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FINE STRUCTURE OF HYDROGEN LINES.
HISTORY AND CURRENT STATE OF THE THEORY1
A. Sommerfeld, Munich
Bohr’s theory in 1913 for the first time gave an explanation of the hydrogen lines \(H_\alpha\), \(H_\beta\), \(H_\gamma\), . . . , which together form the Balmer series. However, it left one point unexplained, namely the doublet structure of these lines found by Michelson. This concerns a very small splitting (Michelson found for \(H_\alpha\) the value \(\Delta \lambda = 0.14\ \text{Å}\)), which can be detected only with the aid of the finest spectroscopic instruments. We therefore speak of the “fine structure of hydrogen lines” in order to distinguish it from the coarser doublet or multiplet structure of hydrogen-like lines (for example, \(\Delta \lambda = 6\ \text{Å}\)—the splitting of the yellow sodium line).
Bohr originally quantized only the circular motion of the electron around the proton, which was expressed in the fact that he used only one quantum number—the “azimuthal quantum number”; in what follows we shall denote it by \(n_\varphi\). In order to explain the doublet character of the hydrogen lines, I had to investigate the general motion of electrons along a Kepler ellipse and also quantize the radial motion, i.e., introduce, alongside \(n_\varphi\), also the radial quantum number \(n_r\). We set
\[ n_r + n_\varphi = n \]
and call \(n\) the principal quantum number. For a given \(n\), the radial quantum number \(n_r\) can take the values
\[ n_r = 0,\ 1,\ 2,\ldots,\ n-1, \tag{1a} \]
and \(n_\varphi\), consequently, the values
\[ n_\varphi = n,\ n-1,\ldots,\ 1. \tag{1b} \]
Thus each simple Bohr energy level splits into \(n\) sublevels; in particular, the fundamental level \(n=2\) of the Balmer lines—into two sublevels. This would already have given an explanation of the doublet character of the Balmer lines, if it had not turned out in calculations by means of classical mechanics that these sublevels coincide with one another—
In fact, the formula for the energy of the electron in the \(n\)-th quantum state is as follows:
\[ E = E_0 - \frac{Rh}{n^2}, \tag{2} \]
where \(E_0\) is the rest energy of the electron and \(R\) is the so-called Rydberg constant:
\[ Rh = E_0 \frac{\alpha^2}{2}, \qquad E_0 = m_0 c^2 . \tag{2a} \]
Here \(m_0\) is the rest mass of the electron and \(\alpha\) is the “fine-structure constant”:
\[ \alpha = \frac{e^2}{\hbar c} = \frac{1}{137}. \tag{3} \]
\(\hbar\) denotes Planck’s constant divided by \(2\pi\); \(e\) and \(c\), as usual, are the charge of the electron and the speed of light.
When \(e\) is expressed in the electrostatic system, then \(\alpha\) represents a universal constant. We shall return later to its value indicated in (3). According to formula (2), the energy \(E\) of the hydrogen electron depends only on the quantum sum \(n = n_r + n_\varphi\), and in \(n\)-states differing in the values of \(n_r\) and \(n_\varphi\), it actually has one and the same magnitude.
However, this holds only in the approximation in which classical mechanics is valid. The latter, after all, does not take into account the finite magnitude of the speed of light, i.e., it assumes \(c = \infty\) and, by (3), consequently, \(\alpha = 0\).
If one calculates more exactly, making use of the theory of relativity, then instead of (2) one obtains
\[ E = E_0 \left\{ 1 + \frac{\alpha^2}{\left(n_r + \sqrt{k^2 - \alpha^2}\right)^2} \right\}^{-1/2}, \tag{4} \]
where \(k\) denotes nothing other than the azimuthal quantum number, which we previously denoted by \(n_\varphi\). If (4) is expanded in powers of \(\alpha^2\) and the higher powers of \(\alpha^2\) are discarded, we arrive again at formula (2). If this is not done, but the exact formula (4) is used, then it is evident that the states \((n_r, k)\) are now energetically somewhat “split” owing to the presence of \(\alpha^2\) in the denominator of (4); “somewhat” here means: “only by an order of magnitude of the small number \(\alpha^2\).”
