MODERN PHYSICS-A
SURVEY.* PART I
SAULDUSHMIW,~ G E N B ~ELECTRIC L COMPANY, SCHBNECTADY, NEW YORK
Introduction In order to appreciate the developments in physics which have occurred during the past twenty-five years, i t is well to consider briefly the achievements in this science a t the end of the nineteenth century.' The first branch of physics t o evolve a s a well-defined unit was mechanics. The genius of Newton gave it a form which could be developed to even greater perfection by succeeding mathematicians. Thus Lagrange was able to replace Newton's laws by a rather simple differential equation, and finally Hamilton demonstrated that all the laws of dynamics could be deduced from a single principle, the Principle of Least Action, which he expressed in a very elegant mathematical form. The same methods were now applied to other branches of physics, such as acoustics, hydrodynamics, and optics. In the latter branch, especially, Newton developed the corpuscular theory of light because of the obvious similarity between the laws of geometrical optics and the motion of material particles; while Fermat, in enunciating his principle of least time for the path of a light ray, stated a generalization which is the analog of Hamilton's Principle of Least Action. It is true that Huyghens favored a wave theory of light propagation, but i t was only a t the beginning of the nineteenth century that Young's experiments on interference led t o the definite overthrow of Newton's theory by Fresnel and others. The last century saw the birth and growth to full stature of the science of electricity with which the names of Volta, Ampere, Faraday, and Maxwell are connected. By the electro-magnetic theory of Maxwell, optics and electricity became united as the science of waves par excellence, and the experimental proof of this point of view by Hertz was all that was needed, apparently, to confirm the whole theory. It seemed as if the growth of physical science had been completed, and that no new epoch-making discoveries would be made. Yet the very last decade of the century saw the discovery of X-rays by Rontgen, of radioactivity by Mme. Curie, Rutherford, Soddy, and others,and the conception of the electron theory by J. J. Thomson. Still these new phenomena and the electron theory could have been comprehended in terms of the classical point of view, but other observations
* An address given before the Physical Science Section of the Tenth Ohio State Educational Conference, held in Columbus, April 3-5, 1930; and printed in the Proceedings; also in Genkral Elec. Rtv., 33, 328-32 (June, 1930). t Assistant Director, Research Laboratory. General Electric Company. The following paragraphs are based largely on the remarks made by L. de Broglie in the introduction to his famous paper entitled "Researches on the Theory of Quanta," Annales de Physique, 3,22-128 (1925). 1778
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were made a t about the same time, which could not be reconciled with any existing theory. In 1900 Lord Kelvin announced that there were apparently two clouds upon the physical horizon. One of these was represented by the experiment of Michelson and Morley, the other involved the failure of classical methods in accounting for the obsemations on the energy distribution in black-body radiation. As we know now, the first difficulty led to Einstein's Theory of Relativity, while the latter led to Planck's Theory of Quanta. In the present connection we shall omit any further discussion of the theory of relativity except to point out its profound philosophical significance inasmuch as i t puts time and space on an equal basis. They are merely two different attributes. Time is simply a fourth coordinate by which to describe the "world p a t h of a particle. The conception of absolute motion disappears, and, what is of even greater importance, the old respectable Principle of Causality is discarded, since, according to Einstein's theory, phenomena which are simultaneous for one observer need not be simultaneous for another observer moving with different velocity. We shall be compelled to return to this point once more in a subsequent section. Origin of the Quantum Theory Turning now to the theory of energy quanta, we find its inception, as mentioned already, in the observations on black-body radiation. These observations led to the conclusion that not only is matter atomic in its nature, but that radiant energy is emitted or absorbed in units, or quanta, whose magnitude is given by the expression hv, where v is the frequency of the radiation and h is a universal constant. The postulation of such a theory meant a complete departure from the highly developed classical concepts. While the latter could always be formulated by differential equations denoting continuity, such a formulation could not be adapted to correlate the observations on black-body radiation, and subsequent discoveries have served only to emphasize the radical difference between the microphysical phenomena to which the quantum theory applies, and the macrophysical, which classical theory "explained" so well. I have used the word "explained in quotation marks because it is necessary, before proceeding further, to point out, as was first done by M a ~ hthat , ~ theories in physics are not "explanations" of observed pbenomena in the philosophical or metaphysical sense of the word. "In other words, theories contain no implicit conclusions regarding some reality behind the phenomena. Rather, the object of a physical theory is to develop a system of concefitions which mdl enable us to describe the results This point is well brought out in a recent address hy P. Jordan, Naturwiss., 16, 765 (1928).
