ATOMIC THEORY AND MECHANICS1

N. Bohr.

CLASSICAL THEORIES.

The study of the equilibrium and motion of bodies not only constitutes the foundation of physics, but also provides extensive material for mathematical investigation, and has proved extraordinarily fruitful for the development of the methods of pure mathematics. The connection between mechanics and mathematics already manifested itself directly in the works of Archimedes, Galileo, and Newton. In their creations the formation of the concepts necessary for the analysis of mechanical phenomena was completed. Since Newton, the development of methods for treating mechanical problems has gone hand in hand with the development of mathematical analysis: it is enough to recall the names of Euler, Laplace, and Lagrange. The further development of mechanics, based on the works of Hamilton, was likewise closely connected with the development of mathematical methods—the calculus of variations and the theory of invariants. In more recent times this connection has been clearly manifested in the works of Poincaré.

Perhaps the greatest successes were achieved by mechanics in the field of astronomy, but the mechanical theory of heat also yielded interesting applications during the last century. The kinetic theory of gases, founded by Clausius and Maxwell, explains the properties of gases chiefly as the result of mechanical interaction between atoms and molecules moving in all possible directions. Let us recall first of all the interpretation of the two principles of thermodynamics given by this theory. While the first principle is a direct consequence of the mechanical law of conservation of energy, the second principle, the law of entropy, may be derived, according to Boltzmann, on the basis of the statistical properties of a large number of mechanical systems. It is interesting to note in this connection that stati-

statistical considerations led to the explanation not only of the average properties of atoms, but also of fluctuation phenomena; and the study of the latter, in particular Brownian motion, unexpectedly made it possible to count atoms. The direct instrument for the systematic development of statistical mechanics was the mathematical theory of canonical systems of differential equations, which owes so much to Gibbs.

The development of electromagnetic theories in the second half of the past century, following the discoveries of Oersted and Faraday, led to a profound generalization of mechanical concepts. Although, for example, mechanical models played an essential role in Maxwell’s electrodynamics, subsequently significant successes were achieved thanks to the fact that, conversely, mechanical concepts were derived from the theory of the electromagnetic field. In this theory the laws of conservation of energy and momentum are based on the fact that energy and momentum are regarded as concentrated in the space surrounding bodies. In particular, in this way an explanation of radiation phenomena can be obtained.

The theory of the electromagnetic field directly led to the discovery of electromagnetic waves, which played such an important role in electrical engineering. Furthermore, the electromagnetic theory of light, founded by Maxwell, provided a profound foundation for the wave theory of light, which goes back to Huygens. By means of atomic theories a general description was obtained of the radiation of light and of those phenomena that occur when light passes through matter. For this it was assumed that atoms consist of electrified particles, which can perform oscillations about positions of equilibrium.

The free oscillations of the particles are the cause of radiation, the composition of which we observe in the atomic spectra of the elements. In addition, the particles can perform forced oscillations under the influence of light waves and in this way themselves become centers of secondary waves, which interfere with the primary waves and produce the familiar phenomena of reflection and refraction of light. If the frequency of the incident waves approaches the frequency of one of the atom’s free oscillations, this produces the phenomenon of resonance, in which the particles enter a state of especially strong oscillations. In this way a simple explanation was obtained of the phenomena of resonance radiation and anomalous dispersion for light close to one of the spectral lines.

Like the kinetic theory of gases, the electromagnetic interpretation of optical phenomena is not limited to the study of the average action of a large number of atoms. Thus, for example, in the scattering of light, the random arrangement of atoms causes such an influence of individual atoms as makes it possible to count their number. Indeed,

Rayleigh succeeded, on the basis of the intensity of the scattered blue light of the sky, in determining the number of atoms in the atmosphere; the results he obtained are in satisfactory agreement with Perrin’s calculation of the number of atoms made in the study of Brownian motion.

A rational mathematical exposition of electromagnetic theory is based on the application of vector analysis, or, in a more general form, tensor analysis for a manifold of several dimensions. This analysis, founded by Riemann, gave Einstein the means for formulating the theory of relativity, which introduces concepts going beyond Galilean kinematics and may be regarded as a natural generalization of the classical theories.

Theory of Quanta.