After this reminder of long-known facts, the left-hand part of our Fig. 1 should be intelligible. It represents the fine structure of the line \(H_\alpha\), i.e., the transition from \(n = 3\) to \(n = 2\) from the point of view of electron orbits. We must still only accept the postulate that, in the transition of the electron from its upper state (upper part of the figure) to the lower state (lower part of the figure), the azimuthal quantum number can change only by one unit:
\[ \Delta k = \pm 1 \quad \text{(selection rule).} \tag{5} \]
The lengths of the arrows shown in the figure indicate the difference of the energies between the initial and final states and at the same time
serve as a measure of the frequency of the oscillations emitted in these transitions; the longer arrows denote the short-wave, the shorter ones the long-wave components of the emitted line. According to the theory set forth, one would expect three components for \(H_{\alpha}\) (the same number of components for \(H_{\beta}, H_{\gamma}, \ldots\), i.e., for the transitions from \(n=4\) to \(n=2\) and \(n=5\) to \(n=2\), etc.).
For testing the theory, however, the decisive factor proved to be the investigation not of the Balmer lines, but of the lines of ionized helium (\(\mathrm{He}^{+}\)). The latter had been studied with respect to their fine structure in Paschen’s classical experimental work, which appeared in the same year, 1915, simultaneously with our theory of fine structure. This theory also covers the \(\mathrm{He}^{+}\) lines; one need only replace \(\alpha\) by \(2\alpha\) in formula (4) (a nucleus with double charge). It must be remembered, however, that the most important \(\mathrm{He}^{+}\) line—\(\lambda = 46.86\)—corresponds to the transition \(4 \to 3\), and not to the transition \(3 \to 2\), as does the \(H_{\alpha}\) line. In accordance with this, the number of structural components increases, namely, according to the theory presented, to five. The number and mutual arrangement of the observed and calculated components coincided, as Paschen showed, almost completely. Thus it seemed that there could be no doubt as to the correctness of this theory.
Fig. 1. Level scheme for \(H_{\alpha}\). On the left—according to the old theory—3 components; on the right—according to Dirac—5 components with quantum numbers \(n_r, k, j\), and comparison with X-ray spectra and spectra of alkali metals
Application of wave mechanics; Dirac’s theory
Nevertheless, serious difficulties soon appeared. The analogy of X-ray spectra with the hydrogen lines was already emphasized in my first paper. In particular, it was found that the X-ray \(L\)-doublet is an enlarged image of the hydrogen doublet. However, in the \(L\) shell \((n=2)\) there are not two energy levels, but three; in the \(M\) shell \((n=3)\), not three, but five, and in general not \(n\), but \(2n-1\) energy levels. How could this increased number of levels be brought under the theory of the hydrogen atom?
On the other hand, Landé repeatedly pointed out the analogy between X-ray spectra and the spectra of alkali metals. Thus the question arose of how the fine structure of the hydrogen lines could be compared with the terms of the alkali metals.
Some authors were inclined to regard the splitting in the hydrogen spectrum not as relativistic, but as magnetic or spin splitting. In particular, the Zeeman effect, whose anomalies gave impetus to the discovery of spin phenomena, reveals (according to Hansen)1 features in hydrogen that point to electron spin. Moreover, it was emphasized that the theory of fine structure uses the theory of relativity not completely, but is limited only to the dependence of the electron mass on velocity, which, as an empirical result, can be singled out from the theory of relativity. We shall return to this observation later.
In the year 1926, critical for atomic theory, in which Schrödinger’s first paper appeared, Unsöld and I gave a comparison of hydrogen states with X-ray spectra, on the one hand, and with the spectra of the alkali metals, on the other. This comparison meant only a new quantum numbering of the energy levels, with the number and position of these levels, indicated by formula (4), fully preserved.
Let us now turn to the right-hand half of our Fig. 1. Here the lower level of the final state and both lower levels of the initial state are depicted as double, whereas the upper level of both states remains single. However, this doubling is only conceptual; the energy values of the double levels, as before, coincide. Only their quantum numbers differ, and not the radial quantum number \(n_r\), to which we assign the same values \(0, 1, 2\), and correspondingly \(0, 1\), as in the left part of the figure, but the azimuthal quantum numbers \(k\). The latter in both parts of the doubled levels differ in sign: \(\pm 2, \pm 1\). That this apparently arbitrary comparison can in fact be justified will soon become clear.