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of an experience-a system which shall be as complete as possible and a t the same time extremely simple and comprehensive." Corpuscular Theory of Light While Planck put forward his quantum hypothesis as a sort of tentative explanation of certain observed phenomena, it remained for Einstein (1905) to suggest a form of quantum theory which emphasized still more the discontinuous nature of energy. Takiig as a fundamental axiom that energy is radiated in the form of light corpuscles or quanta of magnitude, hv, Einstein concluded that in the photoelectric effect the energy represented by this amount must be completely transferred to the electron which is emitted by the action of the light. If W denotes the work required to get the electron through the surface, and V, the maximum kinetic energy (in volts) of the emitted electron, then according to Einstein hv
=
Ve
+W
where e is the charge on the electron. That is, the kinetic energy of the electron emitted by photoelectric action should be proportional to the frequency and not to the intensity of the incident radiation. The confirmation of this relation mainly by Millikan3 and the further observation that the same relation is valid for the inverse photoelectric effect (production of X-rays by electrons incident on a metallic surface) served to strengthen the validity of Einstein's "light-dart" theory. Thus appeared the first sign of dualism in theoretical physics. On the one hand, when measuring wave-lengths of radiation we use the wave theory, but, on the other hand, when we consider the interaction of radiation and matter, we find it more convenient to regard radiant energy as a sort of corpuscular entity, constituted of quanta, "light darts" or photons, traveling in straight lines and obeying the laws of geometrical optics. Classical Theory of Radiation Now according to the wave theory the simplest mechanism for the production of light is an electrical oscillator, that is, an electrically charged particle oscillating about a mean position of equilibrium with a restoring force which varies with the displacement according to some given law. If the restoring force varies directly as the distance, the displacements can be represented on a time axis as a sinusoidal wave of definite frequency, and, according to classical electromagnetic theory, such an oscillator will emit monochromatic radiation of the same frequency as that of the oscillator; the latter is said to be of the linear harmonic type. However, if the restoring force i s not directly proportional to the dis@lacement, but For an account of this work, see R. A. Millikan, "The Electron," Chicago University, 1917.
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varies according to some higher power of the latter, then the oscillator is spoken of as the anharmonic type. The displacement, as a function of the time, is then represented by a combination of a sine or cosine wave of the same frequency as in the harmonic type (fundamental frequency) and other sine or cosine waves whose frequencies are integral multiples of the fundamental. The radiation is then no longer monochromatic, but consists of a whole range of frequencies which may be represented by a series of values, such as v , 2v, 3v, etc., the whole constituting a range of frequencies which resembles from a formal standpoint the fundamental note and overtones which are emitted by a vibrating string in acoustics. Mathematically, the motion of such an oscillator and the resulting intensity of the electromagnetic radiation as a function of the time may be represented by a Fourier series in which each harmonic of the fundamental frequency is represented by the product of a constant (which gives the maximum amplitude of the corresponding radiation) and a cosine or sine function of the particular frequency. Bohr Theory of Energy Levels This picture of the origin of radiation, based on the so-called classical theory of electrodynamics, is of particular interest because it was found to lead to conclusions which are in distinct and irreconcilable contradiction with the actual observations on the frequencies of spectral lines emitted by the various elements. For it had been observed that the lines in the spectrum of an element, such as sodium, could be arranged into a few series in each of which the frequencies of the lines are connected by a rather unexpected but simple relation of the form v = A - B/m2, where A and B are constants for any given series, and m assumes successive integral values for different members of the same series. In other words, while classical theory led to the expectation that the frequencies of lines would be found to be harmonics of one, or at most a few fundamental frequencies, the spectroscopists found that each line in the spectrum could be represented as a difference between two terms. Such an observation obviously cannot be made in general to conform with the representation of the lines by a Fourier series, and hence the mechanism of production of spectral lines cannot be any linear oscillator of the type demanded by classical theory. I t was, however, this very observation that spectral lines may be represented as differences between two terms which was used by Bohr as the most important fact for formulating a completely new theory of the origin of spectral lines. For this purpose, Bohr also incorporated the conclusions reached just previously by Rutherford, Geiger, and Marsden that the atom must be regarded as constituted of a positively charged nucleus and one or more electrons, the number of electrons being equal to the number of
