Abstract The present work is a discussion of a number of simple problems of the free motion of electrons and atoms from the point of view of the wave mechanics. The author has had the advantage of many conversations with Prof. N. Bohr on the subject, and it was composed under this inspiration. Bohr has recently published his views on the foundations of the theory, and it would be out of place to enter here deeply into the matter, but some review of principles is unavoidable if the technical processes are to be understood. Perhaps the chief point of the work is to show how a simple, even old-fashioned, technique is entirely adequate to deal with these very new problems. 1. General Principles. The matrix and the wave mechanics have both been already developed to great lengths as a calculus of stationary states, but they have not yet got so far in what we may call dynamics, a description of the progress of events. More and more complicated phenomena have been fitted into the same scheme, but not much has been done in making this scheme intuitively understandable. One of the most important contributions is a recent work by Heisenberg, who develops an “uncertainty relation” by showing how each observation of a system, say an electron, always itself introduces some disturbance, so that the exact state at the beginning of each new experiment is essentially unknowable. Heisenberg works from the matrix point of view, but Bohr has pointed out that the “uncertainty relation,” exhibited by Heisenberg simply as an experimental result, becomes quite natural on wave principles, and is indeed nothing more than an expression of those principles.
Abstract The present work is a discussion of a number of simple problems of the free motion of electrons and atoms from the point of view of the wave mechanics. The author has had the advantage of many conversations with Prof. N. Bohr on the subject, and it was composed under this inspiration. Bohr has recently published his views on the foundations of the theory, and it would be out of place to enter here deeply into the matter, but some review of principles is unavoidable if the technical processes are to be understood. Perhaps the chief point of the work is to show how a simple, even old-fashioned, technique is entirely adequate to deal with these very new problems. 1. General Principles. The matrix and the wave mechanics have both been already developed to great lengths as a calculus of stationary states, but they have not yet got so far in what we may call dynamics, a description of the progress of events. More and more complicated phenomena have been fitted into the same scheme, but not much has been done in making this scheme intuitively understandable. One of the most important contributions is a recent work by Heisenberg, who develops an “uncertainty relation” by showing how each observation of a system, say an electron, always itself introduces some disturbance, so that the exact state at the beginning of each new experiment is essentially unknowable. Heisenberg works from the matrix point of view, but Bohr has pointed out that the “uncertainty relation,” exhibited by Heisenberg simply as an experimental result, becomes quite natural on wave principles, and is indeed nothing more than an expression of those principles.
Abstract 1. There are two problems which are fundamental to wave mechanics— that of deducing from first principles the properties of the wave field of ψ, by means of which, on modern theory, the mechanical properties of matter are to be described, and that of formulating a logical description of the properties of matter in terms of ψ. The former only of these problems is discussed in this communication. In spite of many valuable papers that have been published on the subject of Dirac’s remarkable wave equation, the derivation of the equation still seemed (to the present writer, at any rate,) to include some obscurities which could perhaps be removed. To some extent this is owing to the development of the equation, as a matter of historical necessity, having come about by successive extensions to classical mechanics; so that some difficulties have been produced by the fundamental relativity considerations having been introduced into the theory at a late stage, instead of in their proper place at the beginning. In what follows, by treating the problem from the beginning as a four-dimensional one, a deduction of the wave equation free from empirical steps is, I think, obtained, while also certain new features of the equation are brought to light. 2. The Principle of Action.—The method of four-dimensional mechanics is to assume that the motions of bodies in the world can be represented by “tracks,” or curved lines, in a “fourfold.” (This last term will be used here to mean a four-dimensional manifold with Euclidian geometry, i. e., in which ds2 = dxi2 (i = 1, . . . 4).) (1) Uniform motion in the world is represented by straight line tracks in the fourfold, and the “classical relativity” laws of motion are derived by formulating the simplest mathematical specification of curved tracks which will represent the non-uniform motions of bodies that are usually observed in the world.
In its beginning, quantum mechanics was approached in two completely different ways. Schrodinger, reasoning that electronic motions could be treated as waves, developed wave mechanics. The stationary states that an electron or molecule might have were analogues to standing waves set up by applying appropriate boundary conditions. Bohr’s theory could not explain the Stark and Zeeman effect, and the spectra of atoms more complex than alkali metals. In 1927, Heisenberg added one more dimension to the problem through the uncertainty principle. The postulates of any theory are a set of fundamental statements that are asked to be believed and draw conclusions from them. In classical mechanics, all dynamical variables are observables, but there are certain fundamental restrictions placed upon simultaneously measurable quantities in quantum mechanics.