As a consequence of our doubling of the levels, the number of states corresponding to a given principal quantum number \(n\) increases from \(n\) to \(2n - 1\), namely, in the lower part of Fig. 1 from \(n = 2\) to 3, in the upper part from \(n = 3\) to 5. But in this way it becomes possible to compare the X-ray levels with the hydrogen levels. As was already noted, the lower part of the figure corresponds to the \(L\) shell, the upper to the \(M\) shell. Their subgroups, according to Bohr, are denoted by the indices \(I, II\), etc. This notation is given in Fig. 1 in the third column from the right; it should also be borne in mind that levels coinciding in the case of the H atom diverge considerably in the X-ray region at higher nuclear charges.
Next, in the fourth and fifth columns of the figure we make a comparison with the alkali metals; generally speaking, atomic states, as the azimuthal quantum number increases, are characterized as \(S\)-, \(P\)-, \(D\)-states; the notation \(2P, 3P\), etc. means that the principal quantum numbers are respectively equal to 2 or 3, etc., but in the alkali metals each \(P\)- and \(D\)-state is double—
(only the \(S\)-states are simple); thus we have a \(P\)-doublet (cf. what was said above concerning sodium), a \(D\)-doublet, etc. To distinguish the two components of these doublets, the quantum number \(j\) is used, which is given in the last column of our figure and which must be thought of as written as a subscript to the symbols \(S\), \(P\), \(D\). I originally called this quantum number the “inner quantum number.” However, the name “quantum number of the total angular momentum of rotation” is preferable, since it denotes the algebraic sum of the orbital and spin angular momenta. The allowed changes of this number \(j\), when the state of the atom changes, are as follows:
\[ \Delta j=\pm 1 \ \text{or}\ 0 \quad \text{(selection rule).} \tag{6} \]
At the same time, rule (5) for our present number \(k\), which has a sign, must therefore be supplemented so that transitions from \(k\) to \(-k\) and conversely are also possible, i.e.
\[ \Delta k=\pm 1 \ \text{and}\ k \leftrightarrow -k. \tag{7} \]
In Fig. 1, on the right, the transitions allowed by rules (6) and (7) are shown by means of arrows. Three transitions are indicated here which lead to the upper simple level and which therefore are also simple transitions. The arrows leading to the lower double level \(n_r=1\) are joined together in pairs, since in fact, owing to the coincidence of the double levels, they have the same length. Thus, in all, there are now \(3+2=5\) components of the fine structure.
Fig. 2. Distribution of intensity for \(H_\alpha\) and, correspondingly, for \(D_\alpha\) (quantitative). Vertical lines: theoretical intensities of the five components. The curve is taken from photographic observations.
We depict this fine-structure pattern once more in Fig. 2, on the scale of “wave numbers” \(\nu\). Since wave numbers in spectroscopy are defined as the reciprocals of wavelengths, the latter increase in the direction opposite to the increase of wave numbers, i.e. in our figure from right to left. We thus obtain a group of three long-wavelength components and a group of two short-wavelength components. The lengths of the vertical strokes show qualitatively the theoretical intensities of the components; in the case of the short-wavelength components they are equal to the sum of the theoretical intensities corresponding to the two modes of occurrence. The meaning of the dotted lines shown in the figure will be explained below.
However, even with the finest spectroscopic instruments (Fabry–Perot standard or Lummer–Gehrcke plate) there is no possibility of observing these five components separately. What can in fact be achieved is shown in our figure by the intensity contours. The three long-wavelength components together give a hump
intensities, two short-wavelength ones—the second hump, which reveals a small protrusion corresponding to the weaker of the two components1).
Let us also indicate what the horizontal arrows shown in Fig. 2 mean. By \(\Delta \nu_H\) is denoted the ideal hydrogen doublet, i.e. the difference in energy, divided by \(h\), of the two states corresponding to \(n=2\). From formula (4) for \(\Delta \nu_H\) one easily obtains
\[ \Delta \nu_H=\frac{R\alpha^2}{6}=0.363. \tag{8} \]
The number obtained, like the numbers in the subsequent equalities (9), (10), (11), and (15), is expressed in reciprocal centimeters.