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positive units of electricity on the nucleus and each equal to the atomic number, N, which represents the place of the element in the periodic arrangement. In such an atomic system the attraction of the electrons by the positive nucleus may be balanced in one of two ways; either we can assume stationary electrons and a force of repulsion which just balances the force of attraction when the electrons are a t certain distances from the nucleus, or we can assume that, as in the solar system, the force of attraction is balanced by the centrifugal force due to rotation of the electrons in orbits. For many reasons Bohr took the latter view, since the problem could then be treated in the same manner as Newton had originally dealt with the analogous problem in celestial mechanics. On this point of view, the model of the hydrogen atom is that of a single electron rotating about a unit positive charge in an elliptic orbit of definite major axis. This is the normal hydrogen atom; but in order to account for the observations on line spectra, Bohr found it necessary to introduce two postulates: firstly, that any atomic system is capable of existing in a series of discrete stationary states, for each of which the energy is diierent; and secondly, that mono chroma ti^ radiation i s emitted or absorbed wkenarer the system passes f r m one state to another, the frequency of the radiation being given by a relation of the f o m
" = E,/h - '?%,/I, where E, and E, represent the energies in the initial and final states, respectively. It will be recognized that this equation for v is of the same form as that observed by spectroscopists and that according to Bohr the spectral terms A and B/mZare to be interpreted as energy larels. In other words, he assumed that these levels or stationary states corresponded, in the case, say, of the hydrogen atom, to different orbits in which the electron may rotate and that the laws governing these orbits are those of ordinary mechanics. Furthermore, in order to define these orbits, Bohr introduced a certain quantum condition, which in the case of a circular orbit may be stated in the form that the angular momentum of the electron in its orbit must always be an integral multiple, n , of the mystic constant, k l ( 2 z ) . The different stationary states thus correspond to different orbits for which the radius increases as n2 where n is known as the quantum number, while the frequency of rotation decreases with increase in radius. Of course, according to classical theory, an electron rotating in orbit must radiate energy continuously, which would lead to a continuous decrease in the radius with corresponding increase in frequency or rotation, until finally the electron would fall into the nucleus. But Bohr merely denies the occurrence of such a phenomenon for electronic orbits, and postulates, as mentioned already, that emission (or the inverse process,
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absorption) of radiation occurs only during a transition of the electron from one orbit to another. The. only defense to be offered for such a set of postulates and assumptions was that they led to remarkable results. Bohr showed that not only could this theory give quantitatively the lines observed in the hydrogen spectrum, but by suitable modification it could be made to account qualitatively for the different series spectra observed for a large number of other elements. As a logical deduction from Bohr's frequency relation it follows that if we have three energy levels, say El, Ez, Eg where v,,, = (El - E d / h and w . ~= (Ex -EJlh then it should be observed that ",,%=
-
"1,s
+
4.3
=
(Ex - E J I h
The validity of this law, &own as the Ritz Combination Principle, had also been observed empirically by spectroscopists, but Bohr first pointed out its significance as a consequence of his theory of energy levels. The extension of Bohr's ideas by Sommerfeld and others led to a fairly consistent picture of atomic structures in which the more complex atoms could be regarded as being built up from elements of lower atomic number, N, by successive addition of electrons rotating in various elliptical and circular orbits. Also the experiments initiated by Franck and Hertz and continued by a number of American investigators, on the excitation and ionization of atoms by collisions with electrons, and the correlation of the results with certain lines in the spectra of these atoms-all these observations coming as direct consequences of Bohr's theory of energy levels-seemed to serve as a proof of the actual existence of electronic orbits. Atomic Mechanics on Classical Basis However, when i t came to the application of the laws of Newtonian mechanics to atomic systems containing more than one electron, difficulties of a more and more serious nature began to be encountered. For one thing, ordinary mechanics deals with continuously varying systems and the existence of discrete energy states with quantum transitions presents a phenomenon totally foreign to the whole of classical theory. In the case of an electric oscillator, the frequencies of radiation emitted are simple multiples of a fundamental, and the intensities of the individual monochromatic radiations are proportional to the squares of the amplitudes of the corresponding displacements in the motion of the oscillator. But in atomic systems the frequency of emission of radiation bears no such simple relation to the frequency of rotation of the electron in its orbit, nor, in general, can the intensity of the light emitted be measured by any
