1.3 THE PHYSICAL CONTENT OF QUANTUM KINEMATICS AND MECHANICS W e r n er H eisenber g First we define the terms velocity, energy, etc. (for example, for an electron) which remain valid in quantum mechanics. It is shown that canonically conjugate quantities can be determined simulta­ neously only with a characteristic indeterminacy (§1). This indeter­ minacy is the real basis for the occurrence of statistical relations in quantum mechanics. Its mathematical formulation is given by the Dirac-Jordan theory (§2). Starting from the basic principles thus obtained, we show how microscopic processes can be understood by way of quantum mechanics (§3). To illustrate the theory, a few special gedankenexperiments are discussed (§4). We believe we understand the physical content of a theory when we.can see its qualitative experimental consequences in all simple cases and when at the same time we have checked that the application of the theory never contains inner contradictions. For example, we believe that we understand the physical content of Einstein’s concept of a closed 3-dimensional space because we can visualize consistently the experimental consequences of this concept. Of course these con­ sequences contradict our everyday physical concepts of space and time. However, we can convince ourselves that the possibility of employing usual space-time concepts at cosmological distances can be justified neither by logic nor by ob­ servation. The physical interpretation of quantum mechanics is still full of internal discrepancies, which show themselves in arguments about continuity versus dis­ continuity and particle versus wave. Already from this circumstance one might conclude that no interpretation of quantum mechanics is possible which uses ordinary kinematical and mechanical concepts. Of course, quantum mechanics arose exactly out of the attempt to break with all ordinary kinematic concepts and to put in their place relations between concrete and experimentally determinable numbers. Moreover, as this enterprise seems to have succeeded, the mathematical scheme of quantum mechanics needs no revision. Equally unnecessary is a revi­ sion of space-time geometry at small distances, as we can make the quantummechanical laws approximate the classical ones arbitrarily closely by choosing sufficiently great masses, even when arbitrarily small distances and times come into question. But that a revision of kinematical and mechanical concepts is necessary Originally published under the title, “Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Zeitschrift for Physik, 43, 172-98 (1927); reprinted in Dokumente der Naturwissenschaft, 4, 9 -3 5 (1963); translation into English by J.A.W. and W.H.Z., 1981. 1.3 PRINCIPLE OF INDETERMINISM 63 seems to follow directly from the basic equations of quantum mechanics. When a definite mass m is given, in our everyday physics it is perfectly understandable to speak of the position and the velocity of the center of gravity of this mass. In quantum mechanics, however, the relation pq —qp = —ih between mass, posi­ tion, and velocity is believed to hold. Therefore we have good reason to become suspicious every time uncritical use is made of the words “position” and “velocity.” When one admits that discontinuities are somehow typical of processes that take place in small regions and in short times, then a contradiction between the con­ cepts of “position” and “velocity” is quite plausible. If one considers, for example, the motion of a particle in one dimension, then in continuum theory one will be able to draw (Fig. 1) a worldline x(t) for the track of the particle (more precisely, its center of gravity), the tangent of which gives the velocity at every instant. In contrast, in a theory based on discontinuity there might be in place of this curve a series of points at finite separation (Fig. 2). In this case it is clearly meaningless to speak about one velocity at one position (1) because one velocity can only be defined by two positions and (2), conversely, because any one point is associated with two velocities. The question therefore arises whether, through a more precise analysis of these kinematic and mechanical concepts, it might be possible to clear up the contradic­ tions evident up to now in the physical interpretations of quantum mechanics and to arrive at a physical understanding of the quantum-mechanical formulas.* * The present work has arisen from efforts and desires to which other investigators have already given clear expression, before the development of quantum mechanics. I call attention here especially to Bohr’s papers on the basic postulates of quantum theory (for example, Zeits. f . Physik, 13, 117 [1923]) and Einstein’s discussions on the connection between wave field and light quanta. The problems dealt with here are discussed most clearly in recent times, and the problems arising are partly answered, by W. Pauli (“Quantentheorie,” Handbuch der Physik, Vol. XXIII, cited hereafter as /.c.); quantum mechanics has changed only slightly the formulation of these problems as given by Pauli. It is also a special pleasure to thank here Herrn Pauli for the repeated stimulus I have received from our oral and written discussions, which have contributed decisively to the present work. §1. C o n c e pt s: P osition, P a t h , V elocity, E nerg y In order to be able to follow the quantum-mechanical behavior of any object one has to know the mass of this object and its interactions with any fields and other objects. Only then can the Hamiltonian function be written down for the quantummechanical system. (The following considerations ordinarily refer to nonrelativistic quantum mechanics, as the laws of quantum electrodynamics are still very incom­ pletely known.)