Despite the considerable successes in the applications of mechanical and electrodynamical ideas to atomistic theory, further development of research encountered very serious difficulties. If mechanics and electrodynamics really do give a complete description of thermal motion and of the radiation connected with this motion, then the general laws of thermal radiation ought to have received a direct explanation. However, contrary to all expectations, it turned out that calculations proceeding from our considerations cannot explain the empirical laws. Planck went further. Basing himself on Boltzmann’s derivation of the second law of thermodynamics, he showed that the laws of thermal radiation compel one to introduce into the description of atomic processes a certain element of discontinuity, completely foreign to the classical theories. Planck found that in determining the statistical properties of particles performing simple harmonic oscillations about positions of equilibrium, only those oscillatory states should be taken into account in which the energy is equal to an integral multiple of the “quantum” \(\omega h\), where \(\omega\) is the frequency of oscillation of the particle, and the quantity \(h\) is a universal constant, the so-called Planck quantum of action.

However, a more complete formulation of the theory of quanta proves to be extraordinarily difficult, if one takes into account that all the constructions of the former theories are based on such ideas according to which continuous changes must exist. This difficulty appears especially sharply in Einstein’s profound investigations. According to the latter, the basic features of the interaction between light and matter lead to the conclusion that light propagates not in the form of waves, but in the form of “light quanta,” which are concentrated in a small part of space and contain an amount of energy \(h\nu\), where \(\nu\) is the frequency of light. The formal character of this assertion is obvious, since the definition and measurement of frequency are based exclusively on the ideas of wave theory.

Component Parts of the Atom

The inadequacy of the classical theories became clearly apparent thanks to the development of our knowledge of the structure of atoms. Formerly it was expected that this knowledge could be gradually extended by analyzing the properties of the elements on the basis of the classical theories, which had proved so fruitful in various respects. These hopes received confirmation shortly before the appearance of the quantum theory, when Zeeman discovered the influence of a magnetic field on spectral lines. As Lorentz showed, this phenomenon corresponds in many cases to precisely that action of a magnetic field on the motion of oscillating particles which can be predicted on the basis of classical electrodynamics. Moreover, from this theory there were derived conclusions concerning the nature of the oscillating particles which were in excellent agreement with the experimental discoveries of Lenard and Thomson in the field of electric discharges in gases. As a result of all these investigations it was established that negatively charged particles, electrons, are component parts common to all atoms.

It is true that the so-called “anomalous” Zeeman effect, observed for many spectral lines, presented considerable difficulties for the classical theory. Similar difficulties were also encountered in attempting to explain, by means of electrodynamic models, the simple empirical regularities of spectral frequencies which had been established in the works of Balmer, Rydberg, and Ritz. Thus, for example, explanations of this kind for spectral laws could not be reconciled with Thomson’s determination of the number of electrons in the atom, although Thomson in doing so directly applied the classical theory to his observations on the scattering of X-rays.

These difficulties could be temporarily explained by the fact that the forces binding the electrons within the atom were not sufficiently well known to us. But the situation changed substantially thanks to experimental discoveries in the field of radioactivity, which provided new means for studying the structure of atoms. Studying the passage through matter of particles emitted by radioactive substances, Rutherford arrived at the idea of the nuclear structure of the atom. According to this conception, the greater part of the mass of the atom is concentrated within a positively charged nucleus, which is very small in comparison with the dimensions of the whole atom. Around the nucleus there moves a definite number of light negative electrons.

Thus the problem of the structure of the atom seemed to have acquired a close resemblance to the problems of celestial mechanics. But a closer examination of this question soon showed that there is also an essen-

essential distinction between an atom and a planetary system. The atom must possess a stability that presents features which do not fit within the framework of mechanical theory. Mechanical laws allow the possibility of continuous changes of motion, and this contradicts the circumstance that each element has quite definite properties. The distinction between an atom and the electrodynamic model becomes obvious if one considers the structure of the emitted radiation. In models of this kind, in which, according to classical theory, the frequency of revolution changes continuously together with the energy, the frequency of the radiation must change continuously during emission. Such radiation has nothing in common with the line spectra of the elements.

Quantum Theory of the Structure of Atoms.

Attempts to find a more precise formulation for the ideas of quantum theory, and moreover one that could overcome the difficulties indicated by us, led to the establishment of the following postulates:

1. An atomic system possesses a certain multiplicity of states, “stationary states,” to which there corresponds a discontinuous series of values of the energy and which possess a definite stability. This is expressed in the fact that every change in the energy of the atom is caused by a “transition” of the atom from one stationary state to another.