Abstract In an article in ‘Nature’ last February I put forward a suggestion, of necessity in so concise a form as to be not very easily intelligible, that when the magnetic properties of the atom are regarded from the point of view of the wave mechanics, they suggest that the electron is to be taken as a wave of two components, like light, not of one like sound. The theory and its mathematical development were only outlined in ‘Nature,’ and the object of the present work is to give them in fuller detail. Recently Pauli has published a paper on the same subject, and arrived at the same mathematical results, but owing to the fact that he is more disposed to regard the wave theory as a mathematical convenience and less as a physical reality, he stops short of the point which was the guiding principle to me, and refuses to interpret the two functions that we both obtain as formed from a vector. I shall therefore here develop somewhat fully the arguments and analogies which seem to me to show that the vector is the right form in which to regard it. The chief part of the paper is concerned with developing the results given in ‘Nature’; owing to other work I have not carried the matter much farther yet. The main new points are general formulæ for the intensities of spectral lines, and for the magnetic moment, and the form the theory must take for several electrons—in which my first suggestion was wrong, and which Pauli has developed from his point of view. In a future paper I hope to discuss the motion of a free electron in a magnetic field, together with other problems. Since writing the account of the theory in ‘Nature’ I have had the immense benefit of a visit to Prof. Bohr’s Institute in Copenhagen, and have thus enjoyed the advantage of discussing the subject with him, Dr. Klein and the other members in detail. I may take the opportunity here to express my thanks to them for their interest in the matter and for many helpful criticisms.
Abstract This chapter reviews quantum mechanics of atoms, ions and molecules and their associated spectra. We have already invoked empirically Planck’s minimum quantum of action ℏ, and Heisenberg’s uncertainty principle. We were able to get so far, but the dynamics of electrons bound to atoms, ions and molecules requires wave or matrix mechanics of Schrödinger and Heisenberg developed in 1925. The present chapter reviews the elements of quantum theory necessary to understand the interaction of classical EM waves with electrons, bound and free. Symmetries are related to conservation equations, to find constants of the motion. The radiation field is treated mostly as a classical EM wave. Spin-orbitals and the quantum harmonic oscillator are identified as quantum building blocks. An explanation of spontaneous emission of radiation is given using the occupation number representation. Different kinds of radiative transitions found in dilute atmospheres and plasmas are discussed, quantum collisions are described, and regularities in spectra are highlighted. In typical plasmas, the important influence of electrons excited into two, not just one, spin-orbitals, is highlighted.
Abstract 1. In a recent paper Dirac* has brilliantly removed the defects before existing in the mechanics of the electron, and has shown how the phenomena usually called the “ spinning electron ” fit into place in the complete theory. He applies to he problem the method of q-numbers and, using non-commutative algebra, exhibits the properties of a free electron, and of an electron in a central field of electric force. In a second paper † he also discusses the rules of combination and the Zeeman effect. There are probably readers who will share the present writer’s feeling that the methods of non-commutative algebra are harder to follow, and certainly much more difficult to invent, than are operations of types long familiar to analysis. Wherever it is possible to do so, it is surely better to present the theory in a mathematical form that dates from the time of Laplace and Legendre, if only because the details of the calculus have been so much more thoroughly explored. So the object of the present work is to take Dirac’s system and treat it by the ordinary methods of wave calculus. The chief point of interest is perhaps the solution of the problem of the central field, which can be carried out exactly and leads to Sommerfeld’s original formula for the hydrogen levels. But it is also of some interest to exhibit the relationship of the new theory to the previous equations which were derived empirically by the present writer.* It appears that those equations were an approximation to the new ones, derived by an approximate elimination of two of Dirac’s four wave functions. We shall also review a few other points connected with the free electron, the emission of radiation from an atom and its magnetic moment, and shall outline a discussion of the Zeeman effect.
In the first two chapters we have examined the basic principles – superposition (p. 18), complementarity (p. 19), quantization (p. 44), statistical algorithm (p. 57), and correspondence (p. 72) (see also Subsec. 2.3.4) – and the basic entities, observables and states, of quantum mechanics, as well as the main differences with respect to classical mechanics. While what we have discussed so far is rather a static picture of observables and states, in this chapter we shall deal with quantum dynamics, i.e. with the time evolution of quantum-mechanical systems. Historically, after Bohr had provided a quantized description of the atom (see Subsec. 1.5.4), Einstein showed the quantized nature of photons (see Subsec. 1.2.1), and de Broglie hypothized the wave-like nature of matter (see Subsec. 1.5.5), the first building block of quantum mechanics was provided by the commutation relations, proposed by Heisenberg in 1925, whose consequence is represented by the uncertainty relation (see Subsec. 2.2.7 and Sec. 2.3). This was the subject of the previous chapters. The dynamical part of the theory was proposed by Schrödinger in 1926, and is known as the Schrödinger equation. It is also known as wave mechanics (see Subsec. 1.5.7). In this chapter we shall show that Heisenberg's and Schrödinger's formulations are only two different aspects of the same theory. We shall also come back to this point in Sec. 8.1.1. Here, first we shall derive the fundamental equation which rules quantum dynamics (Sec. 3.1), and, in Sec. 3.2, we shall summarize the main properties of the Schrödinger equation.