \]

Squaring and averaging over time, we obtain, in the usual way:

\[ (\Delta q)^2_{\mathrm{av}}=\frac{1}{2}\frac{e^2}{m^2}\int \frac{1}{\omega^4}E_\omega^2\, d\omega. \tag{2.27} \]

Further, since the energy of the field is equal to \((E^2+B^2)_{\mathrm{av}}/8\pi=(E^2)_{\mathrm{av}}/4\pi\), and

\[ \frac{1}{4\pi}(E^2)_{\mathrm{av}}=\frac{1}{8\pi}\int E_\omega^2\, d\omega, \tag{2.28} \]

\(E_\omega^2\) can easily be determined if the latter expression is equated to the integral

\[ \int \left(\frac{1}{2}\hbar\omega\right)\frac{\omega^2}{\pi^2c^3}\, d\omega, \tag{2.29} \]

where the first factor is the energy associated with the frequency \(\omega\), and the second is the number of states with frequency \(\omega\) falling in the interval \(d\omega\)*).

Equating (2.29) to the right-hand side of (2.28), we find:

\[ E_\omega^2\, d\omega=\frac{4\hbar\omega^3}{\pi c^3}\, d\omega. \tag{2.30} \]

Finally substituting this expression into (2.27), we obtain:

\[ (\Delta q)^2_{\mathrm{av}}=\frac{2}{\pi}\frac{e^2}{\hbar c}\left(\frac{\hbar}{mc}\right)^2\int \frac{dk}{k}. \tag{2.31} \]

The integral standing on the right-hand side diverges logarithmically. However, this divergence is no longer connected with the fundamental difficulties of radiation theory, but is the result of the assumptions we have made. We have already encountered the divergence at the upper limit above in the derivation of Bethe. Just as there, it is connected with the nonrelativistic treatment of the problem, and in order to eliminate it we must

*) The number of states is equal to the volume of the spherical shell in the space of wave vectors \(4\pi k^2 dk\) \((k=\omega/c)\), multiplied by the number of polarizations \(2\) and divided by the volume of the phase cell corresponding to one state \((2\pi\hbar)^3\).

simply take as the upper limit the quantity

\[ k_{\max} \approx mc/\hbar . \]

The divergence at the lower limit is due to the fact that the electron is treated as free, whereas in considering its interaction with low frequencies it is necessary to take into account the influence of the binding of the electron in the atom.

This effect is considered in Appendix I, where a more rigorous treatment is also given of the fluctuation of the electron coordinate in its interaction with the radiation field.

The result of taking this binding into account will be the appearance in the integral of a certain lower limit \(k_{\min}\), connected, as in Bethe’s derivation, with the mean excitation energy.

Thus, up to a certain factor under the logarithm,

\[ (\Delta q)_{\mathrm{cp}}^{2} = \frac{2}{\pi}\frac{e^{2}}{\hbar c} \left(\frac{\hbar}{mc}\right)^{2} \ln \frac{mc}{\hbar k_{\min}}, \tag{2.32} \]

and, substituting in (2.23), we obtain:

\[ W' = \frac{4e^{2}}{3}\frac{e^{2}}{\hbar c} \left(\frac{\hbar}{mc}\right)^{2} \ln \frac{mc}{\hbar k_{\min}} \,|\psi(0)|^{2}, \tag{2.33} \]

which coincides with Bethe’s expression if one puts

\[ \hbar c k_{\min}=(\Delta E)_{\mathrm{cp}} . \tag{2.34} \]

Thus we see that already in nonrelativistic quantum mechanics one can develop a theory which gives a result in sufficiently good agreement with experiment, despite the fact that in essence the problem is relativistic, since at high field frequencies the electron, upon recoil, will acquire a velocity that is no longer small in comparison with the velocity of light.

However, owing to the fact that in the effect under consideration the main role is played by the intermediate frequencies, the nonrelativistic calculations turn out to lead only to a small inaccuracy. Nevertheless, in order to justify the operation of “cutting off” the high frequencies, it is necessary to construct a relativistic theory of the effect.

As we shall see below, the construction of such a theory leads to the discovery of new effects, some of which have already received experimental confirmation.

3. RELATIVISTIC THEORY

In order to clarify the possibility of constructing a relativistic theory which would be based on an analogous allowance for the electromagnetic mass (infinite), let us consider the Dirac equation for a free electron

\[ H\psi=(c\alpha p+\beta\mu c^{2})\psi=E\psi, \tag{3.1} \]

where, as usual, \(\alpha\) and \(\beta\) are fourth-order matrices, \(p\) is the momentum operator, and \(\psi\) is a four-component wave function.

If we take into account the interaction with the radiation field of the vacuum, then the proper values of the energy turn into infinity. In order that the theory not lose its physical meaning, we shall assume, as before, that the mass of the electron (finite, measured experimentally) consists of two terms

\[ m=\mu+\Delta\mu, \tag{3.2} \]

where \(\mu\) is the parameter entering equation (3.1), and \(\Delta\mu\) is the electromagnetic mass.

In this case both terms in (3.2) are infinite in the existing theory, and only their sum has a finite value.

Then the equation for a free electron must be written not in the form (3.1), but in the form

\[ H\psi=(c\alpha p+\beta mc^2-\beta\Delta\mu c^2)\psi=E\psi. \tag{3.3} \]

When the interaction with the radiation field is taken into account, the term \(\beta\Delta\mu\), by definition, will cancel, and the equation will take the form (3.1), only with the full mass instead of \(\mu\).

Since the interaction with the electromagnetic field is regarded in the theory as small, then, provided that the integral entering \(\Delta\mu\) does not turn out, in its calculation in a future consistent theory, to be anomalously large,* one may regard \(\beta\Delta\mu c^2\) as a perturbation.

But then, when considering by perturbation theory any problem on the interaction of an electron with an external field, we must take as the perturbation in the Hamiltonian the sum of the perturbation associated with this external field and the perturbation \(-\beta\Delta\mu c^2\), and thereafter take into account in the calculations also the interaction with the radiation field of the vacuum. With such a formulation of the problem all infinite terms cancel (at least in the second approximation of perturbation theory with respect to the interaction with the radiation field), and the remaining terms must be regarded as having real physical meaning.

Such a program was proposed and developed by Schwinger\(^{13}\) and Weisskopf\(^{14}\), and was applied to a number of problems\(^{13,15,16}\), whereby the possibility was shown of calculating such effects as had previously been considered not amenable to theoretical treatment.

Unfortunately, the consistent execution of even such a formal program proves impossible. The difficulty encountered by the theory consists in the fact that the expression for \(\Delta\mu\) turns out, as we shall see below, to be relativistically non-invariant.

* Since the divergence is logarithmic, in order for \(\Delta\mu\) to turn out large (greater than \(m\)), it is necessary that the effective upper limit of the frequencies (in units \(mc^3/\hbar\)) be larger, in order of magnitude, than \(e^{137}\), which is scarcely probable.

Let us first consider the expression for the electromagnetic mass (more precisely, for the addition to the energy \(\Delta E\)) arising from interaction with the radiation field. In this, since according to the usual scheme all electronic states with negative energy are regarded as filled, the energy \(\Delta E\) is defined as the difference between the energy of the system consisting of the electron under consideration plus the electrons in the filled negative levels, and the energy of the electrons of the vacuum alone[^11].

Unfortunately, the detailed calculations in computing the relativistic effects are very long, and in what follows we shall omit them to a considerable extent.

It can be shown (cf. [^19]) that the electromagnetic energy in the second approximation is determined as the diagonal matrix element of the following operator:

\[ \frac{e^3}{\hbar c}\frac{\hbar^2 c^2}{4\pi^2}\int \frac{dk}{k} \left[ \frac{2\beta_\mu c^3-E_f+c\alpha p_f}{E_f(E_f-E_0+\hbar ck)} + \frac{2\beta_\mu c^3+E_f+c\alpha p_f}{E_f(E_f+E_0+\hbar ck)} \right], \tag{3.4} \]

where \(p_f\) and \(E_f\) are the momentum and energy of the electron in the “intermediate” state, \(E_0\) is the energy in the initial state, and the integration is carried out over all wave vectors of the virtual quanta. Here

\[ p_f=p_0-\hbar k, \tag{3.5} \]

where \(p_0\) is the momentum of the electron in the initial state.