* About the “Gestalt” (construction) of the object any further assumption is unnecessary; one most usefully employs the word “Gestalt” to designate the totality of these interactions. When one wants to be clear about what is to be understood by the words “position of the object,” for example of the electron (relative to a given frame of reference), then one must specify definite experiments with whose help one plans to measure the “position of the electron”; otherwise this word has no meaning. There is no shortage of such experiments, which in principle even allow one to determine the “position of the electron” with arbitrary accuracy. For example, let one illuminate the electron and observe it under a microscope. Then the highest attainable accuracy in the measurement of position is governed by the wavelength of the light. However, in principle one can build, say, a y-ray microscope and with it carry out the determination of position with as much accuracy as one wants. In this measurement there is an important feature, the Compton effect. Every obser­ vation of scattered light coming from the electron presupposes a photoelectric effect (in the eye, on the photographic plate, in the photocell) and can therefore also be so interpreted that a light quantum hits the electron, is reflected or scattered, and then, once again bent by the lens of the microscope, produces the photoeffect. At the instant when position is determined—therefore, at the moment when the photon is scattered by the electron—the electron undergoes a discontinuous change in momentum. This change is the greater the smaller the wavelength of the light employed—that is, the more exact the determination of the position. At the instant at which the position of the electron is known, its momentum therefore can be known up to magnitudes which correspond to that discontinuous change. Thus, the more precisely the position is determined, the less precisely the momen­ tum is known, and conversely. In this circumstance we see a direct physical interpretation of the equation pq —qp = —ih. Let qx be the precision with which the value q is known (qx is, say, the mean error of q\ therefore here the wavelength of the light. Let p { be the precision with which the value p is determinable; that is, here, the discontinuous change of p in the Compton effect. Then, according to the elementary laws of the Compton effect px and q{ stand in the relation * Quite recently, however, great advances in this domain have been made in the papers of P. Dirac \Proc. Roy. Soc. A114, 243 (1927) and papers to appear subsequently]. 1.3 PRINCIPLE OF INDETERMINISM 65 Pi<7i ~ (1) That this relation (1) is a straightforward mathematical consequence of the rule pq —qp = —ih will be shown below. Here we can note that equation (1) is a precise expression for the facts which one earlier sought to describe by the division of phase space into cells of magnitude h. For the determination of the position of the electron one can also do other experiments—for example, collision experi­ ments. A precise measurement of the position demands collisions with very fast particles, because for slow electrons the diffraction phenomena—which, according to Einstein, are consequences of de Broglie waves (as, for example, in the Ramsauer effect)—prevent a sharp specification of location. In a precise measurement of position the momentum of the electron again changes discontinuously. An ele­ mentary estimate of the precision using the formulas for de Broglie waves leads once more to relation (1). Throughout this discussion the concept of “position of the electron” seems well enough defined, and only a word need be added about the “size” of the electron. When two very fast particles hit the electron one after the other within a very short time interval At, then the positions of the electron defined by the two particles lie very close together at a distance A/. From the regularities which are observed for a-particles we conclude that AZcan be pushed down to a magnitude of the order of 10“ 12cm if only At is sufficiently small and particles are selected with sufficiently great velocity. This is what we mean when we say that the electron is a corpuscle whose radius is not greater than 10“ 12 cm. We turn now to the concept of “path of the electron.” By path we understand a series of points in space (in a given reference system) which the electron takes as “positions” one after the other. As we already know what is to be understood jay “position at a definite time,” no new difficulties occur here. Nevertheless, it is easy to recognize that, for example, the often used expression, the “Is orbit of the elec­ tron in the hydrogen atom,” from our point of view has no sense. In order to mea­ sure this Is “path” we have to illuminate the atom with light whose wavelength is considerably shorter than 10“ 8 cm. However, a single photon of such light is enough to eject the electron completely from its “path” (so that only a single point