Quantum Mechanics¹

Werner Heisenberg.

According to our ordinary “view,” i.e., on the basis of customary space-time concepts, space and matter are, in the final analysis, continuous and in principle divisible into arbitrarily small parts, if one disregards the possible technical impossibility of such division. But, in contrast to this consequence of our immediate intuition, physical and chemical experiments have shown that in phenomena occurring in very small spaces and intervals of time, a certain typically discontinuous element plays an important role. Already the chemical “multiple proportions” compelled one to assume an atomistic structure of matter; the so-called “fluctuations” (Brownian motion, scattering of light, etc.) led to the conception that matter is built of corpuscles of definite, finite magnitude; in experiments with corpuscular rays (cathode rays, $\alpha$-, $\beta$-rays) these particles, of which matter is built, became accessible to direct observation. Owing to such immediate experimental evidence for atomistic conceptions, it was natural to ascribe to the fundamental particles of matter, i.e., in the final analysis to the positive and negative electrons, the same degree of reality as to the objects of everyday life surrounding us. These particles came to be conceived as extremely small bodies of definite (and always one and the same) mass and charge, with an as yet unknown internal structure. These bodies move, according to certain laws still subject to further study, in space and time, in a definite continuous space-time world corresponding to our view. With the passage of time such a conception proved to be incorrect, which is not surprising if one takes into account the fundamental impossibility of visualizing the discontinuous element mentioned. Electrons, or atoms, do not possess that degree of immediate reality which objects

¹ Translation of a lecture delivered at the 89th Congress of German Natural Scientists and Physicians in Düsseldorf in 1926 (Die Naturwissenschaften, No. 45, 1926). Translator’s note.

of everyday experience. The study of the type of physical reality corresponding to atoms and electrons is the subject of atomic physics and, at the same time, of “quantum mechanics” (q. m.). The typical element of discontinuity discussed above is expressed not only by the fact of the atomic structure of matter, but also by the laws governing the structure of the atom. From Bohr’s theory, and experimentally from the collision experiments of Franck and Hertz and the experiment of Stern and Gerlach with a molecular beam, we conclude that there exist discrete stationary states of the atom. The processes of transition from one such state to another have to be regarded as typically discontinuous. Finally, we encounter such a discontinuous element in the phenomena of radiation. This was first found by Planck on the basis of the black-radiation law he had obtained. Einstein showed that the phenomena of fluctuations lead to the conception of “light corpuscles” with quite definite energy and momentum. Experiments on the photoelectric effect, the Compton phenomenon, and, in particular, the Bothe–Geiger experiment with the Compton phenomenon show with complete clarity the fruitfulness of the hypothesis of light quanta. In spite of this, in contrast to particles of matter, light quanta were never assigned the same degree of reality as the objects of the surrounding world. Such a conception would have led to overly great contradictions with the well-tested laws of classical optics. There are, however, indications—this was especially noted by Einstein—that, on the contrary, electrons possess the same degree of reality as light quanta. We shall return to this question later. Here it was important to emphasize that the study of the indicated typical discontinuous element and of the “degree of its reality” is the true task of atomic physics and the content of all the considerations of quantum mechanics.

I. On the basis of the experiments of Lenard and Rutherford and the great successes of Bohr’s theory, it could be considered proven that atoms are constructed of positive and negative electrons, as is assumed in this theory. But Bohr’s fundamental postulates of quantum theory at the same time signify a definitive break with the concepts of classical theory. In spite of this, it was natural to try to use classical concepts and images insofar as this was logically permissible. The form of Bohr’s theory that arose in this way, which made it possible, by means of the correspondence principle, to give a complete qualitative description of the character of atomic structure down to the details, proved unsatisfactory for the quantitative description of atomic processes. Great logical difficulties were also found in applying this form of the theory to certain problems (dispersion, the hydrogen atom in crossed fields). The true cause of these difficulties lay in the transfer of classical con-

Quantum Mechanics

concepts and representations alien to the essence of the fundamental quantum postulates, to the problems of the structure of the atom—the use of simple visual models and images for interpreting physical regularities, the degree of visualizability of which is in reality by no means clear.