Forming the diagonal matrix element of (3.4) and carrying out rather lengthy calculations, we arrive at the following expression for the electromagnetic energy:

\[ \Delta E= \frac{e^3}{\pi\hbar c} \left\{ 2\ln\frac{2\hbar K}{\mu c}(0|\beta_\mu c^2|0) + \left(\frac{1}{2}\ln\frac{2\hbar K}{\mu c}+\frac{1}{12}\right)(0|c\alpha p_i|0) - \left(\frac{1}{2}\ln\frac{2\hbar K}{\mu c}+\frac{1}{4}\right)E_0 \right\}. \tag{3.6} \]

In this expression we have, for simplicity, cut off the divergent integrals at some large wave vector \(K\); the formal (divergent) result is obtained for \(K\to\infty\). Such a cutoff has no physical meaning because of its relativistic noninvariance, and therefore in all formulas we must carry out this limiting transition. The methods presented here are based on the fact that the quantity \(K\) drops out of the final results.

If we now use the Dirac equation and replace \(c\alpha p_0\) by \(E_0-\mu c^2\beta\), then we obtain:

\[ \Delta E= \frac{1}{\pi}\frac{e^3}{\hbar c} \left\{ \frac{3}{2}\ln\frac{2\hbar K}{\mu c}+\frac{1}{12} \right\} (0|\beta_\mu c^2|0) +\frac{1}{6}E_0 . \tag{3.7} \]

We see that the operator of the electromagnetic energy cannot be represented in the form \(\beta\Delta\mu c^2\), as is necessary for invariance

of the scheme set forth above. In addition to a term of this kind, the expression (3.7) also contains the term \(\frac{1}{6}E_0\), which also violates invariance.

One might have hoped that, although in this way it is impossible to obtain a theory of electromagnetic mass, the remaining effects could nevertheless be calculated in an invariant manner, owing to the fact that the non-invariant term \(\frac{1}{6}E_0\) is diagonal and therefore does not cause additional transitions; the infinite term, on the other hand, has precisely the required form and is proportional to \(\beta\).

However, such a point of view also turns out to be incorrect. Indeed, by virtue of the relation

\[ E_0 = c a p - \beta \mu c^2 \tag{3.8} \]

we can include the additional term (or any part of it) in the first term, introducing an additional term proportional to \(c a p\), which will no longer be diagonal. It is important that under such an operation the logarithmically divergent term remains unchanged, while the constant term can, generally speaking, be made arbitrary.*)

It is obvious that such a situation, in which a relativistically non-invariant result has arisen from a relativistically invariant equation, was the consequence of a mathematically illegitimate operation of subtracting two logarithmically divergent integrals. Since such an operation is fundamentally ambiguous, it is not surprising that the result also proves to be so.

At present there exist at least two ways of overcoming this difficulty. The first of them was proposed by Feynman\(^{23}\), the second was developed in the works of Kroll and Lamb\(^{12}\), French and Weisskopf\(^{18}\), and Galanin\(^{19}\). Despite their completely different approaches to the problem, the answer obtained by both methods is the same.

Feynman’s idea is as follows. Since the cause of the appearance of relativistic non-invariance consists in the subtraction of infinite integrals, one may try to change somewhat the law of interaction with the electromagnetic field, in such a way that all effects connected with not too large frequencies remain unchanged, but all integrals become convergent. With such a calculation all results must already be relativistically invariant; and since the integrals that were artificially made convergent are excluded from the final formulae, one may expect that the result of such an operation will not depend on the choice of the method of “cutoff.”

*) Such ambiguity has also been found for particles that do not have spin\(^{21}\), for which there are no states of negative energy, so that this property is not specific to the electron.

As was already said, all divergent integrals diverge logarithmically, i.e. the integrand has the form \(\dfrac{dk}{k}\). It is not difficult to see that, with the aid of the \(\delta\)-function, one can replace the threefold integration over \(dk=(dk_x\,dk_y\,dk_z)\) by a fourfold one over \(dk\,d\omega\) \((\omega=ck)\), putting

\[ \frac{dk}{k}=\frac{2}{c}\,dk\int d\omega\,\delta\left(\frac{\omega^2}{c^2}-k^2\right). \tag{3.9} \]

Indeed, introducing the new variable \(\xi=\omega^2/c^2\) and integrating the right-hand side with respect to \(\xi\), we confirm the equality.

Let us now introduce, in place of \(\delta\left(\dfrac{\omega^2}{c^2}-k^2\right)\), a new function \(g\left(\dfrac{\omega^2}{c^2}-k^2\right)\), defined by the formula

\[ g\left(\frac{\omega^2}{c^2}-k^2\right) = \int_0^\infty \left[ \delta\left(\frac{\omega^2}{c^2}-k^2\right) - \delta\left(\frac{\omega^2}{c^2}-k^2-\lambda^2\right) \right]G(\lambda)\,d\lambda, \tag{3.10} \]

where \(G(\lambda)\) is some (generally speaking, arbitrary) function subject only to the normalization condition:

\[ \int_0^\infty G(\lambda)\,d\lambda=1. \tag{3.11} \]

If great generality is not required, then instead of (3.10) one may simply put \(g\left(\dfrac{\omega^2}{c^2}-k^2\right)\) equal to the integrand

\[ g\left(\frac{\omega^2}{c^2}-k^2\right) = \delta\left(\frac{\omega^2}{c^2}-k^2\right) - \delta\left(\frac{\omega^2}{c^2}-k^2-\lambda^2\right), \tag{3.12} \]

where \(\lambda\) is a very large number \((\widetilde{>}137\,mc/\hbar)\). After such a replacement of the \(\delta\)-function by the \(g\)-function, we find that all results will be finite. Thus, for the self-energy of a free electron in Feynman’s theory one obtains the expression

\[ \Delta E = \frac{1}{\pi}\frac{e^2}{\hbar c} \left\{ \frac{3}{2}\ln\frac{\hbar\lambda_0}{mc} + \frac{3}{8} \right\} (0|\beta\mu c^2|0), \tag{3.13} \]

where

\[ \lambda_0=\int_0^\infty G(\lambda)\ln\lambda\,d\lambda, \tag{3.14} \]

i.e. an expression possessing the required properties (proportional to the matrix \(\beta\)).

Therefore all further calculations can already be carried out according to the scheme set forth at the beginning of the paragraph, namely, by introducing the quantity \(-\Delta \tilde E\) as a perturbation into the original Hamiltonian. In this case, since

the function \(G(\lambda)\) enters only into the mass renormalization, then it drops out of the final expression.

In order to avoid an infrared catastrophe—the divergence at small wave vectors of the quantum—Feynman introduces the following formal device, which is equivalent to a simple stitching together of the relativistic formula with the nonrelativistic one calculated by Bethe. In the integrals that already determine the desired effect (after removal of the terms associated with the electromagnetic mass), instead of the \(\delta\)-function
\[ \delta\left(\frac{\omega^3}{c^2}-k^2\right) \]
one introduces the function
\[ \delta\left(\frac{\omega^3}{c^2}-k^2-\lambda_{\min}\right), \]
where \(\lambda_{\min}\ll mc/\hbar\). By this operation the small frequencies are automatically cut off. In order to obtain the correct formula, which passes at small frequencies into Bethe’s result, one must set
\[ \ln \lambda_{\min}=\ln 2k_{\min}-\frac{5}{6}, \tag{3.15} \]
where the value \(k_{\min}\) is determined by formula (2.34): \(\hbar c k_{\min}=(\Delta E)_{\mathrm{av}}\).

Considering the interaction with an external electromagnetic field, described by the potentials \(\varphi\) and \(\mathbf A\) or by the vectors \(\mathbf E\) and \(\mathbf B\), by means of the method set forth above, Feynman) arrives at the following expression for the correction to the energy (for a nonrelativistic electron):
\[ \Delta E=\frac{e^2}{2\pi\hbar c}\left(-\frac{\hbar}{2\mu c}\right)[\boldsymbol{\beta}\mathbf B-i\boldsymbol{\beta}\boldsymbol{\alpha}\mathbf E]+ \]
\[ +\frac{1}{3\pi}\frac{e^3}{\hbar c}\left(\frac{\hbar}{mc}\right)^2(e\Delta\varphi-e\boldsymbol{\alpha}\Delta\mathbf A) \left(\ln\frac{\mu c}{2\hbar k_{\min}}+\frac{11}{24}-\frac{1}{5}\right). \tag{3.16} \]
The last term leads to the relativistic analogue of Bethe’s formula; we shall examine it in more detail in the next paragraph*).