of such a path can be defined). Therefore here the word “path” has no definable meaning. This conclusion can already be deduced, without knowledge of the recent theories, simply from the experimental possibilities. In contrast, the contemplated measurement of position can be carried out on many atoms in a Is state. (In principle, atoms in a given “stationary” state can be selected, for example, by the Stern-Gerlach experiment.) There must therefore exist for a definite state—for example, the Is state—of the atom a probability function for the location of the electron which corresponds to the mean value for the classical orbit, averaged over all phases, and which can be determined through 66 HEISENBERG the measurement with an arbitrary precision. According to Born,* this function is given by 0 all values of q are equally probable; that is, the probability that q lies in any finite region is quite nil. Physically this is already clear without further investigation. Thus the exact determination of q0 leads to an infinitely large Compton recoil. The same would naturally apply for an arbitrary mechanical system. If, however, q0is known at the time t = 0 only within the range ql9 and p0 in the range px [see equation (3)], then S(q,q0) = const exp [-(^ o - q')2/2q] - (ijh)p'{q0 - and the probability [amplitude] function for q is to be calculated according to the formula, S(q, q) = J S(ti, ,q)dq0. The result is S(rj, q) = const Jexp {(im/ht)[q0(q - tp'/m) - ql/2] - {q - q0)2l2q\\dq0. (14) With the abbreviation j3 = th/mq\ , (15) the exponent in (14) becomes * The word “representation,” not employed by Heisenberg himself, is introduced here for clarity. He uses the phrase “matrix scheme.” (Translators’ note.) HEISENBERG -{(n + 1) At J n At S(p,E)= f <"+1)4' S(p, t ) e ^ hdt. If the determination “state n = 2” is made at the time (n + 1) At, then for every­ thing later one must ascribe to the atom not the eigenfunction (18) but one which results from (18) when t is replaced by t —(n + 1) At. If, on the contrary, one finds “state n = 1,” then from that point on one has to attribute to the atom the eigen­ function il/{Eu p)e~iElt/\ Thus one will first find for a series of intervals At “state n = 2,” then steadily “state n = 1.” In order that a distinction between the two states will still be possible, At cannot be shrunk below h/AE. Thus the instant of the transition is determinable within this spread. We imply an experiment of the kind just sketched quite in the spirit of the old formulation of quantum theory founded by Planck, Einstein, and Bohr when we speak of the discontinuous change of the energy. As such an experiment can in principle be carried out, an agreement about its outcome must be possible. In Bohr’s basic postulates of quantum theory, the energy of an atom has the Compare W. Pauli, l.c., p. 12. HEISENBERG advantage—just as do the values of the action variables J—over other deter­ minants of the motion (position of the electron, etc.) in that its numerical value can always be given. This preferred position which the energy has over other quantum-mechanical quantities it owes only to the circumstance that it represents an integral of the equations of motion for closed systems (the energy matrix E is a constant). For open systems, in contrast, the energy is not singled out over any other quantum-mechanical quantity. In particular, one will be able to devise experiments in which the phases, w, of the atom are precisely measurable, but in which then the energy remains in principle undetermined, corresponding to the relation Jw — wj = —ift or Jxwx ~ h. Resonance fluorescence is such an experi­ ment. If one irradiates an atom with an eigenfrequency, say v12 = (E2 — EJ/h, then the atom vibrates in phase with the external radiation. Then, even in principle, it makes no sense to ask in which state, E x or E2, the atom is thus vibrating. The phase relation between atom and external radiation may be determined, for example, by the phase relations of large numbers of atoms with one another ([R. W.] Wood’s experiments). If one prefers to avoid experiments with radiation then one can also measure the phase relation by carrying out exact position determinations on the electron in the sense of§l at different times relative to the phase of the light impinging (on many atoms). A “wave function,” say, of the form, S(q,t) = c2^ 2(E2, qXe-HE2t + m + (1 _ ih(£i, (1 can be ascribed to the individual atom. Here c2 depends on the strength and /? on the phase of the incident light. The probability of a definite position q is thus S(q, t)S(q, + t) = c ty 2$ 2 + (I - c2)4 c2(1 - d ) 1/2{^2ipie - ‘HE2 -Ei),+m + > The periodic term in (20) is experimentally distinguishable from the unperiodic ones, as the determinations of position can be carried out for different phases of the incident light. In a well-known idealized experiment proposed by Bohr, the atoms of a SternGerlach atomic beam are first excited to a resonance fluorescence at a definite state by incident radiation. After a little way they go through an inhomogeneous magnetic field. The radiation emerging from the atoms can be observed during the whole path, before and after the magnetic field. Before the atoms enter the magnetic field, ordinary resonance radiation takes place; that is, as in dispersion theory, it must be assumed that all atoms send out spherical