The program of quantum mechanics, therefore, had first of all to consist in liberation from visual images and in establishing simple relations between experimentally given quantities, instead of the laws of classical kinematics and mechanics that had been used hitherto. The former theory combined the advantage of immediate visualizability and the application of tested physical principles with the drawback that one had to operate with relations fundamentally not amenable to verification, which could lead to internal contradictions. The new theory, on the contrary, is compelled first of all to abandon visualizability completely, but in return it must contain only concrete relations accessible to direct experimental verification and free from the danger of internal contradictions.

To achieve this aim, of course, it was necessary to depart far from classical conceptions. Let us turn, for example, to the spectrum of hydrogen. As is known, the contradiction consisted in the fact that, according to classical kinematics, the spectrum of any periodic motion of a particle must consist of equidistant lines. In reality a line spectrum is observed, with lines condensing in a finite region. Nevertheless, on the basis of the correspondence principle, we speak of the periodic motion of the electron. If, in general, the corpuscular conception is to be preserved, then the difficulty can be avoided only by refusing to ascribe to the electron, or to the atom, a definite point in space as a function of time. To justify this it is necessary to assume that such a point cannot be directly observed. Such a refusal signifies the first decisive restriction in considering the question of the reality of corpuscles.

Instead of the discarded concept of the “position of the electron,” quantum mechanics attempts to introduce a set of physically well-defined quantities which in the classical theory are mathematically equivalent to the position of the electron. In the classical theory the complete radiation of an electron is given by expanding the electron’s motion in a Fourier series; this expansion may also be regarded as a representation of the electron’s motion. The frequency, amplitude, and polarization of a spectral line are in any case well-defined quantities accessible to observation. Therefore in quantum mechanics the set of radiation quantities accessible to observation and corresponding to the classical Fourier series is regarded as the equivalent of the “position of the electron.” According to the fundamental postulates of the theory of quanta, the emission of a line is connected with a transition from one stationary state to another. Therefore each radiation quantity corres-

1*

created by two terms, or states. Instead of the classical “coordinates of the electron,” quantum mechanics consequently has a two-dimensional “table” of radiation quantities, a so-called “matrix.”

It was natural to take the next step as well, i.e. to introduce such tables of concrete observable quantities in place of concepts borrowed from classical theory and, perhaps, not directly observable, such as momentum, energy, etc. For example, the total energy of an atom corresponds to a table of energy values of the stationary states of the atom. The proposal just described means a further restriction of the character of reality of atoms considered above. On the other hand, the same proposal made it possible to establish a remarkable close connection between quantum mechanics and those discontinuities that are contained in Bohr’s theory. First of all it was found that the existence of discrete energy values is as natural for quantum mechanics as, say, the existence of a discrete series of natural vibrations of a membrane is in classical theory. Further, it turned out that simple relations, obtained, for example, from considerations concerning the frequency of transitions or concerning the mean values of discontinuous variable quantities in time, follow as a mathematical result of calculation with such tables composed of radiation quantities. It seems to me that this contains one of the most important properties of quantum mechanics. Unfortunately, precisely this feature of quantum mechanics has so far been little investigated.

In order to obtain a complete theory it is still necessary to find the mathematical relations between the indicated tables of radiation quantities that would correspond to the analogous relations of classical mechanics. It turned out that, from the formal side, these relations are very simple. On the basis of conclusions from physical analogy it was discovered that addition and multiplication of these tables must be performed according to the well-known rules of matrix algebra. From the purely formal side, the difference between the new and the old theory consisted above all in the inapplicability of the commutative law in multiplication. The Hamiltonian equations of mechanics can, in form, be completely rewritten in the new theory. Owing to the noncommutativity of the factors in multiplication, it was necessary to find still certain rules of permutation in order to give the mathematical scheme of the theory a completed form. These rules correspond, in a certain sense, to the quantum conditions of the former theory and contain Planck’s constant.

Thus the mathematical scheme of quantum mechanics is completely finished. In its detailed implementation, of which I shall now speak, it turns out that in many respects quantum mechanics is quite similar to classical theory. The laws of conservation of energy and momentum remain, as in classical theory. One can develop the theory of canonical transformations and thereby obtain a complete theory of perturbations, entirely corresponding to the methods of astronomy. In this

relations, quantum mechanics is even much simpler than classical mechanics. Perturbation series in the many-body problem are associated in classical theory with well-known difficulties of convergence. In quantum mechanics these series, generally speaking, converge, and therefore the many-body problem presents no difficulties of principle.