However, this \(\Delta \nu_H\) in Fig. 2 appears not as the distance between the two principal components in the long-wavelength and short-wavelength groups, but, as can be seen from Fig. 1, as the distance between the short-wavelength principal component and the long-wavelength mean component. The distance between the two principal components is correspondingly smaller, namely, theoretically
\[ \Delta \nu=0.328. \tag{9} \]
In addition, in Fig. 2 the theoretical distance between the two short-wavelength components from one another is also indicated:
\[ \Delta \nu=0.108. \tag{10} \]
By introducing negative values of \(k\) in numbering the levels of Fig. 1, we, going beyond the proposal of Sommerfeld and Unsöld, anticipated Dirac’s theory of the electron. This brilliant theory, which should be regarded as the crown of Schrödinger’s wave mechanics, arose from the requirement that wave mechanics be made invariant with respect to Lorentz transformations. Here, therefore, the special theory of relativity, the content of which coincides with the requirement of invariance with respect to Lorentz transformations, is taken as the basis in its full scope, and not merely in the form of a single law of the relativistic dependence of mass on velocity, as was done in the old orbital theory of electrons.
The success of Dirac’s equation was decisive. It automatically gives the spin of the electron and its correct magnetic moment. From the value of the latter there follows, mathematically quite naturally, the explanation of the anomalous Zeeman effect. In exactly the same way the double sign of the azimuthal quantum number is obtained by itself and, finally, by integrating Dirac’s equation—the fine-structure formula (4).
From the fact that this last depends only on \(k^2\), there follows the coincidence of both levels corresponding to \(\pm k\), which previously represented
proved arbitrary for us. Thus the objections mentioned above are also eliminated: that the hydrogen doublet is not of relativistic origin, but must be regarded as the result of magnetic or spin splitting. In fact, as we have indicated, spin and its orientation in a magnetic field in Dirac’s theory are a mathematical consequence of the postulates of the theory of relativity. Thus spin and relativistic splittings do not contradict one another. In any case, our confidence in the fine-structure formula has increased since its derivation from Dirac’s equation.
Comparison with Experiment
On this question, in my book Atomic Structure and Spectra, vol. II, it is said: “As regards the experimental verification of the fine-structure formula, the question has not yet been finally settled. While the careful American investigations, especially those of Houston and his students, apparently reveal small deviations from the theory, the work of Mary Heyden, carried out under Kopfermann’s direction, confirms the formula within the limits of error. The ideal arrangement of the experiment, according to a kind personal communication from C. W. Meissner, would consist in transverse observation of canal rays in hydrogen, since in this case the Doppler effect is almost completely excluded. In the end the question comes down to whether, in addition to the Coulomb forces, there also plays a role here an interaction between proton and electron of the type considered in nuclear physics.”
I think that in this passage the situation that existed at the time my book was completed is correctly presented. In an addendum, at the end of the book, moreover, a note by Pasternak is cited (Phys. Rev., 54, 1013; cf. also Phys. Rev., 55, 421). Pasternak relies, among other things, on the Williams measurements mentioned above, according to which the distance in wave numbers between the short-wavelength and long-wavelength principal components is not 0.328, as indicated in (9), but for H and D
\[ \Delta \nu = 0.319, \tag{9a} \]
and between the two short-wavelength components not 0.108, as given in (10), but in the case of D:
\[ \Delta \nu = 0.135. \tag{10a} \]
To explain these deviations Pasternak assumes that the \(2S\) level is displaced toward higher energies, i.e., in Fig. 1—upward. All the other levels remain unchanged.
The displacement of the \(2S\) level entails, as a consequence, that both pairs of arrows leading to \(2S\) and joined by braces are somewhat shortened. The picture of the fine structure in this case consists no longer of five, but actually of seven components, of which, however, two pairs of components, caused solely by a displacement of \(2S\), are so close that they can practically be replaced
lines passing through the center of gravity. The determination of the center of gravity must, of course, be carried out taking into account the intensities of both lines of the pair in such a way that, in the pair in which the components leading to \(2S\) have the greatest intensity, the displacement of the center of gravity is greatest.
In Fig. 2 the lines of both pairs passing through the center of gravity are drawn with a dashed line. From the figure it is seen that the former distance \(\Delta \nu = 0.328\) decreases only slightly, while the former distance \(\Delta \nu = 0.108\) correspondingly increases more strongly.