2. The possibility of the emission and absorption of radiation by an atom is conditioned by the possibility of a change in the energy of the atom, and the frequency of the radiation is connected with the difference of the energy values in the initial and final states by the following relation:

\[ h\nu = E_1 - E_2. \]

These postulates, which cannot be explained on the basis of classical conceptions, apparently provide a sufficient foundation for a general description of the observed physical and chemical properties of the elements. In particular, they gave an immediate explanation of the fundamental feature of the empirical spectral laws. The principle of combination of spectral lines—the Ritz principle—establishes that the frequency of each spectral line can be represented in the form of the difference of two terms from the multiplicity of spectral terms characterizing the given element. Indeed, we see that these terms can be identified with the energy values of the stationary states of the atom, divided by \(h\). Moreover, this description of the origin of spectra also gives a direct explanation of the essential difference between absorption and emission spectra. According to the postulates, the condition for the selective absorption of a frequency corres-

...of the corresponding combination of two terms, consists in the fact that the atom must be in the state with the lower energy, whereas for emission it must be in a stationary state with the greater energy. In short, the picture described is in very close agreement with the results of experiments on the excitation of spectra. This is proved especially clearly by the discovery of Franck and Hertz, concerning collisions between atoms and electrons. Franck and Hertz found that the transfer of energy from an electron to an atom can occur only in amounts exactly equal to the differences between the energy values of the stationary states, as calculated from the spectral terms. The excitation of an atom to radiation generally occurs instantaneously. On the other hand, an excited atom can, according to Klein and Rosseland, lose its radiating capacity upon being struck by an electron whose energy is increased by the corresponding amount.

As Einstein showed, these postulates also provide a sufficient basis for a rational development of statistical problems, in particular for a very clear derivation of Planck’s radiation law. Einstein’s theory assumes that an atom which can make a transition between two stationary states and is in the higher state has a certain “probability,” depending only on the atom, of jumping spontaneously (of its own accord) in a given interval of time to the lower state. In addition, it assumes that when the atom is illuminated from outside by radiation with a frequency corresponding to the transition, the atom acquires a probability, proportional to the intensity of the radiation, of passing from the lower state to the higher. An essential feature of the theory is also the assumption that if the atom is illuminated by such radiation while in the lower state, then, in addition to its spontaneous probability, it acquires a further probability¹) of jumping to the lower state.

Quantum Theory of Radiation.

Einstein’s theory of heat radiation, while confirming the postulates, at the same time emphasizes the formal nature of the frequency condition. From the conditions of thermal equilibrium Einstein draws the conclusion that every process of absorption and emission is accompanied by the transfer of an amount of motion

\[ \frac{h\nu}{c} \]

(where \(c\) is the speed of light), as might be expected on the basis of the conception of light quanta. The significance of this conclusion was especially noted thanks to Compton’s discovery, who found that the scattering of homogeneous

¹) This latter probability is also proportional to the intensity of the radiation.
Translator’s note.

the X-rays is accompanied by a change in the wavelength of the scattered radiation, and this change depends on the direction in which the scattered rays are observed. Such a change in frequency follows directly from the theory of light quanta, if in calculating the deflection of the quantum one takes into account the laws of conservation of energy and momentum.

The contradiction between the wave theory of light, as applied to the explanation of optical phenomena, and the theory of light quanta, which explains well many features of the interaction between light and matter, continued to grow and led to the idea that the insufficiency of the classical theory might even affect the validity of the laws of conservation of energy and momentum. These laws, which occupy such a central position in classical theory, might have only statistical significance in the description of atomic processes. However, such an assumption does not give a satisfactory resolution of the dilemma, as is shown by experiments on the scattering of X-rays, carried out recently with the aid of refined methods that make it possible to observe individual processes directly. Geiger and Bothe succeeded in showing that the “recoil” electrons accompanying the scattered radiation and the photo-electrons appearing upon its absorption correspond to one another in pairs, as should be expected on the basis of the theory of light quanta. Compton and Simon proved, with the aid of Wilson’s cloud chamber, that there exists not only a pairwise correspondence of electrons, but also the dependence, required by the theory of light quanta, between the direction in which the scattered radiation is observed and the direction of the velocity of the recoil electrons accompanying this scattering.

From these results it apparently follows that, in the general problem of quantum theory, one has to deal not only with a modification of the mechanical and electrodynamical theories, which can be expressed by means of ordinary physical conceptions, but also with an essential deficiency of the space-time images on which the description of natural phenomena has hitherto been based. This deficiency becomes apparent upon the closest examination of collision phenomena. In particular, for collision phenomena in which the duration of the collision is small in comparison with the natural periods of the atom, and for which very simple results might have been expected on the basis of ordinary mechanical conceptions, it turns out that the postulate of stationary states is apparently incompatible with a space-time description of the collision based on the modern doctrine of the structure of atoms1.