waves in phase with the incident light. The latter view at first sight contradicts the result that a crude application of the quantum theory of light or the basic rules of quantum theory would give. Thus from this view one would conclude that only a few atoms are raised to the 1.3 PRINCIPLE OF INDETERMINISM 79 “upper state” through absorption of the light quantum, and that therefore the entire resonance radiation arises from a few intensively radiating centers. It therefore seemed natural in earlier times to say that the concept of light quanta ought to be brought in here only to account for the balance of energy and momen­ tum, and that “in reality” all atoms in the ground state radiate weak and coherent spherical waves. After the atoms have passed the magnetic field, however, there can hardly be any doubt that the atomic beam has divided into two beams of which one corresponds to atoms in the upper state, the other in the lower. If now the atoms in the lower state were to radiate, then we would have a gross violation of the law of conservation of energy. For all the energy of excitation resides in the beam with atoms in the upper state. Still less can there be any doubt that past the magnetic field only the “upper state” beam sends out light, and indeed incoherent light, from the few intensively radiating atoms in the upper state. As Bohr has shown, this idealized experiment makes it especially clear that care is often needed in applying the concept of “stationary state.” The formulation of quantum theory developed here allows a discussion of the Bohr experiment to be carried through without any difficulty. In the external radiation field the phases of the atoms are determined. Therefore it is meaningless to speak of the “energy of the atom.” Also, after the atom has left the radiation field one is not entitled to say that it is in a definite stationary state, insofar as one enquires about the coherence properties of the radiation. However, one can set up an experiment to find out in what state the atom is. The result of this experiment can be stated only statistically. Such an experiment is really performed by the inhomogeneous magnetic field. Beyond the magnetic field the energies of the atoms are well determined and therefore the phases are indeterminate. The resulting radiation is incoherent and comes only from atoms in the upper state. The magnetic field determines the energies and therefore destroys the phase relation. Bohr’s idealized experiment is a very beauti­ ful illustration of the fact that the energy of the atom “in reality” is not a number but a matrix. The conservation law holds for the energy matrix and therefore also for the value of the energy as precisely as it can be measured. In mathematical terms the lifting of the phase relation can be traced out as follows, for example. Let Q be the coordinates of the center of gravity of the atom, so that one ascribes to the atom, instead of (19), the eigenfunction S(Q,t)S(q9t) = S(Q9q9t). (21) Here S(Q, t) is a function that, like S(rj, q) in (16), differs from zero only in a small neighborhood of a point in Q-space and propagates with the velocity of the atoms in the direction of the beam. The probability of a relative amplitude q regardless of Q is given by the integral of S(Q, q, t)S(Q, q9t) HEISENBERG over Q—that is, through (20). However, the eigenfunction (21) will change by a calculable amount in a magnetic field and, on account of the different deviation of atoms in the upper and lower state, will have changed beyond the magnetic field into S(Q, q, t) = c2S2(Q, tW2(E29 q ) e - ^ t+^ + (1 - c2)1/2S ,(& # i ( £ i , q)e~iE'tlh. (22) Here t) and S2(Q, t) will be functions in Q-space which differ from zero only in the small neighborhood of a point; but this point is different for Sx and S2. The product S XS2 is therefore zero everywhere. The probability of a relative amplitude q and a definite value Q is therefore S{Q, q,t)S(Q, q, t) =clS2S2il/2ij/2+ (1 - c|)Sj The periodic term of (20) has disappeared and with it the possibility of measuring a phase relation. The statistical result of position determinations will always be the same, whatever the phase of the incident radiation. We may assume that experiments with radiation, the theory of which has not yet been developed, will give the same results about phase relations between atoms and incident radiation. Finally let us examine the connection* of equation (2), E it1 ~ h, with a complex of problems which Ehrenfest and other investigators have discussed on the basis of Bohr’s correspondence principle in two important papers.