A very interesting fact is the appearance of the first term, proportional to the fields \(\mathbf B\) and \(\mathbf E\). Its existence was first discovered by Schwinger, and it represents the interaction of the additional magnetic moment of the electron with an external (magnetic and electric) field.

The appearance of the anomaly in the magnetic moment of the electron is the second remarkable effect that is a result of the interaction of the electron with the vacuum field. A characteristic feature of this effect is that here the result is obtained unambiguously and is not connected with the subtraction of infinite integrals***).

*) In Feynman’s paper an incorrect coefficient is given; the error was corrected in the work of French and Weisskopf\({}^{18}\).

**) In Feynman’s paper the constant term in the last parentheses is equal to \(5/8\). This occurred, first, because of the incorrect definition of \(\lambda_{\min}\), noted above, and, second, because of the neglect of terms associated with vacuum polarization, which led to the loss of the term \(-\frac{1}{5}\).

***) The effect was calculated by Schwinger\({}^{13}\), Galanin\({}^{19}\), Feynman\({}^{20}\), and Luttinger\({}^{34}\). On the last paper see below.

As is seen from (3.16), the magnetic moment of the electron is equal to

\[ \mu=\mu_0\left(1+\frac{1}{2\pi}\frac{e^2}{\hbar c}\right), \tag{3.17} \]

or, numerically,

\[ \mu=\mu_0(1+0.001162). \tag{3.18} \]

The uniqueness of the calculation of the effect is connected with the fact that the terms arising in the Hamiltonian in the second approximation of perturbation theory and proportional to the field \(H\) turn out not to be connected with the electromagnetic mass. (Terms of this kind do not arise from a perturbation of the form \(\beta\Delta\mu c^2\).) Therefore the problem of the anomalous magnetic moment of the electron is not connected with the formalism of subtracting the infinite electromagnetic energy.

Thanks to this, there exists still another way of calculating the anomalous magnetic moment, indicated by Luttinger \(^{24}\).

Luttinger calculated the difference between the energies, in a weak magnetic field, of a system consisting of filled levels with negative energy and one electron situated on a level with positive energy, and of a system consisting only of electrons filling the levels with negative energy in the same field. In doing so, he calculated only the terms proportional to \(H\). These terms turned out to be finite and gave the value of the magnetic moment (3.17)*).

Feynman’s method has the shortcoming that, by actually changing the law of interaction of charges with the field, it violates the system of quantum electrodynamics, introducing still other fictitious fields characterized by the function \(G(\lambda)\). Therefore this method must be regarded as a formal relativistic cut-off of divergent integrals, not affecting the final result.

Another path was developed in the works of French and Weisskopf \(^{25}\), Kroll and Lamb \(^{12}\), and Galanin \(^{19}\). The new point of view is set forth most consistently in the first of these works.

In this work the authors do not require that the Hamiltonian be corrected by the addition of a relativistically invariant term of the type \(\beta\Delta\mu c^2\). Instead, they set the problem of finding such an operator \(M\) (no longer having the form \(\beta\Delta\mu c^2\)) that its value, averaged with the aid of the proper function of the free electron, would give the expression for the proper energy of the latter.

*) Let us note that Welton’s nonrelativistic calculations \(^{10}\) give an incorrect result for this effect.

Having found such an operator, one can then define the observable effect as the difference of the mean values of the operators*)

\[ \Delta E=[H]_{\rm av}-[M]_{\rm av}, \tag{3.19} \]

where \([H]_{\rm av}\) is the mean value of the ordinary Hamiltonian. It is obvious that the calculation of (3.19) with the eigenfunctions of a free electron gives an identical zero.

In such a formulation, however, the method possesses an ambiguity in the definition of the operator \(M\), connected with the fact that mean values determine only the diagonal elements of the operator, while the nondiagonal elements remain arbitrary.

This ambiguity can be eliminated if one makes use of the independence, already noted above, of the anomalous magnetic moment from the method of calculation.

Having defined in some way the operator \(M\), we can obviously add to it any operator having no diagonal elements, whose mean value, taken for a free electron, is equal to zero, whereas the mean value for a bound electron is different from zero. It can be shown (see \(^{12}\)) that, up to terms of order \(1/c^4\), the only such operator will be the operator (or an operator proportional to it)

\[ T=\frac{\alpha}{3\pi m}(\beta p^2-\alpha p). \tag{3.20} \]

The mean value of such an operator can be calculated with the aid of Pauli functions and is equal to the mean value of the operator

\[ T\to \frac{\alpha}{6\pi}\frac{\hbar}{mc}X+\frac{\alpha}{3\pi}eaA. \tag{3.21} \]

The second term, proportional to \(\alpha A\), has no physical meaning and leads simply to a renormalization of the charge (in the Hamiltonian there is already a term \(\sim \alpha A\), so that, by changing the charge, one can collect all terms of this type and renormalize the charge, just as was done with the mass).**)

The operator \(X\) is the operator which arises in the nonrelativistic approximation from the operator of interaction of the magnetic moment with the electric field, \(-i\beta\alpha E\) (see (3.16)), and is equal in a static field to

\[ X=\frac{1}{mc}\nabla(\varphi\sigma\times p)-\frac{1}{2}\frac{\hbar}{mc}\Delta\varphi. \tag{3.22} \]

*) The mean values must be calculated with the wave functions of the electron in the Coulomb field, i.e. taking as the perturbation only the interaction with the electromagnetic field of the vacuum.

**) Charge renormalization, generally speaking, is possible only in the case when, in an arbitrary electromagnetic field, terms of the type \(c_1e\varphi-c_2e\alpha A\) appear in the Hamiltonian, with \(c_1=c_2\). In Feynman’s theory this is indeed the case (\(c_1=c_2=0\)). In the other variants \(c_1\ne c_2\), and these terms are simply discarded, without attention being paid to their noninvariant character.

We see that the addition of the operator \(T\) leads to the addition of a term \(\sim \Delta \varphi\), i.e., of a constant term to the logarithm in the formula for the displacement of the term (as we have already said above), and to a simultaneous change in the magnitude of the interaction of the anomalous magnetic moment with the electric field (vanishing in the \(s\)-state).

However, the relativistic invariance of the theory requires that the magnitude of the interaction of the magnetic moment with the electric field be determined by the magnitude of its interaction with the magnetic field*).

But the magnitude of the interaction with the magnetic field is determined uniquely. Therefore the coefficient of the term with the operator \(X\) (or \(-i\beta\boldsymbol{\alpha}\mathbf{E}\)) is fixed. Thereby the operator \(M\) also proves to be fixed. Namely, it must be such that the terms with \(\mathbf{B}\) and \(\mathbf{E}\) are the same as in formula (3.16), obtained in Feynman’s theory.

The final expression for the operator \(M\) has the form:

\[ M=-\frac{e^{2}}{\hbar c}\,\frac{\hbar c^{3}}{4\pi^{2}} \int \frac{d\mathbf{k}}{k} \left[ \sum_{\lambda}^{\prime} \frac{\alpha_{\lambda}\Lambda_{+}\alpha_{\lambda}} {E_{f}-H+\hbar ck} - \frac{\alpha_{\lambda}\Lambda_{-}\alpha_{\lambda}} {E_{f}+H+\hbar ck} \right], \tag{3.23} \]

where

\[ \Lambda_{\pm}=\frac{1}{2} \left(1\pm \frac{c\boldsymbol{\alpha}\mathbf{p}+\beta\mu c^{2}}{E_{f}}\right); \qquad H=c\boldsymbol{\alpha}\mathbf{p}+\mu c^{2}\beta \tag{3.24} \]

is the projection operator which usually arises in summation over spins (cf., for example, \({}^{32}\), p. 166). The prime on the summation sign means that the term with \(\lambda=4\) is to be taken with a minus sign, while \(\alpha_i\) \((i=1,2,3)\) are the Dirac matrices, and \(\alpha_4=1\) (\(\alpha_3\) is the projection on \(\mathbf{k}\)).

If one substitutes the value of \(\Lambda_{\pm}\) into (3.23) and sums over \(i\), then, since

\[ \sum_{\lambda}^{\prime}\alpha_{\lambda}\alpha_{\lambda}=2,\qquad \sum_{\lambda}^{\prime}\alpha_i\alpha_{\lambda}\alpha_i=-2\alpha, \qquad \sum_{\lambda}^{\prime}\alpha_{\lambda}\beta\alpha_{\lambda}=-4\beta, \]

as a result we arrive at the operator for the proper mass of the electron (3.4).