Thus quantum mechanics, at least in principle, corresponds within very broad limits to our factual knowledge about atoms. For macroscopic processes quantum mechanics formally passes into classical mechanics; consequently, the character of reality comes extremely close to our ordinary view. For microscopic processes there remain only relations among observable, experimentally given quantities. For the time being it is impossible to give a directly intuitive interpretation of the physical processes underlying this.

For an experimental verification of the theory there is unusually extensive material: the spectra of all elements, measurements of energies, etc. To compare the theory with experience it was first necessary to develop the theory mathematically. This has been accomplished in three different independent ways.

1. Born–Jordan. Quantum quantities are given from the very beginning in the form of matrices, and therefore the known methods of higher algebra can be applied. The solution of the quantum problem is reduced to the problem of characteristic numbers (Eigenwertproblem), namely to the transformation of principal axes in the Hermitian form. In this way the following have been analyzed: the hydrogen atom, the Zeeman effect, the Stark effect (Pauli), the intensities of lines in the Zeeman effect, multiplets and the anomalous Zeeman effect, the question of the fine structure of spectral lines, band spectra (oscillator and rotator), and the theory of dispersion.

2. Dirac. An algebra and an analysis have been developed for quantities for which the commutative law is not satisfied ($q$-numbers), independently of the interpretation of these quantities as “matrices.” The calculation has been developed to such an extent that it is possible to find simple methods of calculation for mechanical problems, very similar to the methods of the classical theory (the introduction of action variables and angle variables, Fourier series; cf. also the works of London). The following have been considered: the hydrogen atom, intensity formulae for multiplets, $q$-values, relativistic quantum mechanics, the Compton effect, recoil $\frac{h\nu}{c}$, and dispersion.

3. Schrödinger. I shall speak later of the physical foundations of Schrödinger’s theory; for the moment it is necessary to point out its significance for the mathematical development of quantum mechanics. By the known principles of higher linear algebra and analysis, the problem of the characteristic numbers of an infinite quadratic form, as in Born–Jordan, is in general equivalent to the analytic problem of characteristic

numbers, determined by a linear differential equation and boundary conditions. When it proved possible to find this linear differential equation and this problem of characteristic numbers, the mathematical treatment of quantum problems was reduced to widely developed mathematical methods. This was carried out by Schrödinger, who, however, arrived at such a mathematical scheme by another route than the one indicated here, and independently of earlier work in quantum mechanics. The following have been considered: the hydrogen atom (without fine structure in a magnetic field), the Stark and Zeeman effects, band spectra, dispersion. The advantage of Schrödinger’s method consists chiefly in the fact that it makes it possible simply to determine transition probabilities. In other mathematical methods this is, generally speaking, very difficult. The following have been considered: the intensities of the components of the Stark effect for the hydrogen atom, formulas for the intensities for the Zeeman effect, intensities in the Lyman and Balmer series (Pauli).

II. Let us now return from the mathematical development of the theory once more to the physical meaning of its formal side, i.e. to an analysis of those assertions which can be made concerning the reality and laws of corpuscles. De Broglie, in his theory, approached this problem from an entirely new side. From de Broglie’s theory arose Schrödinger’s considerations and the application of statistics by Bose and Einstein. The restrictions made by us in Part I concerning the reality of corpuscles, in particular the assertion that it is impossible to assign to a corpuscle a definite place as a function of time, a definite energy, etc., make it possible to suppose that the reality of material corpuscles is in many respects similar to the reality of light quanta. The phenomena of interference and diffraction of light waves do not permit one to assign to light quanta a definite path and position. The analogy between material corpuscles and light quanta becomes especially clear if one investigates, according to the laws of quantum mechanics, the reflection of a material particle, for example, from a diffraction grating. Jordan, on the basis of one of Duane’s considerations, has recently shown that the reflection of material particles from a grating according to quantum mechanics must occur in perfectly definite discrete directions, just as the diffraction of a light ray does. Analogies of this kind led de Broglie, long before the emergence of quantum mechanics, to the following supposition1. In the theory of light there exists at present a remarkable dualism; many phenomena are described by the wave theory, others by the theory of light quanta, and some by both theories. It is permissible to assume a dualism of the same kind also for material particles. On this basis de Broglie assigned to every material particle a wave of a definite frequency. This frequency, as in the case of light quanta, is determined by