Pasternak finds that the assumption
\[ \Delta \nu_{2S}=0.003\ \text{cm}^{-1} \tag{11} \]
leads to empirically correct values (9a) and (10a). Moreover, he showed that the same assumption (11), also for the lines \(\mathrm{H}_{\beta}, \mathrm{H}_{\gamma}, \ldots\) of the Balmer series, gives empirically correct magnitudes of the hydrogen doublets (the distances between the principal lines). Let us note in passing that from this one may conclude that the displacement (11), derived from the Williams structure for \(D\), also takes place for the H atom, which (cf. below) is by no means self-evident.
What cause can produce the displacement of the \(2S\) level and the unchanged position of the remaining levels?—It may be thought, as was indicated in the quotation cited, that this is a change of the Coulomb field as a consequence of interactions of the nuclear type. From the well-known experiments on the scattering of protons by protons, performed by Geiger, Hafstad, and Tuve, and recently further improved, Breit and his collaborators have concluded that the radius of action of nuclear forces is of the order of the magnitude of the “classical radius of the electron”
\[ r_0=\frac{e^2}{m_0c^2}=2.81\cdot 10^{-13}\ \text{cm}. \tag{12} \]
When protons approach to this distance, then, besides Coulomb repulsion, much stronger nuclear forces appear, in a certain sense corresponding to the mutual impenetrability of nuclei. One half of the distance (12) should, therefore, correspond to the effective proton radius for scattering. We may imagine, at this half-distance around the proton, a potential barrier which prevents the further approach of the two protons. The action of such a potential barrier on the Schrödinger wave function is replaced in (nonrelativistic) wave mechanics by the boundary condition \(\psi=0\).
We shall transfer these ideas to the interaction between a proton and an electron. We shall move the place of the potential barrier to the distance \(\dfrac{r_0}{q}\), where \(q\) is the corresponding number. Thus we write the boundary condition
\[ \psi=0\quad \text{for } r=r_q=\frac{r_0}{q}. \tag{13} \]
Thus, we arrive at a modified Kepler problem: in the usual Kepler problem one considers the wave function between the boundaries \(r=0\) and \(r=\infty\) and writes the “natural boundary conditions”: \(\psi\) is finite for \(r=0\) and \(\psi=0\) for \(r=\infty\).
Now we restrict the interval of applicability of \(\psi\) to the region between \(r_q\) and \(\infty\) and write, at the point \(r=r_q\), the artificial boundary conditions (13), while preserving the condition \(\psi=0\) for \(r=\infty\).
In the 1938 Planck issue of Annalen der Physik, I considered, together with Welker, the Kepler problem with artificial boundary conditions, which are of interest for astrophysical questions. In that consideration the \(\psi\)-function was confined by an external potential barrier to the region \(0 \le r \le r_q\). In that paper we spoke of a “confined electron” and investigated the influence of confinement on the energy of the ground state of hydrogen. Now, with our potential barrier at \(r=r_q\), we may speak of a “released electron” and study the influence of release on its energy level.
It is immediately clear that this influence in the case of \(P\)- and \(D\)-states, as well as in all states to which large values of \(k\) correspond, must be small. Indeed, the eigenfunctions for these states vanish at \(r=0\) already in the normal Kepler problem; the condition \(\psi=0\) at small \(r_q\) therefore has little effect. The situation is different in the case of \(S\)-states. The corresponding eigenfunctions, under the usual normalization, have the value
\[ \psi=\frac{1}{n\sqrt{n}} \]
for \(r=0\). Therefore they are strongly changed by condition (13), especially for small \(n\), as a result of which one may expect a noticeable change also in the corresponding energy levels.
In order to calculate this rigorously, we must consider the analytic character of the eigenfunctions. Instead of the so-called Laguerre polynomials, which in the case of the ordinary Kepler problem constitute an essential part of the eigenfunctions, there now appear transcendental solutions of the Laguerre differential equation, namely nonterminating confluent hypergeometric functions. Since their asymptotic behavior for \(r=\infty\) is in our hands, they can first of all be adapted to the condition \(\psi=0\) for \(r=\infty\). In order to satisfy the other boundary condition, namely \(\psi=0\) for \(r=r_q\), one may make use of the energy parameter \(E\) entering into the wave function.