Having thus determined the operator \(M\), French and Weisskopf obtain for the change in the energy an expression exactly coinciding with Feynman’s expression (and also with those of other authors).

Thus it turns out that the condition of relativistic invariance is sufficient to determine uniquely the corrections to the Hamiltonian.

*) That is, in order that in the addition to the Hamiltonian

\[ c_3\,\frac{1}{2}\,\frac{\hbar}{mc}\,X + c_4\,\frac{1}{2}\,\frac{e\hbar}{mc}\,\boldsymbol{\beta}_0\mathbf{B} \]

one have \(c_3=c_4\), in other words, that the magnitude of the magnetic moment measured in an electric field and in a magnetic field be one and the same.

4. FORMULA FOR THE SHIFT OF A TERM

As a result of the calculations briefly described in the preceding paragraph, the following formula is obtained for the operator determining the shift of a level in an electric field:

\[ \frac{1}{3\pi}\frac{e^2}{\hbar c}\left(\frac{\hbar}{mc}\right)^2 \left\{(-e)(\Delta\varphi)\left[\int_{k_{\min}}^{\mu c/\hbar}\frac{dk}{k} -\ln 2+\frac{11}{24}-\frac{1}{5}\right] -\frac{3}{4}X\right\}. \tag{4.1} \]

For the hydrogen atom and atoms similar to it one can show that

\[ \left[\left(\frac{\hbar}{mc}\right)^2(-e)\Delta\varphi\right]_{\mathrm{av}} = \begin{cases} 8\left(\dfrac{e^3}{\hbar c}\right)^2\dfrac{Z^4}{n^3}R, & \text{for } l=0,\\[6pt] 0, & \text{for } l\ne 0, \end{cases} \]

\[ \left[\left(\frac{\hbar}{mc}\right)^2X\right]_{\mathrm{av}} = \begin{cases} -4\left(\dfrac{e^3}{\hbar c}\right)^2\dfrac{Z^4}{n^3}R \dfrac{1}{(l+1)(2l+1)} \left(j=l+\dfrac{1}{2}\right),\\[8pt] +4\left(\dfrac{e^3}{\hbar c}\right)^2\dfrac{Z^4}{n^3}R \dfrac{1}{l(2l+1)} \left(j=l-\dfrac{1}{2}\right). \end{cases} \]

Let us note that for all terms, except \(S\), the shift is determined only by the magnetic moment (the operator \(X\)).

Thus, for the shift of the hydrogen \(S\)-term we obtain:

\[ W'=\frac{8}{3\pi}\left(\frac{e^3}{\hbar c}\right)^3R\,\frac{1}{n^3} \left[\ln\frac{mc}{\hbar k_{\min}}-\ln 2+\frac{19}{30}\right], \tag{4.2} \]

for the term \(P_{1/2}\)

\[ W'=-\frac{1}{3\pi}\left(\frac{e^2}{\hbar c}\right)^3R\,\frac{1}{n^3}, \tag{4.3} \]

for the term \(P_{3/2}\)

\[ W'=\frac{1}{6\pi}\left(\frac{e}{\hbar c}\right)^3R\,\frac{1}{n^3}. \tag{4.4} \]

Using the value of the logarithm calculated by Bethe (see \(^{13}\), p. 1246), equal to 7.6876, we obtain numerically:

\[ W'(2S_{1/2})=1034\ \text{Mc}, \]

\[ W'(2P_{1/2})=-17\ \text{Mc}, \]

\[ W'(2P_{3/2})=8\ \text{Mc}, \]

and for the splitting \(2S_{1/2}-2P_{1/2}\) we obtain the value

\[ 1034+17=1051\ \text{Mc}. \]

5. MEASUREMENTS OF THE MAGNETIC MOMENT OF THE ELECTRON

The first indications of discrepancies between the theories of the interaction of the electron with a magnetic field were obtained in the works of Nafe, Nelson, and Rabi \(^{25}\) and of Nagley, Julian, and Zacharias \(^{26}\).

These authors measured the magnitude of the hyperfine splitting of the levels of hydrogen and deuterium. According to the theory of this effect, the magnitude of the splitting should be equal (in frequencies) to

\[ \nu=\frac{4}{3}\,\frac{2i+1}{i}\,\mu_N\mu_e|\psi(0)|^2, \tag{5.1} \]

where \(i\) is the spin of the nucleus \(\left(\tfrac{1}{2}\right.\) for H and \(1\) for D), \(\mu_N\) is the magnetic moment of the nucleus, \(\mu_e\) is the magnetic moment of the electron, which is taken to be equal to the Bohr magneton \(\mu_0\), and \(\psi(0)\) is the value of the electron wave function at the center of the atom [formula (2.16)]. As it turned out, neither the magnitudes themselves \(\nu_{\mathrm H}\) and \(\nu_{\mathrm D}\), nor their ratio \(\nu_{\mathrm H}/\nu_{\mathrm D}\) (from which \(\mu_e\) drops out*) agree with the usual theory.

Table I **)

	Experiment	Theory	Experiment / Theory
\(\nu_{\mathrm H}\)	\(1420.410 \pm 0.006\)	\(1416.97 \pm 0.54\)	\(1.00242 \pm 0.0004\)
\(\nu_{\mathrm D}\)	\(327.384 \pm 0.003\)	\(326.53 \pm 0.12\)	\(1.00262 \pm 0.0003\)
\(\nu_{\mathrm H}/\nu_{\mathrm D}\)	\(4.33867 \pm\) \(\pm 0.00004\)	\(4.339385 \pm\) \(\pm 0.00003\)	\(0.999835 \pm\) \(\pm 0.000001\)

Table I gives the theoretical and experimental (from a later and more precise work \(^{27}\)) values of these quantities.

The errors indicated for the theoretical values are connected with the inaccuracy of the constants used ***). In the last column the ratio of the experimentally measured and calculated values is given.

Leaving aside for the moment the very small discrepancy (less than \(0.017\%\)) in the values of the ratio \(\nu_{\mathrm H}/\nu_{\mathrm D}\), let us consider the quantities \(\nu_{\mathrm H}\) and \(\nu_{\mathrm D}\) themselves. The measured values of these quantities show that the magnetic moment of the electron indeed exceeds \(\mu_0\).

\[ \text{*) When forming the ratio } \nu_{\mathrm H}/\nu_{\mathrm D} \text{ one should remember that into } \psi(0) \]
enter the reduced masses of the electrons in the hydrogen atom \(m_{\mathrm H}\) and deuterium \(m_{\mathrm D}\). Therefore this ratio is equal to

\[ \frac{\nu_{\mathrm H}}{\nu_{\mathrm D}} = \frac{4}{3} \left(\frac{m_{\mathrm H}}{m_{\mathrm D}}\right)^3 \frac{\mu_p}{\mu_d}. \tag{5.2} \]

**) All quantities are given in Mc/sec.

***) The small error in \(\nu_{\mathrm H}/\nu_{\mathrm D}\) is connected with the very high degree of accuracy in the measurements of the ratio \(\mu_p/\mu_d\) \((\sim 10^{-4}\%)\).

In order to compute, from the experimental data, the value of the magnetic moment of the electron, it is necessary to take into account the fact that in experiments on measuring the magnetic moment of the nucleus by the molecular-beam method (the resonance method) what is actually measured is the ratio of the magnetic moment of the nucleus to the magnetic moment of the electron*).

Therefore, for comparison with experiment, (5.1) must be rewritten in the form:

\[ \nu=-\frac{4}{3}\,\frac{2i+1}{i}\,\mu_e^2\,\frac{\mu_N}{\mu_e}\,|\psi(0)|^2 . \tag{5.3} \]

If we put

\[ \mu_e=\mu_0(1+\delta), \tag{5.4} \]

then the ratio of the measured value \(\nu_{\text{expt}}\) to the theoretical value \(\nu_{\text{theor}}\) (computed for \(\mu_e=\mu_0\)) will be equal to

\[ \frac{\nu_{\text{expt}}}{\nu_{\text{theor}}}=1+2\delta . \tag{5.5} \]

From Table I it is seen that, if the values of \(\nu_H\) are used, then

\[ \delta=0.0012\pm0.0002 \tag{5.6} \]

in excellent agreement with the theoretical value

\[ \frac{1}{2\pi}\,\frac{e^2}{\hbar c}=0.00116 . \]

Within the framework of the theory set forth, there still remains unexplained the discrepancy between the experimental and theoretical values of the ratio \(\dfrac{\nu_H}{\nu_D}\). This discrepancy was explained by O. Bohr \(^{28}\) and is connected with the finite size of the deuteron. Since the latter problem is not directly connected with the anomalous magnetic moment of the electron, its discussion is placed in Appendix II.