of the particle’s energy from the relation \(E=h\nu\). According to Einstein, these waves, like light waves, can interfere. Consequently, the stream of electrons upon reflection from the grating must proceed only in certain discrete directions. As was indicated, the same result was later derived from quantum mechanics. Hence it is natural to suppose that quantum mechanics and de Broglie waves are closely connected. De Broglie had already shown that one can obtain the Bohr orbits of the hydrogen atom if one imposes the condition that the waves corresponding to the motion of the nucleus around the electron are a single-valued function of space. The true connection between de Broglie’s theory and quantum mechanics was discovered by Schrödinger. This investigator further developed the ideas of de Broglie and Einstein; he established the differential equation of the de Broglie waves and showed that the problem of the characteristic numbers of this equation coincides with the problem of the characteristic numbers of quantum mechanics. It turned out, however, that in the theory of matter one cannot find a three-dimensional wave equation as in the theory of light, since the velocity of the waves always depends on the presence of other particles. For a problem concerning the motion of \(f\) particles, it is possible to establish a wave equation in a coordinate space of \(3f\) dimensions, which is mathematically quite equivalent to the problem of quantum mechanics. So far it has not been possible, in the general case, to find a direct connection between Schrödinger waves in phase space and de Broglie waves in ordinary space, which must be similar to light waves. Thus waves in \(q\)-space as yet have a formal significance. In his time Hamilton discovered a great formal analogy between classical mechanics and geometrical optics in many-dimensional spaces. This circumstance served as a powerful mathematical method for solving classical problems. In exactly the same way, according to Schrödinger, there is a great formal similarity between quantum mechanics and wave optics in many-dimensional spaces, and a powerful mathematical method is obtained for solving the problems of quantum mechanics. Recently it has repeatedly been suggested (Schrödinger, Flamm) that, on the basis of Schrödinger’s differential equation, a purely continuous description of quantum data is possible, approximately in the same way as in classical theory; that is, that quantum theory in the form in which it has existed up to now is illusory. For the consistent pursuit of such a point of view, however, one must abandon precisely the foundations of de Broglie’s theory, and consequently of quantum mechanics and of quantum theory in general. Thus, in my opinion, a complete contradiction with experiment is inevitable (the law of black radiation, the theory of dispersion). Consequently this path is unacceptable. The actual reality of de Broglie waves is revealed in the interference phenomena indicated above, which elude any interpretation on the basis of classical concepts. The exceptional physical significance of Schrödinger’s results consists in the assertion that

the expectation that the intuitive interpretation of the formulas of quantum mechanics contains typical features of both the corpuscular and the wave theory.

III. On the basis of Bose’s statistics of light quanta, Einstein proposed a statistics of particles of matter that reveals the reality of de Broglie waves from yet another side. Bose showed that one can obtain the statistics of light corpuscles, in agreement with experiment, if one refrains from using the position of a particle in phase space to define the “state”; instead, the “state,” according to Bose, is determined by the number of identical particles found in a given cell of phase space. This basic assumption of Bose’s, apparently quite incompatible with corpuscular theory, is in any case a noteworthy restriction on the reality of corpuscles. It becomes, however, somewhat intelligible if one passes from light corpuscles to certain light waves, “corresponding” to them in a way still unknown. Instead of the “number of corpuscles,” the state will then be determined by the “energy of the proper oscillation,” which is entirely consistent with ordinary statistics. Einstein transferred this basic assumption of Bose’s statistics directly to the statistics of material particles. From the point of view of de Broglie’s wave theory such a statistics is intelligible. For corpuscular theory it means that, generally speaking, it is impossible to trace a corpuscle along its path and distinguish it from others, i.e. that the individuality of the corpuscle may be lost. Such an assumption is in complete agreement with the restrictions on such concepts as the position of an electron, etc., which were made above in deriving the foundations of quantum mechanics. Despite this, at first quantum mechanics remained incompatible with Einstein’s statistics. Quantum mechanics calculates by corpuscles, whereas Schrödinger’s calculation is by waves in \(3f\)-dimensional spaces; therefore the counting of states, for example of an atom, at first always gives a result corresponding to classical statistics.