The Correspondence Principle

Nevertheless, it proved possible to construct such mechanical images of stationary states as are based on the conception of the nuclear atom and have played an essential role in explaining the special properties of the elements. In the simplest case of an atom with one electron, such as the neutral hydrogen atom, the orbit of the electron is, according to classical mechanics, a closed ellipse obeying Kepler’s laws. According to these laws, the major axis and the frequency of revolution are connected by a simple dependence with the work that must be expended for the complete separation of the particles composing the atom. If one assumes that the spectral terms of the hydrogen spectrum characterize this work, then the spectrum gives us an indication of the existence of a series of successive processes during which the electron becomes bound to the atom more and more strongly, passing onto orbits of ever smaller dimensions and emitting radiation in the process. When the electron is bound most strongly, and the atom therefore can no longer radiate, the normal state of the atom has been reached. The dimensions of the orbits, calculated from the spectral terms, have a magnitude of the same order as the dimensions of atoms obtained on the basis of the mechanical properties of the elements. But, according to the very character of the postulates, such mechanical features as the frequency of revolution and the form of the electronic orbits cannot be subjected to comparison with the results of experiments. The symbolic character of these images is best seen from the circumstance that an atom in the normal state does not radiate at all, although, according to mechanical conceptions, the electron continues to move.

Despite this, the representation of stationary states by means of mechanical conceptions led to a far-reaching analogy between quantum theory and mechanical theory. This analogy was carried through in determining those initial states of the binding process described above in which the motions corresponding to neighboring stationary states differ comparatively little from one another. Here it proved possible to note an asymptotic correspondence between the spectrum and the motion. On the basis of this correspondence there is derived a quantitative relation in which the constant appearing in Balmer’s formula for the hydrogen spectrum is expressed in terms of Planck’s constant and the values of the mass and charge of the electron. The important role of this formula is evident from the fact that, on the basis of the theory, it was possible to predict the dependence between the spectrum and the charge of the nucleus. The latter result may be regarded as the first step toward carrying out the program outlined by the doctrine of the nuclear atom, whose aim is to express the interrelations between the properties of the elements solely with the aid of a single integer,

denoting the number of unit positive charges of the nucleus, the so-called “atomic number.”

The proof of the asymptotic correspondence between the spectrum and the motion led to the formulation of the “correspondence principle,” according to which the possibility of any transition process associated with radiation is conditioned by the existence of the corresponding harmonic components in the motion of the atom. The frequencies of the corresponding harmonic components asymptotically coincide with the values obtained from the frequency condition, in the limiting case when the energy values of the final states converge with one another. But, in addition, the amplitudes of the mechanical components of the oscillation give, in the limit, an asymptotic measure of the probabilities of the transition process; and the intensities of the observed spectral lines depend on these probabilities. The correspondence principle expresses the tendency, in the systematic development of quantum theory, to make use of every feature of the classical theory. But an expedient transcription of this kind is carried out with due regard for the essential difference between the postulates of the two theories.

Quantization Rules

A considerable step forward was made when it proved possible to formulate certain general laws, the so-called “quantization rules,” by means of which, from the continuous manifold of mechanical motions, one can select the motions inherent in stationary states. These rules apply to atomic systems for which the solutions of the mechanical equations are periodic or multiply periodic. In these cases the motion of each particle can be represented as a sum of discrete harmonic oscillations. The quantization rules may be regarded as a rational generalization of Planck’s original conclusions concerning the possible energy values of a harmonic vibrator. According to these rules, known components of the action characterizing the solutions of the mechanical equations of motion are set equal to integral multiples of Planck’s constant. Thanks to the quantization rules, a classification of stationary states has been developed in which to each state there corresponds several integers, “quantum numbers”; the number of the latter is equal to the degree of periodicity of the mechanical motion.

In the formulation of the quantization rules, an essential role was played by the modern development of mathematical methods as applied to mechanical problems. It is sufficient to recall the theory of phase integrals, used in particular by Sommerfeld, or the property of adiabatic invariance of these integrals, indicated by Ehrenfest. The theory acquired a very elegant form thanks to the introduction of Stäckel’s uniformizing variables. In such a formulation

the fundamental frequencies characterizing the periodicity of the mechanical solution are represented as partial derivatives of the energy with respect to those components of the action which are subject to quantization1. Hence there is obtained a justification for the asymptotic correspondence between the motion and the spectrum, which is calculated from the frequency condition.

By means of the quantization rules an explanation was obtained for various detailed properties of spectra. Of particular interest was Sommerfeld’s proof that such a structure of the spectral lines of hydrogen is explained by small deviations from Keplerian motion, which depend on the modification of Newtonian mechanics introduced by the theory of relativity. In addition, let us recall the explanation given by Epstein and Schwarzschild for the splitting of spectral lines under the influence of an external electric field, discovered by Stark.