Further measurements of the magnetic moment of the electron were carried out by Kusch and Foley \(^{29}\), who studied the Zeeman effect in atoms with one valence electron—gallium, indium, and sodium (resonance method)**).

In order to avoid measuring the magnetic field, in this work the ratio of the splitting of two terms was determined; from this the ratio of the Landé factors for these terms was found.

The results are given in Table II, in which the terms studied, the theoretical value of the ratio of the Landé factors, the measured ratio, and, finally, the value of the correction to the magnetic moment \(\delta\) calculated from this are indicated.

*) In this method the constant field is calibrated by measuring the fine structure. In computing the magnitude of the field from the observed splitting, the value of the magnetic moment of the electron is used.

**) We note that in these spectra the corrections for deviations from Russell–Saunders coupling turn out to be smaller than the observed effect.

Table II

Quantity studied	Theoretical value for $\mu_e=\mu_0$	Measured ratio	$\delta$
$\dfrac{g({}^{2}P_{3/2},\ \mathrm{Ga})}{g({}^{2}P_{1/2},\ \mathrm{Ga})}$	2	$2(1.00172 \pm 0.00006)$	$0.00114 \pm 0.00004$
$\dfrac{g({}^{3}S_{1/2},\ \mathrm{Na})}{g({}^{2}P_{1/2},\ \mathrm{Ga})}$	3	$3(1.00242 \pm 0.00006)$	$0.00121 \pm 0.00003$
$\dfrac{g({}^{3}S_{1/2},\ \mathrm{Na})}{g(P_{1/2},\ \mathrm{In})}$	3	$3(1.00243 \pm 0.00010)$	$0.00121 \pm 0.00005$
		Average value:	$0.00119 \pm 0.00005$

The result is again in excellent agreement with theory. Attention is drawn to the fact that the last two values coincide exactly, whereas the first is somewhat smaller. Since this discrepancy lies within the limits of experimental error, there is as yet no reason to regard this discrepancy as real.

6. OTHER EFFECTS

The question naturally arises as to what other effects exist whose calculation becomes possible. First of all, this is the determination of corrections to various cross sections calculated within the framework of ordinary perturbation theory (the magnitude of these corrections should differ from the cross section itself by a factor of $e^2/\hbar c$).

Up to now there have been no experiments in which cross sections have been measured with such accuracy, and therefore these corrections at present have only theoretical significance.

Only two effects have been studied theoretically, namely the scattering of an electron in a Coulomb field and the Compton effect. The first problem is, obviously, a complete analogue of the problem of the displacement of levels. Their difference consists only in the fact that, instead of diagonal matrix elements, the scattering problem contains non-diagonal elements of the same operator. Lewis[^15] and Epstein[^16] first showed that all infinite terms arising in the calculation of the correction to the scattering cross section, associated with virtual emission and absorption

quanta arise from the electromagnetic mass and can be eliminated by the corresponding mass renormalization. In this case, as in the case of level shifts, it turns out that the result has only logarithmic accuracy.

The origin of the correction to scattering can be explained more or less visually as follows (not quite precisely (cf. 19)).

In the course of scattering the electron can virtually emit and absorb quanta. If both the emission and the absorption occur before the scattering process (or after it), then this is equivalent to a simple change of mass—the corresponding terms are infinite and disappear under mass renormalization. If, however, the emission of a virtual quantum occurs before the scattering and the absorption after it, then such an effect is no longer reducible to a mass effect and leads to the correction to the cross section under consideration.

The radiative correction to the scattering cross section was calculated by Schwinger1. He showed that scattering is always accompanied by the emission of quanta of small energy. If, in calculating the cross section, one takes into account the possibility of emission of (virtual and real) quanta with energies from \(0\) to \(\Delta E\) (\(\Delta E\) small compared with the kinetic energy of the electron), then the ordinary scattering cross section is multiplied by the factor \((1-\delta)\), where, in the case of nonrelativistic electron velocities (for a given scattering angle \(\vartheta\)),

\[ \delta= \frac{8}{3\pi}\frac{e^2}{\hbar c} \left(\frac{v}{c}\right)^2 \left[ \ln \frac{mc^2}{\Delta E}+\frac{19}{30}-\ln 2 \right]\sin^2\frac{\vartheta}{2}. \tag{6.1} \]

In the limiting relativistic case \(\delta\) is determined by the expression

\[ \delta= \frac{4}{\pi}\frac{e^2}{\hbar c} \left[ \left(\ln \frac{E}{\Delta E}-\frac{13}{12}\right) \left(\ln \frac{2E}{mc^2}\sin\frac{\vartheta}{2}-\frac{1}{2}\right) +\frac{17}{72}+\varphi(\vartheta) \right], \tag{6.2} \]

where

\[ \varphi(\vartheta)= \frac{1}{2}\sin\frac{\vartheta}{2} \int_{\cos \frac{\vartheta}{2}}^{1} \left( \frac{\ln \frac{1}{2}(1+x)}{1-x} - \frac{\ln \frac{1}{2}(1-x)}{1+x} \right) \times \frac{dx}{\left(x^2-\cos^2\frac{\vartheta}{2}\right)^{1/2}}. \tag{6.3} \]

For \(\vartheta=\pi\), \(\varphi(\vartheta)=\dfrac{\pi^2}{24}\); for small angles \(\vartheta\),

\[ \varphi(\vartheta)\sim \frac{1-\cos\frac{\vartheta}{2}} {\left[ 2\cos\frac{\vartheta}{2}\left(1+\cos\frac{\vartheta}{2}\right) \right]^{1/2}} \left[ -\ln 2\left(1-\cos\frac{\vartheta}{2}\right) +\frac{1}{2}\left(1-\cos\frac{\vartheta}{2}\right)+1 \right]. \tag{6.4} \]

In the general case of arbitrary energies the formula has a very complicated form, and we shall not give it here. Let us note that (6.2) has a noticeable inaccuracy only when the electron energy is several MeV.

The magnitude of the correction (for \(\Delta E\) of several tens of KeV) reaches \(5\)—\(10\%\). Thus, for an electron with energy \(3.1\) MeV at \(\vartheta=\dfrac{\pi}{2}\), \(\Delta E=10\) KeV, \(\delta=8.6\%\).

These formulas are incorrect for very small \(\Delta E\), since they diverge logarithmically. This difficulty (connected, as always, with the emission of many quanta of small energy) can be formally eliminated if the correction factor \((1-\delta)\) is replaced by \(e^{-\delta}\) and the higher powers of \(\delta\) are regarded as a correction for multiple processes.

However, such corrections have no practical significance.

Schafroth\({}^{34}\) calculated the magnitude of the correction to the cross section of the Compton effect on the electron. Accurate up to terms \(\sim k^2\) (\(\mathbf{k}\) is the wave vector of the photons), the effective cross section for processes in which, in addition to the scattered photon (into the solid angle \(d\Omega\)), at least one more photon with energy \(<\hbar\omega\) is emitted has the form:

\[ d\sigma'= \left(\frac{e^2}{\hbar c}\right) \left(\frac{e^2}{mc^2}\right)^2 \frac{d\Omega}{4\pi} \left(\frac{\hbar k}{mc}\right)^2 \left\{ (1+\cos^2\vartheta) \left( 2\ln\frac{\hbar k}{mc} +\frac{8}{3}\ln\frac{\hbar\omega}{mc^2} \right) \right. \]
\[ \left. -\frac{8}{3}(1+\cos^2\vartheta)\ln\frac{\hbar\omega}{mc^2} -\frac{4}{3}(1-\cos^2\vartheta)\cos\vartheta \ln\frac{\hbar k}{mc} \right\}. \tag{6.5} \]

The formula for the general case of scattering of a photon of arbitrary energy has not yet been published.

ADDENDA

I. Convergence of integrals at low frequencies

In calculating the term shift of hydrogen, a difficulty was noted connected with the divergence of integrals at low frequencies. This difficulty has nothing in common with the fundamental questions of relativistic effects and is encountered in all nonrelativistic processes connected with the emission of quanta of small energies.