To clarify this contradiction, many-body problems occurring in atomic systems were investigated in detail. It is true that here one is dealing with a mechanical problem different from the one that underlies Einstein’s statistics. But one could hope to find here as well essential features that distinguish Einstein’s statistics from classical statistics. First of all it turned out (with the reservation indicated) that the experimentally found number of stationary states in atoms with many electrons definitely argues in favor of a reduction of statistical weights in Einstein’s sense and not in favor of classical statistics.

If classical statistics were correct, the number of stationary states would be many times greater than that observed. It further turned out that the solution of the many-body problem in quantum mechanics exhibits a characteristic indefiniteness: the complete system of terms, or spectrum,

terms in the problem decomposes into various partial systems. Among these partial systems there is one that contains no “equivalent orbits.” It differs from the other systems in that from its terms there can be no transitions to the terms of the remaining partial systems. The general system of terms, and equally the indicated partial system, may be regarded as a complete quantum solution of the problem. For the solution in quantum mechanics is subject only to the requirement that the system of terms be “closed”; in a closed system, transitions are possible only within the system. If, without any justification, the indicated partial system is chosen as the final quantum solution, then the statistical weights are reduced precisely in Einstein’s sense; on the other hand, Pauli’s prohibition of equivalent orbits is automatically fulfilled. On the basis of these investigations it cannot be decided to what extent Pauli’s prohibition is inversely connected with Einstein’s modification of statistics. The existence of a close connection with Einstein’s statistics is evident from the fact that the indicated reduction of statistical weights is possible only when there is complete equality of the particles in the many-body problem (for electrons such equality, of course, is present). If there are even the slightest differences among the particles, there must be transitions between the partial systems named above. The quantum solution of the problem will then be the entire system of terms, which corresponds to the classical count. The mechanical basis of the subdivision of the spectrum of terms, considered above, into partial systems that do not combine with one another is the characteristic resonance phenomenon. It consists in the fact that in all many-body problems the particles forming the system continuously exchange places. For example, it is meaningless to distinguish inner and outer electrons in atoms. If, as indicated above, one chooses a partial system as the quantum solution, then in the course of the calculation this means that only symmetric functions of the electrons in the atom have physical meaning; one cannot speak of the motion of a single electron, or of a matrix representing this motion. Here there is a new limitation in the question of the reality of corpuscles. As examples of the many-body problem, the spectra of atoms with two electrons, i.e. He and Li\(^+\), have so far been investigated; the results agree satisfactorily with experiment.

Let us briefly summarize once more the various assertions that can be made, on the basis of the considerations set forth, concerning the typical discontinuous element in processes occurring in small regions of space and concerning the degree of their reality.

1. On the basis of all experiments with \(\alpha\)- and \(\beta\)-rays, Wilson photographs, experiments with molecular rays, etc., there follows direct experimental evidence for material corpuscles. In the same way, experiments with the photoelectric effect, the Geiger—Bothe experiment (cf. also Bothe’s new experiment and Kirchner’s investigations)

directly reveal the reality of light quanta. The existence of discrete stationary states in atoms has been revealed by the experiments of Franck and Hertz and by the experiments of Stern and Gerlach with a molecular beam.

1. There is no possibility of assigning to a corpuscle a definite position as a function of time, but there can be associated with it a set of radiation quantities corresponding to a Fourier series in the classical theory. Furthermore, in a series of identical corpuscles it is in principle impossible to distinguish or identify any particular corpuscle.

2. In our visual interpretations of physical processes and in mathematical formulas there is a dualism of the wave and corpuscular theories. This dualism is of such a kind that many phenomena are most naturally described by the wave theory of light, or of matter—especially the phenomena of interference and diffraction. Other phenomena can be interpreted only on the basis of the corpuscular theory.

These propositions were intended to outline in broad contours the present state of our knowledge of the typical discontinuous element that reveals itself in processes occurring in very small regions of space. The contradictions in the visual interpretations of various phenomena, present in the schemes used up to now, are wholly unsatisfactory. For a visual interpretation of experience that is free from contradictions—experience which in itself, of course, is free from contradictions—there is still lacking some essential feature in our conception of the structure of matter.

1. On de Broglie’s theory see Ya. I. Frenkel, UFN, 4, 1925. ↩