Here we are dealing with a mechanical problem whose treatment had been considerably advanced in the hands of such mathematicians as Euler and Lagrange, after whom Jacobi found his famous elegant solution with the aid of Hamilton’s partial differential equation. Especially after the application of the correspondence principle—by means of which not only the polarization of the components of the Stark phenomenon was explained, but also, as Kramers showed, the distribution of intensities among the separate components—we may say that in this phenomenon it is possible to discern every feature of Jacobi’s solution, though under the cover of quantum theory. In this connection it is interesting to mention that, with the aid of the correspondence principle, the influence of a magnetic field on the hydrogen atom can be investigated in such a way that this method proves very similar to Lorentz’s calculation of the Zeeman effect, carried out on the basis of classical electrodynamics, especially in the form of Larmor’s equations.

Stability of the Structure of Atoms

The problems we have mentioned are a direct application of the quantization rules. But in the problem of the structure of atoms with several electrons we encounter a case in which the general solution of the mechanical problem does not possess the periodic properties that appear necessary for the mechanical representation of stationary states. It is nevertheless natural to suggest that this further limitation on the applicability of mechanical pictures to the study of the properties of atoms with several electrons, in comparison with atoms containing a single electron, is not—

is directly connected with the postulate on the stability of stationary states. Indeed, the interaction of electrons in an atom presents a problem analogous to the problem of a collision between an atom and a free electron. Just as no mechanical explanation can be given for the stability of an atom in such a collision, so too, in any description of the stationary states of an atom, one must assume that, in the interaction of electrons, the share of participation of each of them is introduced in a completely non-mechanical way.

This point of view is in agreement with spectroscopic data. One of the most important data of this kind is the fact, established by Rydberg, that the same constant as in Balmer’s formula enters into the empirical formulas for the series spectra of all elements, despite the more complex structure of the spectra of different elements in comparison with the spectrum of hydrogen. This discovery receives a simple explanation if the series spectra are regarded as a reflection of the process of attachment of an electron to an atom, in which the electron becomes bound more and more strongly, step by step, and emits radiation. The nature of the binding of the other electrons remains unchanged during this time, and the gradual strengthening of the binding of the given electron occurs in orbits which at first are large in comparison with the ordinary dimensions of the atom, and then become smaller and smaller, until the normal state of the atom is reached. In the case when the atom possesses a single positive charge before capturing the electron, the attraction of the electron by the rest of the atom has, from this point of view, a great similarity with the mutual attraction of the parts of the hydrogen atom. It is therefore clear why the spectral terms representing the binding of electrons exhibit an asymptotic coincidence with the terms of the hydrogen spectrum. In the same way one can obtain an immediate explanation of the general dependence of series spectra on the state of ionization of the atom, which was established by the remarkable works of Fowler and Paschen.

Characteristic indications of the manner in which electrons are bound in the atom are provided by the study of X-ray spectra. On the one hand, Moseley’s fundamental discovery of the striking similarity between the X-ray spectrum of an element and the spectrum corresponding to the binding of a single electron by a nucleus can easily be explained if one takes into account that, within the atom, the influence of the nucleus on the nature of the binding of each individual electron considerably exceeds the mutual influence of the electrons. On the other hand, X-ray spectra reveal a characteristic difference from series spectra. This difference is explained by the circumstance that, in an X-ray spectrum, we are concerned not with the binding of a newly attaching electron, but with a transformation of the binding of the remaining electrons after the removal of one of the electrons which had previously been bound. Thanks to this

circumstance, which was especially noted by Kossel, it proved possible to shed light on new important aspects of the question of the stability of the structure of atoms.

Analysis of Spectra.

In order to explain the detailed structure of spectra, it is, of course, necessary to study in detail the interaction between the electrons within the atom. In developing this problem one has to depart from the strict application of mechanics. To each electron there is ascribed a motion with such periodic properties that it becomes possible to classify spectral terms by means of quantum numbers. In the works of Sommerfeld a considerable number of spectral regularities received, in this way, a simple interpretation. Moreover, these considerations opened a wide field for the application of the correspondence principle. Indeed, with their help it proved possible to explain certain restrictions among the possible combinations of spectral terms, the so-called selection rules.