As was shown by Pauli and Fierz\({}^{31}\), at low frequencies the probability of emitting one quantum becomes small in comparison with the probability of emitting many quanta, and therefore ordinary perturbation theory proves inapplicable in this region.

In order to eliminate this difficulty, Pauli and Fierz pass to new variables (perform a canonical transformation), so that at low frequencies the quanta turn out to be, in a certain sense, bound to the electrons. This should lead to the appearance of a lower limit in the corresponding integrals, depending on the possible states of the electron \((k_{\min}=(\Delta E)_{\mathrm{cp}}\) in the problem of the level shift.) The meaning of this operation is clearest if

consider the Pauli and Fierz transformation in the form given in the work cited above.^11

Let us consider the Hamiltonian of an electron in an electromagnetic field in the nonrelativistic approximation with respect to the electron (see, for example,^32):

\[ H=\frac{p^{2}}{2m}+V(\mathbf q)+\frac{1}{8\pi}(E^{2}+H^{2})-\frac{e}{2m}(\mathbf A\mathbf p), \tag{D.1} \]

where \(\mathbf p\) and \(\mathbf q\) are the momentum and coordinate of the electron (we omit the term with \(A^{2}\), regarding the field as weak).

Expanding the vector potential in a Fourier series:

\[ \mathbf A=(8\pi\hbar c)^{\frac12}\sum_{k}\frac{a_k}{k^{1/2}}e^{i(\mathbf k\mathbf r-\omega t)}\mathbf e_k, \tag{D.2} \]

where \(\mathbf e_k\) is the polarization vector, and the coefficients are chosen so that the field energy has the usual form (i.e., so that the operator \(a_k^{+}a_k\) has eigenvalues \(0,1,2,\ldots\)), and substituting into (D.1), we obtain in the usual way:

\[ H=\frac{p^{2}}{2m}+V(\mathbf q)+\sum_{k}a_k^{+}a_k\hbar ck -\frac{e}{mc}\sum_{k}\left(\frac{2\pi\hbar c}{k}\right)^{\frac12} (\mathbf e_k\mathbf p)(a_k+a_k^{+}). \tag{D.3} \]

Let us complete the sum to a perfect square, rewriting (D.3) as follows:

\[ H=\frac{p^{2}}{2m}+V(\mathbf q)+\sum_{k} \left(a_k^{+}-\frac{e}{mc}\left(\frac{2\pi}{\hbar c k^{3}}\right)^{1/2}\mathbf e_k\mathbf p\right)\times \]

\[ \times\left(a_k-\frac{e}{mc}\left(\frac{2\pi}{\hbar c k^{3}}\right)^{1/2}\mathbf e_k\mathbf p\right)\hbar ck -\frac{e^{2}}{m^{2}c^{2}}\sum_{k}\frac{2\pi}{k^{2}}(\mathbf e_k\mathbf p)^{2}. \tag{D.4} \]

The last term is nothing other than the electromagnetic energy of the electron (nonrelativistic):

\[ -\frac{e^{2}}{m^{2}c^{2}}\sum_{k}\frac{2\pi}{k^{2}}(\mathbf e_k\mathbf p)^{2} = \frac{2e^{2}}{3}\frac{p^{2}}{m^{2}c^{2}}\int dk. \tag{D.5} \]

Discarding this term, we arrive at the Hamiltonian

\[ H=\frac{p^{2}}{2m}+V(\mathbf q)+\sum A_k^{+}A_k\hbar ck, \tag{D.6} \]

where

\[ A_k=a_k-\frac{e}{mc}\left(\frac{2\pi}{\hbar c k^{3}}\right)^{1/2}(\mathbf e_k\mathbf p). \tag{D.7} \]

In order for the new \(A_k\) to be able to serve as canonical variables, it is necessary to introduce also new canonical variables for the electron (only in this case will \(A_k\) commute with the coordinate and momentum of the electron):

\[ \mathbf P=\mathbf p, \]

\[ \mathbf Q=\mathbf q+\frac{ie}{mc}\sum_k\left(\frac{2\pi\hbar}{ck^3}\right)^{1/2}\mathbf e_k\left(A_k-A_k^+\right); \tag{D1.8} \]

then the Hamiltonian is transformed to the form:

\[ H=\frac{p^2}{2m}+V\left[\mathbf Q-\frac{ie}{mc}\sum_k\left(\frac{2\pi\hbar}{ck^3}\right)^{1/2}\mathbf e_k\left(A_k-A_k^+\right)\right]+ \]

\[ +\sum_k A_k^+A_k\,\hbar ck. \tag{D1.9} \]

We see that, as a result of the operations performed, we have arrived at a Hamiltonian in which the connection between the electron and the field is expressed by the fact that the coordinate entering into the potential energy becomes dependent on the state of the field, i.e. precisely what serves as the starting point for deriving the formula for the level shift (see Section 2).

Let us introduce the usual coordinates and momenta \((Q_k\) and \(P_k)\) of the oscillators of the electromagnetic field*:

\[ P_k=\left(\frac{1}{2}\hbar ck\right)^{1/2}\left(A_k^+ + A_k\right), \tag{D1.10} \]

\[ Q_k=i\left(\frac{\hbar}{2ck}\right)^{1/2}\left(A_k^+ - A_k\right). \]

We finally obtain:

\[ H=\frac{P^2}{2m}+V(\mathbf q+\Delta \mathbf q)+\sum_k\frac{1}{2}\left(P_k^2+c^2k^2Q_k^2\right), \tag{D1.11} \]

where

\[ \Delta \mathbf q=\frac{e}{mc}(4\pi)^{1/2}\sum_k \mathbf e_k\,\frac{Q_k}{k}. \tag{D1.12} \]

Thus, the interaction of the electron with the radiation field includes an averaging of the potential over the coordinates of all oscillators of the field. Such averaging will take place differently for small and large wave vectors.

Let us now show, first of all, that at large (but still nonrelativistic) frequencies one obtains a result coinciding with that obtained in Section 2.

*) The coefficients are chosen so that the commutator \([P_k Q_k]=-i\hbar\).

If the frequencies are large, these quanta may be regarded as free, and the averaging of the potential is carried out with the aid of the wave functions of the free quanta.

Expanding \(V(q)\) in a three-dimensional Fourier integral:

\[ V(q)=\frac{1}{(2\pi)^{3/2}}\int d\lambda\, U(\lambda)\exp(i\lambda q). \tag{D1.13} \]

Then

\[ V(q+\Delta q)= \]

\[ =\frac{1}{(2\pi)^{3/2}}\int d\lambda\, U(\lambda)\exp(i\lambda q) \prod_k \exp\left[\frac{ie}{mc}(4\pi)^{1/2}(e_k\lambda)\frac{Q_k}{k}\right]. \tag{D1.14} \]

The wave function of a quantum (oscillator), as is known, is a Gaussian function (the first Hermite function)

\[ \varphi(Q_k)=\left(\frac{ck}{\pi\hbar}\right)^{1/4} e^{-\frac{ck}{2\hbar}Q_k^2}; \tag{D1.15} \]

the wave function of all quanta is the product of all \(\varphi(Q_k)\):

\[ \varphi=\prod_k \varphi(Q_k). \tag{D1.16} \]

Averaging (D1.14) with its aid, we find:

\[ [V(q+\Delta q)]_{\mathrm{av}}= \]

\[ =\frac{1}{(2\pi)^{3/2}}\int d\lambda\, U(\lambda)\exp(i\lambda q) \prod_k \exp\left(-\pi(e_k\lambda)^2\frac{e^2}{\hbar c} \left(\frac{\hbar}{mc}\right)^2\frac{1}{k^3}\right). \tag{D1.17} \]

Comparing with the expression for \((\Delta q^2)_{\mathrm{av}}\) obtained in Section 2, this expression can be rewritten as follows:

\[ [V(q+\Delta q)]_{\mathrm{av}}= \frac{1}{(2\pi)^3}\int\!\!\int d\lambda\,dq'\, e^{i\lambda(q-q')}e^{-\frac{1}{6}\lambda^2(\Delta q^2)_{\mathrm{av}}}. \tag{D1.18} \]

This expression can be integrated with respect to \(\lambda\). As a result we obtain:

\[ [V(q+\Delta q)]_{\mathrm{av}}= \frac{3}{2\pi(\Delta q^2)_{\mathrm{av}}} \int dq'\, V(q')\exp\left[-\frac{3}{2}\frac{|q-q'|^2}{(\Delta q^2)_{\mathrm{av}}}\right]. \tag{D1.19} \]

(D1.17) can formally be written in the form\(^*\):

\[ [V(q+\Delta q)]_{\mathrm{av}}= \exp\left[\frac{1}{6}(\Delta q^2)_{\mathrm{av}}\Delta\right]V(q). \tag{D1.20} \]

\[ \overline{\phantom{xxxxxxxxxxxxxxxx}} \]

\(^*\) This expression should be understood as follows. Let us expand \(V(q)\) in a Fourier integral, and formally represent the exponent also in the form of a series:

\[ 1+\frac{1}{6}(\Delta q^2)_{\mathrm{av}}\Delta+\ldots; \]

after applying the operator series to \(V(q)\), collecting again the expressions that arise into an exponent, we arrive at (D1.18).