Instead of being interested in the values of \(E\), one may also, while formally observing relation (2), consider the corresponding values of \(n\). Whereas previously \(n\) was an integer, for example \(n=2\) for the state \(2S\), now \(n\) becomes nonintegral. We therefore write \(n+\Delta n\) and find from boundary condition (13), applied to our hypergeometric function, the simple result:
\[ \Delta n=2\frac{r_q}{a} \tag{14} \]
for sufficiently small \(r_q\). Here \(a\) is the so-called hydrogen radius
\[ a=\frac{\hbar^2}{m_0 e^2} \]
and, according to (12) and (13), we obtain
\[ \frac{r_0}{a}=\frac{e^4}{\hbar^2 c^2}=\alpha^2 . \]
Therefore, instead of (14), we may also write
\[ \Delta n=\frac{2\alpha^2}{q}. \tag{14a} \]
Taking (2) into account, let us write the changed energy
\[ E+\Delta E=E_0-\frac{Rh}{\left(n+\dfrac{2\alpha^2}{q}\right)^2} =E_0-\frac{Rh}{n^2}\left(1-\frac{4\alpha^2}{qn}\right). \]
Hence the following change of the wave number is obtained:
\[ \Delta \nu=\frac{\Delta E}{h}=\frac{4R\alpha^2}{qn^3}, \tag{15} \]
From this relation we obtain first of all
\[ \Delta \nu_{1S}:\Delta \nu_{2S}:\Delta \nu_{3S}:\ldots =1:\frac{1}{8}:\frac{1}{27}:\ldots \tag{16} \]
Consequently, using Pasternak’s result \(\Delta \nu_{2S}=0.03\), we obtain
\[ \Delta \nu_{1S}=8\Delta \nu_{2S}=0.24; \qquad \Delta \nu_{3S}=\frac{8}{27}\Delta \nu_{2S}=0.009. \]
Since we may regard the last value as inaccessible to observation because of its smallness, we thereby confirm the postulated above unchanged position of all initial levels of \(\mathrm{H}_{\alpha}\) (as well as of \(\mathrm{H}_{\beta}\)), after we have already earlier explained the insensitivity of the \(P\), \(D\), etc. levels (namely, both for the initial and for the final states). We also now understand why no discrepancy was established in the fine structure of the lines \(\mathrm{He}^{+}=4686\) 1.
In fact, here the lower state is precisely the insensitive level \(3S\). On the other hand, one might have thought of checking the influence on the most sensitive level \(1S\), by detecting this influence on the lines of the Lyman series. However, this expectation is illusory, since the final level of the Lyman series is simple and consists of only one term \(1S\): its displacement \(\Delta \nu_{1S}\) therefore has no effect on the fine structure, but only on the general position of the lines of the series, and moreover in percentage terms in an imperceptible way.
Finally, let us compute from (15) the quantity \(q\), substituting \(n=2\) and \(\Delta \nu=0.03\). We obtain
\[ q=\frac{1}{2}\frac{R\alpha^2}{0.03}=84, \tag{17} \]
since
\[ R=1.10\cdot 10^5,\qquad \alpha^2=5.27\cdot 10^{-5}. \]
This value is strikingly large. On the basis of the results with proton scattering we might have expected approximately \(q=2\). In order to make the small values of \(r_q\) following from \(q=84\) plausible, one could imagine that, owing to the Coulomb attraction in the Kepler problem, the electron’s approach to the proton may be closer than in the case of Coulomb repulsion in proton scattering.
The remarks given above arose under the influence of a recently published interesting paper\(^{1}\), in which the proton field in the Kepler problem is calculated on the basis of meson theory. It turned out that the repulsion superposed on the Coulomb attraction appears rather abruptly at \(\frac{r_0}{6}\). Since the authors idealized this repulsion by an infinitely steep potential barrier, they arrived at the boundary condition indicated above, \(q=6\). The resulting changes in the energy level are calculated by the authors by an approximate method based on Green’s theorem\(^{2}\), in which the investigation of the analytic character of the altered eigenfunctions is left aside. The result agrees with our equation (15); in particular, the correct ratio \(\Delta \nu_{1S}:\Delta \nu_{2S}=8:1\) is obtained. The absolute value of \(\Delta \nu_{2S}\), of course, is obtained, because of \(q=6\), much too large. However, we do not wish from this to conclude that meson theory is incorrect, or that the work of Fröhlich, Heitler, and Kahn itself is incorrect. We shall only draw attention to one objection which applies both to the work of these authors and to our own considerations outlined above: Schrödinger wave mechanics is valid only down to quantities of order \(a\). If, as in fact occurs in the case of our problem, quantities of this order result, the calculation should be carried out according to Dirac. This, however, would take us too far here and will be done elsewhere.