Following this path, in recent times it has proved possible, on the basis of data from series spectra, as well as X-ray spectra, to draw conclusions about the groupings of electrons in the normal state of the atom. These groupings provide explanations of the principal features of the periodic system of the elements in accord with the ideas about the chemical activity of atoms developed by J. J. Thomson, Kossel, and Lewis. Progress in this field has recently been closely connected with the accumulation of new spectroscopic data. The investigations of Lyman and Millikan played no small role; thanks to them a bridge was thrown across the gulf between optical spectra and the region of X-rays. In the latter region great successes have been achieved thanks to the labors of Siegbahn and his collaborators. It is also necessary to mention Coster’s work on the X-ray spectra of heavy elements, which contributed significantly to clarifying the fundamental features of the periodic system.

However, the study of the fine details of spectra revealed such features as could not be explained with the aid of mechanical conceptions on the basis of the theory of periodic systems. These include, for example, the multiplet structure of spectral lines and the influence of a magnetic field on this structure. This phenomenon, known under the name of the anomalous Zeeman effect, presents, as we have already mentioned, serious difficulties for the classical theory. True, it fits into the scheme of the basic postulates of the quantum theory. As Landé showed, the frequencies of the components into which each spectral line is split under the influence of a field can be represented in the form of a combination of terms, similarly to the principal lines. The totality of these magnetic terms can be obtained if one replaces

each main spectral term into several quantities which differ little from it, the differences depending on the field intensity. Indeed, the splendid experiments of Stern and Gerlach established a direct connection between the force acting on an atom in an inhomogeneous magnetic field and the energy values of the stationary states in the field, calculated on the basis of the magnetic terms. These experiments may be regarded as one of the direct proofs of the fundamental propositions of quantum theory.

However, the analysis carried out by Lande reveals a strange difference between the interaction of electrons in the atom and the combination of mechanical systems. Indeed, one is forced to assume that the interaction of the electrons in the atom is connected with a certain “tension,” which does not admit of a mechanical description and does not yield a one-to-one correspondence with the quantum numbers on the basis of mechanical conceptions1. In the discussion of this problem an essential role was played by the general condition of thermodynamic equilibrium established by Ehrenfest. In its application to quantum theory this condition indicates that the statistical “weight” belonging to a stationary state is not changed under a continuous transformation of the atomic system. It has recently been established that this same condition leads, even for atoms with only one electron, to such difficulties as indicate the necessity of restricting the limits of applicability of the theory of periodic systems. Indeed, the problem of the motion of point charges admits certain special solutions which must be excluded from the totality of stationary states. This exclusion artificially restricts the rules of quantization, but does not stand in obvious contradiction to experimental data. Especially serious difficulties were brought to light by an interesting investigation of the problem of the hydrogen atom in crossed electric and magnetic fields, carried out by Klein2 and Lenz3. In this case it proved impossible to satisfy Ehrenfest’s condition, since the corresponding modification of the external forces can gradually transform orbits describing stationary states and not subject to exclusion from such states into orbits of such a kind that, moving along them, the electron falls onto the nucleus.

Despite these difficulties, the analysis of the fine details of the spectrum has considerably advanced the quantum interpretation of the laws of inter-

relations between the elements. In the works of Dauvillier¹), Main Smith²), and Stoner³), on the basis of various experimental data, representations of quantum theory concerning groupings of electrons in atoms have been developed. Despite the formal character of these considerations, they reveal a close connection with the spectral regularities disclosed in Landé’s investigations. In this direction significant successes have recently been achieved, especially by Pauli⁴). Although these results constitute a considerable step forward toward carrying out the program outlined above (the explanation of the properties of the elements exclusively on the basis of the atomic number), they nevertheless do not yet give an unambiguous correspondence with mechanical representations.

Quantum theory and optical phenomena.

A new epoch in the development of quantum theory has recently begun thanks to a deeper study of optical phenomena. At first, as we have already mentioned, the classical theory achieved significant successes in this field, whereas the postulates did not give a direct solution. It is true that one could conclude on the basis of experiments that an illuminated atom produces a scattering of light essentially analogous to that scattering which, according to the classical theory, is produced by elastically bound electric particles. The frequencies of the natural oscillations of these particles are equal to the frequencies corresponding to the transition processes which the atom can perform under the influence of external radiation. Indeed, according to the classical theory, such harmonic vibrators would, under the influence of excitation, emit radiation of the same properties as an atom that has passed into a higher stationary state.