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properties of a single orbit, since the radiation originates in a transition between two orbits. Under such conditions how can one modify the laws of ordinary mechanics to obtain results in agreement with actual observations? Bohr had recognized these difficulties early in the development of the theory, but had pointed out that under certain conditions the deductions obtained by classical theory for macroscopic systems must be valid for atomic systems as well. It is well known that the laws of classical electrodynamics apply to radiation of long wave-length; but such radiation is obtained, on the point of view of the new theory, by transitions between orbits of very large radius, where the frequency of rotation is much smaller than for orbits of smaller radius (smaller values of the quantum number, n). For such large orbits, the frequency of the radiation emitted in passing from one orbit to a neighboring one is approximately equal to a simple multiple of the frequency of rotation in either orbit, in accordance with classical theory. Hence the laws of ordinary mechanics may be apqlied to orbits of large quantum number and the deductions thus obtained may then be extrapolated to the case of orbits of small quantum number. This idea, that in the limit the laws of quantum mechanics.must approach asymptotically those of ordinary mechanics, was formulated by Bohr as his famous Correspondence Princiqle, and still furnishes the background for even the most recent developments in quantum mechanics. It will be observed that the essential purpose of this principle is to retain in atomic mechanics as much as possible of that classical mechanics which found its most important formulation in the so-called Hamiltonian canonical conditions and the Hamilton-Jacobi partial differential equation. Difficulties in Applying Classical Mechanics to Atoms However, while this quantum mechanics, based, as it was, on a hybridization of ordinary mechanics and quantum discontinuities, gave quantitative results in the case of the normal hydrogen atom and ionized helium, i t met with real difficulties in the treatment of the spectrum of neutral helium, where two electrons revolve around the nucleus. In the case of atoms of higher atomic number, the problem obviously becomes still more complex. Bohr attempted to deal with these atoms as perturbed states of the simple hydrogen-like system. By introducing the notion of penetrating and non-penetrating orbits a certain measure of success was obtained in deriving a general form of expression similar to that which had been found to apply to spectroscopic terms. Also, by assigning to each electron in the atom four different kinds of quantum numbers, Sommerfeld, Land6, Pauli, Heisenberg, and others were able to develop semi-empirical formulas for the separation of multiplets and for the effects of magnetic and electric fields on spectral lines. In order to account for
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the absence of certain classes of transitions between levels, it was also found necessary t o postulate certain Principles of Selection regulating the nature of the possible transitions. A further important generalization was derived by Pauli according to which: "There cannot exist in any one atom two electrons having the four quantum numbers respectively the same for both." Utilizing this rule and the different selection principles, Hund was able t o deduce a correlation between the arrangement and nature of the electrons in any atom, on the one hand, and the spectroscopic normal term on the ~ t h e r . ~ As stated already, all this development was based on more or less empirical considerations. All attempts t o derive the same results by the methods of Newtonian or classical mechanics proved unsuccessful, and hence the question arose as to whether there is any reason whatever for expecting successful results by this method. Physicists like Bohr, Born, Heisenberg, and others began to consider the whole prohlem from a more fundamental point of view. As stated by Born,' "After all, no other result could he expected, for the validity of the frequency condition is sufficient to show conclusively that in the realm of atomic processes the laws of classical theories (geometry, kinematics or mechanics, electrodynamics) are not right. That in certain simple cases, as for a single electron, they give partially correct results is, in fact, more astonishing than that they fail in the more complicated cases of several electrons. This failure of the theory in the case of interactions among several electrons is evidently connected with the following fact. We know that electrons react quzte unclassically to light wanes, because the latter produce quantum jumps. In a system made up of several electrons, each electron is in the oscillating field due to all other electrons and the periods of these fields are of the same order of magnitude as those of light waves, therefore we have no reason to expect that the electron should react classically to this oscillating field. This point of view gives grounds for understanding why we obtain, by the classical theory, correct results in many cases of the one electron prohlem." Failure of Principle of Causality Also, i t is of importance to realize the difficulties involved in the simple model of electronic transitions between levels. In passing, for instance, from a higher to a lower level, the atomic system radiates monochromatic energy. When does this radiation occur? If it occurs during the transition then the electron knows the ultimate level to which it shall pass.