SHIFT OF TERMS OF HYDROGEN-LIKE ATOMS

Expression (ДI.20) coincides with (2.20)*).

Let us now pass to small quantum energies. We apply to the new Hamiltonian (ДI.11) perturbation theory, considering as the perturbation (to within terms of order \(e^2\))

\[ V(\mathbf q+\Delta \mathbf q)-V(\mathbf q) = \frac{e}{mc}(4\pi)^{\frac12}\sum_k \frac{1}{k}\, Q_k(\mathbf e_k\nabla)V(\mathbf q) + \frac12\left(\frac{e}{mc}\right)^2 4\pi \left[\sum_k Q_k(\mathbf e_k\nabla)\right]^2 V(\mathbf q). \tag{ДI.21} \]

The correction to the energy in second order (in \(e\)) perturbation theory has the form (the notation is the same as in Section 2):

\[ W' = 4\pi\left(\frac{e}{mc}\right)^2 \sum_k\sum_n \frac{\hbar}{2ck^3}\, \frac{\left|(0|\mathbf e_k\nabla V|n)\right|^2} {E_0-E_n-\hbar ck} + \]

\[ + \frac12\,4\pi\left(\frac{e}{mc}\right)^2 \sum_k \frac{\hbar}{2ck^3}\, (0|(\mathbf e_k\nabla)^2 V|0). \tag{ДI.22} \]

After replacing the sum over \(k\) by an integral, integrating over angles, and summing over the polarizations of the quantum, this expression takes the form:

\[ W' = \frac{1}{3\pi}\frac{e^3}{\hbar c} \left(\frac{\hbar}{mc}\right)^2 \int \frac{dk}{k} \left\{(0|\Delta V|0)+ \right. \]

\[ \left. +\,2\sum_n \frac{(0|\nabla V|n)(n|\nabla V|0)} {E_0-E_n-\hbar ck} \right\}. \tag{ДI.23} \]

If \(k\) is very small, so that \(\hbar ck \ll E_0-E_n\) (for all \(E_n\)), then, expanding in a series, we obtain:

\[ 2\sum_n \frac{(0|\nabla V|n)(n|\nabla V|0)} {E_0-E_n-\hbar ck} = 2\sum_n(0|\nabla V|n)(n|\nabla V|0)\times \]

\[ \times \left\{ \frac{1}{E_0-E_n} + \frac{2\hbar ck}{(E_0-E_n)^3} + O(k^2) \right\}. \tag{ДI.24} \]

It is not difficult to see that the following relation between operators is valid:

\[ \Delta V = -\frac{i}{\hbar}(H\mathbf p-\mathbf pH) = +\frac{d\mathbf p}{dt}. \tag{ДI.25} \]

* Thanks to the fact that here the Laplace operator enters the exponent, the question of terms of higher order, which remained unclear when deriving expression (2.20), does not arise. As \((\Delta q^2)_{\mathrm{cp}}\to\infty\), (ДI.19) tends to zero, and (2.20) to infinity.

Using the fact that for the matrix elements

\[ \left(0\left|\frac{d\rho}{dt}\right|n\right) =\frac{d}{dt}(0|\rho|n) =-\frac{i}{\hbar}(E_0-E_n)(0|\rho|n), \tag{ДI.26} \]

we transform (ДI.24) to the form:

\[ 2\sum_n \frac{(0|\nabla V|n)(n|\nabla V|0)} {E_0-E_n-\hbar ck} = \]

\[ =-(0|\Delta V|0)+\frac{2kc^2}{\hbar}(0|p^2|0)+O(k^2), \tag{ДI.27} \]

where by \(O(k^2)\) are denoted terms proportional to \(k^2\) and to higher powers of \(k\).

Substituting into (ДI.23), we see that the logarithmic divergence disappears. By more exact calculations one can show that for the bound electron one obtains Bethe’s result \([\,\text{effective cutoff at } k_{\min}=\frac{1}{\hbar c}(\Delta E)_{\mathrm{av}}\,]\).

II. Hyperfine Structure of Deuterium

In Section 5 it was indicated that, in measuring the hyperfine splitting of deuterium and hydrogen, it was found that the ratio of the splittings \(\nu_{\mathrm H}/\nu_{\mathrm D}\), independent of the magnetic moment of the electron, turns out to be less than the theoretical value by \(0.017\%\). This discrepancy may be explained by the fact that, in the interaction of the electron with the magnetic moment of deuterium, it is necessary to take into account the finite dimensions of the deuteron\(^{27}\).

Let us consider this question in more detail.

The interaction of the electron with the magnetic moment may be described by means of the operator

\[ H'=\frac{8}{3}\,\frac{2i+1}{i}\,\frac{e\hbar}{2mc}\,(\boldsymbol{\mu}\mathbf S)\,\delta(\mathbf r), \tag{ДII.1} \]

where \(\boldsymbol{\mu}\) and \(i\) are the spin and magnetic moment of the nucleus, \(\mathbf S\) is the spin of the electron, and \(\delta(\mathbf r)\) is the \(\delta\)-function of the vector \(\mathbf r\): \(\int \delta(\mathbf r)\,d\mathbf r=4\pi\int \delta(r)r^2dr=1\).

The correctness of formula (ДII.1) follows from the fact that if one multiplies \(H'\) by \(\psi^2\) and integrates over space, one obtains the usual formula for the hyperfine structure (5.1).

Considering the magnetic interaction of the electron with the deuteron, one must take the sum of two operators of the form (ДII.1)—one for the interaction with the proton, the other for the interaction with the neutron. In doing so, since the electron interacts electrically only with the proton, it may be assumed that the electron in its motion

MIXING OF TERMS OF HYDROGEN-LIKE ATOMS

one must follow the motion of the proton; the interaction with the neutron, however, must be averaged over the volume of the deuteron*).

It is obvious that under such averaging the interaction of the electron with the neutron will decrease in absolute value, and since the magnetic moment of the neutron is negative, the resulting value of the hyperfine splitting will be greater than that obtained directly from formula (DII.1), if in it one substitutes the spin and magnetic moment of the deuteron. Let us note that in calculating the correction we must consider only the coordinate parts of the wave functions, and therefore the factor \(\dfrac{2i+1}{i}\) in (DII.1) must be omitted.

In order to calculate the magnitude of the correction**), let us denote by \(\mathbf r\) the distance from the neutron to the electron and by \(\mathbf R\) the distance from the proton to the neutron. Then

\[ \rho=\mathbf R+\mathbf r \tag{DII.2} \]

will be the distance from the electron to the proton.

The wave function of the system electron plus neutron is the product

\[ (4\pi)^{-1/2}\frac{\chi(R)}{R}\,\varphi(\rho), \tag{DII.3} \]

where \((4\pi)^{-1/2}\dfrac{\chi(R)}{R}\) is the wave function of the deuteron, and \(\varphi(\rho)\) is the hydrogen wave function (in view of what has been said, we regard the electron as bound only to the proton). We assume both functions to be normalized:

\[ \int \chi^2(R)\,dR=1;\qquad \int \varphi^2(\rho)\,d\rho=4\pi\int \varphi^2(\rho)\rho^2\,d\rho=1. \tag{DII.4} \]

Substituting now in (DII.1) \(i=\dfrac12\) and \(\boldsymbol\mu=\boldsymbol\mu_n\) (the magnetic moment of the neutron), we obtain, with the aid of function (DII.2), for the interaction energy the expression

\[ \varepsilon=A_n\iint \delta(\mathbf r)\chi^2(R)\varphi^2(\rho)\,dR\,d\rho, \tag{DII.5} \]

where

\[ A_n=\frac{4}{3}\frac{e\hbar}{mc}(\boldsymbol\mu_n\mathbf S). \]

*) It is obvious that for the magnetic interaction the wave function inside the nucleus is of substantial importance. The frequency corresponding to the motion of the electron inside the nucleus, \(\dfrac{\hbar}{mr_0^2}\), is \(\gg\) the frequency of the neutron, \(\dfrac{\hbar}{Mr_0^2}\) (\(r_0\) is the radius of the nucleus). Hence the correction factor appears in the magnetic interaction. At the same time, the frequency of motion of the electron in the hydrogen orbit is \(\ll \dfrac{\hbar}{Mr_0^2}\), and therefore such a correction does not appear when considering the problem of the levels of the deuterium atom.