The possibility of combining the description of optical phenomena with the representation of vibrators connected with transition processes was brought nearer to realization thanks to Slater’s idea⁵), according to which the emission of radiation by an activated atom may be regarded as the “cause” of spontaneous transitions, by analogy with the way transitions are caused by radiation falling from outside. Ladenburg made the first subsequent step on the path toward a quantitative description of the phenomenon of dispersion,

¹) A. Dauvillier, C. R., 177, 476, 1924.
²) J. D. Main Smith, Journ. Chem. Ind., 44, 944, 1925.
³) E. C. Stoner, Phil. Mag. 48, 719, 1924.
⁴) W. Pauli jr., ZS. f. Phys. 31, 765, 1925. See also H. N. Russell and F. A. Saunders, Astrophys. Journ. 61, 38, 1925; S. Goudsmit, ZS. f. Phys. 32, 794, 1925; W. Heisenberg, ZS. f. Phys. 32, 841, 1925; F. Hund, ZS. f. Phys. 33, 345; 34, 296, 1925.
⁵) J. C. Slater, Nature, 113, 307, 1924; see also N. Bohr, H. A. Kramers and J. C. Slater, Phil. Mag. 47, 785, 1925 (the same article in ZS. f. Phys. 24, 69, 1924). — Translator’s note.

stating the assumption that there exists a definite relation between the “scattering” ability of vibrators and the probability of the corresponding transitions in Einstein’s theory. But decisive success in this direction was achieved by Kramers1. The latter, in accordance with the correspondence principle, gave an ingenious interpretation of those phenomena which, according to classical theory, take place in an electrodynamic system illuminated by light waves. What is characteristic in this interpretation is the following: just as radiation frequencies are calculated, on the one hand, according to classical theory and, on the other hand, according to quantum theory, so in the present case the derivatives of classical theory are replaced by ratios of finite differences. Only quantities accessible to direct observation enter into the final formulae. In Kramers’ theory, the scattering produced by an atom in a definite stationary state depends quantitatively on the frequencies corresponding to the processes of transition into other stationary states, and also on the probabilities of the occurrence of such transitions under the influence of illumination.

The essential feature of the theory is that, in calculating anomalous dispersion near a spectral line, two different kinds of resonance phenomena are taken into account, depending on whether the spectral line corresponds to the transition of the atom into a state of greater or lesser energy. Previously, when calculating dispersion on the basis of classical theory, only the resonance phenomena corresponding to the first transition were taken into account2. It is interesting to note that, in the further development of the theory, Kramers and Heisenberg3 gave a quantitative explanation of additional phenomena of scattering with changed frequency, the existence of which had been predicted by Smekal4 on the basis of the theory of light quanta5. This points to the fruitfulness of the latter theory.

The description of optical phenomena was in complete agreement with the fundamental ideas of quantum theory. But it soon turned out that it was in strange contradiction with the mechanical concepts which had previously been used for the analysis of stationary states—

...of knowledge. First of all, it proved impossible, on the basis of the scattering power of illuminated atoms required by the dispersion theory, to establish an asymptotic correspondence between the reaction of an atom to an alternating field of ever decreasing frequency and the reaction of an atom to a constant field, calculated on the basis of the quantization rules from the theory of periodic systems. These difficulties still further emphasize those doubts concerning the theory which are raised, as we have already mentioned, by the problem of the hydrogen atom in crossed electric and magnetic fields. Moreover, an unsatisfactory aspect of the theory of periodic systems must be regarded as the fact that it is apparently useless in solving the problem of the quantitative determination of transition probabilities, if one applies the mechanical conceptions of stationary states. This defect became still more noticeable after it proved possible in some cases to obtain a quantitative formulation of the most important propositions of the correspondence principle concerning these transition probabilities, by making use of an analysis of the optical properties of electrodynamic models1. These results are in excellent agreement with measurements of the relative intensities of spectral lines made at Utrecht, but they can be incorporated only in a very artificial way into schemes determined by the quantization rules2.

An attempt to construct a rational quantum mechanics.

Recently Heisenberg3, who has drawn special attention to these difficulties, has made a very significant, apparently, step forward on the path toward a new formulation of the problems of quantum theory. One may hope that this formulation will help to overcome the difficulties connected with the use of mechanical conceptions. In Heisenberg’s theory an attempt is made to express mechanical concepts and all their applications in such a way that they correspond to the nature of the theory of quanta and, moreover, so that at every stage of the calculation there enter only quantities accessible to direct observation. In pro-

ATOMIC THEORY AND MECHANICS

In contrast to ordinary mechanics, the new mechanics does not deal with the description of the motion of atomic particles in space and time1. It operates with collections of quantities which replace the components of harmonic oscillatory motion and symbolize the probabilities of transitions between stationary states, in accordance with the correspondence principle. These quantities satisfy known relations which replace the mechanical equations of motion and the rules of quantization.