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For a detailed account of this work see the writer's paper on "Line Spectra and the Periodic Arrangement of the Elements," Chemical Reuiews, 5, 109-71 (1928). 5 M. Bom, "Problems of Atomic Dynamics " Lectures delivered at M . I. T,in 1926.
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This obviously means a doctrine of determinism and is therefore utterly absurd. Bohr, Kramers, and Slater suggested in 1925 a theory which would permit the electron t o radiate while in its orbit without the occurrence of the transition until a definite amount of energy had been emitted. On this point of view, the law of conservation of energy as applied to atomic system would assume the nature of a statistical principle, that is, one which is true for a large number of transitions, but not necessarily valid for each individual transition. But more recent investigations, especially those connected with the Compton effect (see below), showed the untenability of this point of view. It was demonstrated that the transitions are governed by the existence of certain probabilities for their occurrence. Hence, it i s no longer possible to assume a causal mechanism governing the transitions. I n fact, i t was perceived that the failure to attain successful results by the application of Newtonian mechanics is in reality bound up with the failure of the so-called Principle of Causality when applied to microphysical phenomena. As will be shown in a subsequent section, this principle has been found to he just as unnecessary in the new quantum physics as the hypothesis of an ether in the theory of relativity.
Compton Effect Mention has already been made of the difficulties encountered by physicists owing to the necessity of assuming two different hypotheses for the mechanism by which radiant energy is propagated. Waves or corpuscles? This was the most important question, and numerous attempts were made to settle it. In 1922-23, A. H. Compton discovered a new phenomenon which served to emphasize still more the corpuscular nature of radiation. When an X-ray impinges on matter, a secondary X-ray is produced of slightly longer wave-length or lower frequency. This is contrary to what would be expected on the basis of classical theory. In order to explain this observation, Compton utilized the theory of light darts or "photons," as we prefer to designate them now. On this theory, the X-rays consist of streams of energy quanta. While each quantum carries the energy equivalent to hv, we may also specify each of these "photons" (light units) by the momentum, which, according to the theory of relativity, is equal to hvlc. When this photon collides with a free or loosely hound electron there is an interchange of both energy and momentum in accordance with the laws of conservation of energy and momentum. Consequently, the photon suffers a recoil in one direction with loss of momentum, while the electron moves off in another direction with added momentum. The collision is, indeed, analogous to that of a golf hall striking a perfectly elastic football. The decrease
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in momentum of the scattered X-ray photon corresponds to an increase in wave-length. Compton found that the actually observed changes in wave-length were in agreement with the theoretical predictions based on the above assumption. A further point that was brought out by subsequent experiments is that this conservation of momentum and energy is not the result of a statistical state of affairs, but must hold valid for every individual collision between a photon and electron. Here, then, we have a phenomenon which must be recognized as indicating almost conclusively the corpuscular nature of light energy. Yet in these very experiments Compton made use of the wave theory in measuring the wave-length of the secondary radiation. (To be continued.)