**) The derivation differs from that given in N. Bohr’s work and gives a more visual representation of the connection of the effect with the properties of the deuteron.

Introducing, instead of \(\rho\), the variable \(\mathbf r\), according to (DII.2), we obtain:

\[ \varepsilon=A_n\iint \delta(\mathbf r)\chi^2(R)\varphi^2(|\mathbf R+\mathbf r|)\,dR\,dr, \tag{DII.6} \]

or, integrating with respect to \(\mathbf r\),

\[ \varepsilon=A_n\int \chi^2(R)\varphi^2(R)\,dR. \tag{DII.7} \]

If the dimensions of the deuteron could be neglected, then the magnitude of the interaction would be equal to

\[ \varepsilon_0=A_n\varphi^2(0). \tag{DII.8} \]

(This expression is obtained from the preceding one if \(\chi^2(R)\) is replaced by a \(\delta\)-function.) Using the normalization condition for \(\chi(R)\) (DII.4), one may write

\[ \varepsilon-\varepsilon_0 = A_n\varphi^2(0)\int \chi^2(R) \left(\frac{\varphi^2(R)}{\varphi^2(0)}-1\right)dR. \tag{DII.9} \]

The hydrogen wave function of the \(s\)-electron is equal to

\[ \varphi(R)=\varphi(0)e^{-\frac{R}{a}}, \tag{DII.10} \]

where

\[ a=\frac{\hbar^2}{me^2}. \]

Expanding the exponential in a series and retaining only the first nonvanishing term, we obtain from (DII.9):

\[ \Delta\varepsilon = A_n\varphi^2(0)\frac{2}{a}\int \chi^2(R)R\,dR. \tag{DII.11} \]

This correction must be referred to the value of the hyperfine splitting for deuterium \(\varepsilon_0\), computed without taking account of the finiteness of the dimensions of the deuteron. It is determined by formula (DII.9), where in \(A_n\) the magnetic moment of the neutron must be replaced by the magnetic moment of the deuteron. \(\varphi(0)\) may in both cases be regarded as the same, since the difference of the reduced masses of the electron may here be neglected.

Finally, for the magnitude of the relative correction we obtain the expression (the sign of the effect was explained above)

\[ \frac{\Delta\varepsilon}{\varepsilon} = -\frac{|\mu_n|}{\mu_d}\frac{2}{a}\int \chi^2(R)R\,dR. \tag{DII.12} \]

Thus, the magnitude of the correction is determined by the mean distance between the particles in the deuteron.

This quantity can be determined approximately (with an error of less than \(10\%\)) by means of the elementary theory of the deuteron.

It can be shown\(^ {33}\) that, with great accuracy, for not too small \(R\), the deuteron wave function has the form:

\[ \chi(R)=\sqrt{\frac{3}{d}}\,e^{-R/d}, \tag{DII.13} \]

where \(d\) is the deuteron radius:

\[ d=\frac{\hbar}{(M\varepsilon_d)^{1/2}}; \tag{DII.14} \]

\(\varepsilon_d\) is the binding energy of the deuteron, \(M\) is the proton mass\(^*\).

Since, in the expression for the mean distance, small \(R\) do not play a noticeable role, this quantity can be calculated by means of the function (DII.13). As a result we obtain

\[ \int R\chi^2(R)\,dR=\frac{3}{4}d, \tag{DII.15} \]

whence, finally, for the required magnitude of the correction we shall have:

\[ \frac{\Delta\varepsilon}{\varepsilon} = \frac{3}{2}\frac{d}{a}\frac{|\mu_n|}{\mu_d}. \tag{DII.16} \]

Substituting the values of the quantities entering into this formula,

\[ \frac{d}{a} = \frac{e^2}{\hbar c} \sqrt{\frac{m}{M}\frac{mc^2}{\varepsilon_d}} = 0.81\cdot 10^{-1}, \]

we obtain

\[ \frac{\Delta\varepsilon}{\varepsilon}=2.6\cdot 10^{-4}, \tag{DII.17} \]

which, although it agrees in order of magnitude, is one and a half times greater than the observed effect\(^ {**}\). The reason for this discrepancy may be that the magnetic moment of the neutron and of the proton is not pointlike, but is spread out in space (for example, over a distance of the order of the range of nuclear forces).

CITED LITERATURE

1. W. H. Houston, Phys. Rev. 51, 446 (1937).
2. R. C. Williams, Phys. Rev. 54, 558 (1938).
3. W. E. Lamb, Jr., and R. C. Retherford, Rhys. Rev. 72, 241 (1947).
4. R. C. Retherford and W. E. Lamb, Jr., Phys. Rev. 75, 1325 A (1949).
5. J. E. Mack, Phys. Rev. 72, 972 (1948).
6. G. R. Fowles, Phys. Rev. 73, 639 (1948).

\(^*\) The function (D II.13) does not satisfy the normalization condition (D II.4), since (D II.13) is invalid for small \(R\).

\(^ {**}\) O. Bohr, in his article, using an incorrectly normalized deuteron wave function, obtains agreement with experiment and merely notes that, with a correct derivation, such a discrepancy may arise.

1. M. Skinner and W. E. Lamb, Jr., Phys. Rev. 75, 1325A (1949).
2. H. A. Bethe, Phys. Rev. 72, 339 (1947).
3. H. A. Bethe, Phys. Rev. 73, 1271 A (1948).
4. T. A. Welton, Phys. Rev. 74, 1157 (1948).
5. V. Weisskopf, Phys. Rev. 56, 72 (1939).
6. N. M. Kroll and W. E. Lamb, Jr., Phys. Rev. 75, 388 (1949).
7. J. Schwinger, Phys. Rev. 73, 416 (1948).
8. V. Weisskopf, Phys. Rev. 73, 1272 (1948).
9. H. W. Lewis, Phys. Rev. 73, 173 (1948).
10. S. T. Epstein, Phys. Rev. 73, 177 (1948).
11. J. Schwinger, Phys. Rev. 75, 898 (1949); 76, 790 (1949).
12. J. B. French and V. F. Weisskopf, Phys. Rev. 75, 1240 (1949).
13. A. D. Galanin, ZhETF, 19, 521 (1949).
14. R. P. Feynman, Phys. Rev. 74, 430 (1948).
15. F. J. Dyson, Phys. Rev. 73, 617 (1948).
16. Z. Koba, T. Tati and S. Tomonaga, Prog. Theor. Phys. 2, 101, 198, 218 (1947).
17. J. Schwinger, Phys. Rev. 74, 1439 (1948); 75, 651 (1949).
18. J. M. Luttinger, Phys. Rev. 74, 893 (1948).
19. J. E. Nafe, E. B. Nelson and I. I. Rabi, Phys. Rev. 71, 914 (1947).
20. D. E. Nagle, R. S. Julian and J. R. Zacharias, Phys. Rev. 72, 971 (1947).
21. J. E. Nafe and E. B. Nelson, Phys. Rev. 73, 718 (1948).
22. A. Bohr, Phys. Rev. 73, 1109 (1948).
23. P. Kusch and H. M. Foley, Phys. Rev. 74, 25 (1948).
24. E. A. Uehling, Phys. Rev. 48, 55 (1935).
25. W. Pauli and M. Fierz, Nuovo Cimento 15, 167 (1938).
26. V. Heitler, The Quantum Theory of Radiation, GTTI, Moscow—Leningrad, 1940, p. 108.
27. Ya. Smorodinskii, DAN, 40, 217 (1948).
28. R. Schafroth, Phys. Rev. 75, 1111 (1949).

1. Reference number as printed in the source. ↩