Remarks on the fundamental constants of atomic physics
In equation (3) we gave for the fine-structure constant the value equal to the reciprocal of the integer 137. Against this there is the following objection. In 1929 Eddington developed a bold theory
\(^{1}\) H. Fröhlich, W. Heitler and B. Kahn, Proc. Roy. Soc., 171, 269, 1939.
\(^{2}\) H. Fröhlich, Phys. Rev., 54, 945, 1938.
based on the Dirac equation, according to which he arrived at the following value of \(a\):
\[ a=16+\frac{16\cdot 15}{2}=136. \]
In order to explain this result it is necessary to set forth the following considerations. In order to satisfy the requirement of four-dimensional invariance, Dirac considered it necessary to introduce into his equation four matrices with four rows, the use of which is rather cumbersome. Eddington, Sauter, and others observed—and in my book I consistently carried out—the replacement of these matrices by four hypercomplex numbers, which behave anticommutatively and, under multiplication, form a group of sixteen hypercomplex units. The number of pairs of these units is as follows:
\[ 16 \quad \text{(a combination of two identical units),} \]
\[ \frac{16\cdot 15}{2} \quad \text{(a combination of two different units).} \]
Eddington attempted to justify the fact that the sum 136 should be equal to the number of degrees of freedom of the electron and coincide with the reciprocal value of \(a\). Soon afterward he decided to add one more degree of freedom, as a result of which the agreement with the empirical value of \(a\) was improved. The latter, on the basis of the then available values of \(e\) and \(h\), was equal to 137.3. It is undoubtedly remarkable that with further improvement, especially of the value of \(e\), for \(a\) there was obtained a number lying closer to the integer 137. The present best value is
\[ \frac{1}{a}=137.02. \]
It cannot, however, be passed over in silence that in another case a prediction made by Eddington on the basis of analogous considerations was not confirmed. According to Eddington, the magnetic moment of the proton should have been
\[ \frac{5}{2} \]
in units of the so-called nuclear magnetons. However, the very precise value of the magnetic moment of the proton known at present is \(2.785\pm 0.2\), i.e., in no case rational and, of course, quite different from \(\frac{5}{2}\).
At the congress dedicated to the memory of Galvani, in Bologna in 1937, after a report by M. Siegbahn, the question of the exact values of the fundamental constants \(e\), \(h\), \(m_0\), and of their agreement with the much more accurately known value of the Rydberg constant \(R\), was discussed. We then strongly urged Siegbahn, as a master of precision measurements, to check, by means of X-ray measurements, the value of \(h\), which at that time had been measured least accurately, and he promised to do so. However, the final results are still unknown. A highly careful and clear discussion of the difficulties currently existing has been given by DuMond [Phys. Rev., 56, 153, 19391].
Already in 1919, in the first edition of my book, I proposed determining the values \(e\), \(m\), \(h\) purely spectroscopically, with one
on the one hand, from the quantum-theoretical formulas for the Rydberg constant for H and He\(^+\), and, on the other hand, from very precise measurements of the fine structure. In connection with the latter, we again emphatically point to Meissner’s proposal, mentioned at the beginning of the preceding section.
However, the situation has now changed. As we have seen, the measurement of the fine structure provides information not only about fundamental constants, but also about the still quite mysterious nuclear forces. It would, of course, be very naive to think that the Coulomb field must hold all the way into the region in the immediate vicinity of the nucleus and even to the center of the nucleus, \(r = 0\). But a deviation from the Coulomb field, as we have shown, even if it occurs at the very smallest values of \(r\), has an effect on the fine structure, especially in H\(_\alpha\). Interest in exact knowledge of the fine structure is thereby shifted, but certainly not in the least diminished.