* Ehrenfest and Tolman speak of “weak quantization” when the quantized periodic motion is interrupted through quantum jumps or rather perturbations in intervals of time which can be regarded as not very long compared to the periods of the system. These cases should reveal not only the exact quantum energy values but also energy values which do not differ too much from the quantum values, and these with a smaller and qualitatively predictable a priori probability. In quantum mechanics this behavior is to be interpreted in these terms. As the energy is really changed by external perturbations or quantum jumps, every energy measurement, insofar as it is to be unique, must be done in the time between two perturbations. In this way an upper bound is specified for in the sense of §1. Therefore we measure the energy value E0 of a quantized state also only within a spread E 1 ~ h/tl. Here the question is meaningless even in principle whether the system “really” takes on with the correspondingly lower statistical weight such energy values E as deviate from E0, or whether their experimental realization is to be attributed only to the inaccuracy of the measurement. If t t is smaller than the period of the system then it is no longer meaningful to speak of discrete stationary states or discrete energy values. * W. Pauli drew my attention to this connection. f P. Ehrenfest and G. Breit (Zeits. f . Physik, 9, 207 [1922]) and P. Ehrenfest and R. C. Tolman (Phys. Rev., 24, 287 [1924]). See also the discussion in N. Bohr, Grundpostulate der Quantentheorie, l.c. 1.3 PRINCIPLE OF INDETERMINISM 81 Ehrenfest and Breit in a similar connection draw attention to the following paradox. A rotator, which we will visualize as a gear-wheel, is provided with an attachment which after / revolutions of the wheel exactly reverses the direction of its rotation. For example, let the gear-wheel mesh with a toothed sliding member which moves on a straight line between two stops. The slider hits a stop after a definite number of rotations and in that way reverses the rotation of the gear­ wheel. The true period T of the system is long in comparison with the rotation period t of the wheel. The discrete energy levels are densely packed—and the denser the packing, the greater the value of T. From the standpoint of consistent quantum theory all stationary states have the same statistical weight. Therefore, for sufficiently great T, practically all energy values occur with equal frequency, in opposition to what would be expected for the rotator. We may sharpen this paradox a little before we treat it from our standpoint. Thus, in order to determine whether the system takes on the discrete energy values belonging to the pure rotator exclusively or particularly often, or whether it assumes with equal prob­ ability all possible values (that is, values which correspond to the small energy interval h/T), a time t1 suffices which is small relative to T (but » t). In other words, although the long period plays no part at all in such measurements, it appears to express itself in the fact that all possible energy values can occur. We are of the view that, in reality also, such experiments for the determination of the total energy of the system would give all possible energy values with equal prob­ ability. The factor responsible for this outcome is not the big period T, but the sliding member. Even if the system sometimes happens to have an energy identical with the quantized energy value of the simple rotator, it can be modified easily—by external forces acting on the stop—to states which do not correspond to the quantization of the simple rotator.* The coupled system, rotator-plus-slider, indeed shows a periodicity entirely different from that of the rotator. The solution of the paradox lies rather in a different circumstance. When we want to measure the energy of the rotator alone, we must first break the coupling between the rotator and the slider. In classical theory, when the mass of the slider is sufficiently small, the coupling can be broken without a change in energy; and there, conse­ quently, the energy of the entire system can be equated to that of the rotator (for small slider mass). In quantum mechanics the energy of interaction between slider and rotator is at least of the same order of magnitude as the level spacing of the rotator (even for small slider mass there is a high zero point energy associated with the elastic interaction between rotator and slider). On decoupling, the slider and the rotator individually take their characteristic quantum energies. Conse­ quently, insofar as we are able to measure the energy values of the rotator alone we always find the quantum energy values with experimental accuracy. Even for * According to Ehrenfest and Breit this cannot happen, or can happen only rarely, through forces which act on the gear-wheel. HEISENBERG vanishing mass of the slider, however, the energy of the coupled system is different from the energy of the rotator. The energy of the coupled system can take on all possible values (consistent with the T-quantization) with equal probability. Quantum kinematics and mechanics show far-reaching differences from the ordinary theory. The applicability of classical kinematics and mechanical concepts, however, can be justified neither from our laws of thought nor from experiment. The basis for this conclusion is relation (1), pxqx ~ h. As momentum, position, energy, etc. are precisely defined concepts, one does not need to complain that the basic equation (1) contains only qualitative predictions. Moreover, as we can think through qualitatively the experimental consequences of the theory in all simple cases, we will no longer have to look at quantum mechanics as unphysical anid abstract.