A method of this kind leads to a self-sufficient theory having sufficient analogy with classical mechanics. This is seen from the fact that, as Born and Jordan have shown, in Heisenberg’s quantum mechanics there is a conservation theorem analogous to the law of conservation of energy in classical mechanics. The theory is constructed in such a way that it is in automatic agreement with the postulates of quantum theory. In particular, the frequency condition is fulfilled because the values of energy and frequency are derived from the quantum-mechanical equations of motion. Although the fundamental equations replacing the rules of quantization include Planck’s constant, the quantum numbers nevertheless do not enter into them explicitly. The classification of stationary states is based exclusively on consideration of the transition probabilities, which determine the successive formation of the totality of these states one after another. In short, the whole apparatus of quantum mechanics may be regarded as an exact formulation of the tendencies embodied in the correspondence principle. It should be added that the theory satisfies the requirements of Kramers’ dispersion theory.

In view of the great difficulties of a mathematical nature, it has not yet been possible to apply Heisenberg’s theory to the question of the structure of atoms. But even from our brief exposition one may conclude that in the new theory those results retain their significance which had earlier been derived on the basis of mechanical representations with the aid of the correspondence principle, such as, for example, the expression for Rydberg’s constant2. Moreover, it is extremely interesting

note that even in those simple cases which have so far been considered on the basis of Heisenberg’s theory, the new theory leads to a quantitative calculation of transition probabilities and of the energy values of stationary states which differs systematically from the calculation carried out by means of the quantization rules of the old theory. It may therefore be hoped that Heisenberg’s theory will prove useful in the struggle with the complex difficulties that arise in the study of the fine details of spectra.

Above we mentioned those profound difficulties which are connected with conceptions of interaction between atoms both through the medium of radiation and in collisions.

These difficulties apparently require the same abandonment of mechanical models in space and time as is characteristic of the new quantum mechanics. But the presently existing formulation of this mechanics does not yet consider the pairwise connected transition processes that occur in such interaction. In the new theory one encounters only those quantities which depend on the existence of stationary states and on transition probabilities between them, and the time during which these transitions take place is not considered at all.

This limitation, which is characteristic of the treatment of the question of the structure of atoms on the basis of quantum theory, makes it possible to discover only certain aspects of the analogy between the quantum theory and classical theories. Analogies of this kind relate chiefly to the properties of atoms in radiation, and here Heisenberg’s theory can render real assistance. It gives, for example, the possibility of establishing, for the processes of scattering, the existence of electrons bound in the atom, by means of a method analogous to the methods of the classical theory1, which, as we have already mentioned, led Thomson to the calculation of the number of electrons in an atom from the scattering of X-rays.

This application of the conservation laws to the interaction between atoms reveals quite different aspects of the correspondence between the quantum theory and classical theory. The latter are very important for the general formulation of the quantum theory, and discussion of them is inevitable in a more detailed investigation of the interaction between atoms and rapidly moving particles. It is precisely in this field that the classical theories have proved substantially important for our knowledge of the structure of atoms.

For mathematical circles it will be of interest that the mathematical methods created by higher algebra play an essential role in the formulation of the new quantum mechanics. Thus, for—

For example, the general proof of the conservation theorems in Heisenberg’s theory, given by Born and Jordan, is based on the application of matrix theory, which goes back to Cauchy and was developed especially by Hermite. One may hope that a new era has begun of mutual stimulation between mathematics and mechanics. Physicists will probably, above all, regret that in the problems of atomistics we apparently encounter a limitation of our general modes of representation. But this regret, one must think, will give way to gratitude for the fact that mathematics, even in this domain, provides us with instruments for further progress.

1. Cf. H. A. Kramers, Physica, Dec. 1925. ↩↩↩↩↩↩↩↩

2. Author’s note to the proof. Dr. Pauli kindly informed me that he has succeeded in deriving quantitatively from the new theory Balmer’s formula for the hydrogen spectrum, and also in calculating the influence of electric and magnetic fields on the spectrum. This result is of great importance, since Pauli’s analysis has shown that the new theory, in explaining spectral data, is free from the former difficulty, which consisted in the necessity of excluding stationary states corresponding to special solutions of the equations of motion of the electrons. ↩↩↩↩

3. W. Heisenberg, ZS. f. Phys. 33, 879, 1925. ↩↩↩

4. A. Smekal, Die Naturwissenschaften, 11, 873, 1923. ↩

5. Proceeding from the postulates on the energy and quantity of motion of the quantum, Smekal comes to the conclusion that, in scattering light, an atom emits quanta not only with a frequency equal to that of the incident light, but also with a greater or lesser frequency, depending on the transition into another stationary state. Another article by Smekal on this question: ZS. f. Phys., 32, 241, 1925. Translator’s note. ↩