* Of course we would also like to be able to derive, if possible, the quan­ titative laws of quantum mechanics directly from the physical foundations—that is, essentially, from relation (1). On this account Jordan has sought to interpret the equation, S(q, q")= J % , q')S(q, q")dq\ as a probability relation. However, we cannot accept this interpretation (§2). We believe, rather, that for the time being the quantitative laws can be derived out of the physical foundations only by use of the principle of maximum simplicity. If, for example, the X-coordinate of the electron is no longer a “number,” as can be concluded experimentally, according to equation (1), then the simplest assumption conceivable [that does not contradict (1)] is that this X-coordinate is a diagonal term of a matrix whose nondiagonal terms express themselves in an uncertainty or—by transformation—in other ways (see for example §4). The prediction that, say, the velocity in the X-direction is “in reality” not a number but the diagonal term of the matrix, is perhaps no more abstract and no more unvisualizable than the statement that the electric field strengths are “in reality” the time part of an antisymmetric tensor located in space-time. The phrase “in reality” here is as much and as little justified as it is in any mathematical description of natural processes. As soon as one accepts that all quantum-theoretical quantities are “in reality” matrices, the quantitative laws follow without difficulty. If one assumes that the interpretation of quantum mechanics is already correct * Schrodinger describes quantum mechanics as a formal theory of frightening, indeed repulsive, abstractness and lack of visualizability. Certainly one cannot overestimate the value of the mathematical (and to that extent physical) mastery of the quantum-mechanical laws that Schrodinger’s theory has made possible. However, as regards questions of physical interpretation and principle, the popular view of wave mechanics, as I see it, has actually deflected us from exactly those roads which were pointed out by the papers of Einstein and de Broglie on the one hand and by the papers of Bohr and by quantum mechanics on the other hand. 1.3 PRINCIPLE OF INDETERMINISM 83 in its essential points, it may be permissible to outline briefly its consequences of principle. We have not assumed that quantum theory—in opposition to classical theory—is an essentially statistical theory in the sense that only statistical con­ clusions can be drawn from precise initial data. The well-known experiments of Geiger and Bothe, for example, speak directly against such an assumption. Rather, in all cases in which relations exist in classical theory between quantities which are really all exactly measurable, the corresponding exact relations also hold in quantum theory (laws of conservation of momentum and energy). But what is wrong in the sharp formulation of the law of causality, “When we know the present precisely, we can predict the future,” is not the conclusion but the assumption. Even in principle we cannot know the present in all detail. For that reason every­ thing observed is a selection from a plenitude of possibilities and a limitation on what is possible in the future. As the statistical character of quantum theory is so closely linked to the inexactness of all perceptions, one might be led to the pre­ sumption that behind the perceived statistical world there still hides a “real” world in which causality holds. But such speculations seem to us, to say it explicitly, fruitless and senseless. Physics ought to describe only the correlation of observa­ tions. One can express the true state of affairs better in this way: Because all experiments are subject to the laws of quantum mechanics, and therefore to equation (1), it follows that quantum mechanics establishes the final failure of causality. A ddition in Proof After the conclusion of the foregoing paper, more recent investigations of Bohr have led to a point of view which permits an essential deepening and sharpening of the analysis of quantum-mechanical correlations attempted in this work. In this connection Bohr has brought to my attention that I have overlooked essential points in the course of several discussions in this paper. Above all, the uncertainty in our observation does not arise exclusively from the occurrence of discontinuities, but is tied directly to the demand that we ascribe equal validity to the quite different experiments which show up in the corpuscular theory on one hand, and in the wave theory on the other hand. In the use of an idealized gamma-ray microscope, for example, the necessary divergence of the bundle of rays must be taken into account. This has as one consequence that in the observation of the position of the electron the direction of the Compton recoil is only known with a spread which then leads to relation (1). Furthermore, it is not sufficiently stressed that the simple theory of the Compton effect, strictly speaking, only applies to free electrons. The consequent care needed in employing the uncertainty relation is, as Professor Bohr has explained, essential, among other things, for a comprehensive discussion of the transition from micro- to macromechanics. Finally, the discussion of resonance fluorescence is not entirely correct because the connection between HEISENBERG the phase of the light and that of the electronic motion is not so simple as was assumed. I owe great thanks to Professor Bohr for sharing with me at an early stage the results of these more recent investigations of his—to appear soon in a paper on the conceptual structure of quantum theory—and for discussing them with me. Copenhagen, Institute for Theoretical Physics of the University.