Abstract It is well known that de Broglie’s wave mechanics is based on the analogy between the mechanical principle of least action and the optical principle of least time. Schrodinger’s original theory of the hydrogen atom was likewise based on a variational principle, and the method has been used extensively in subsequent developments of the subject. Very recently Darwin has given the form of the principle appropriate to Dirac’s theory and has shown that expressions for the current vector and electric density can be deduced from it. In Dirac’s work the electron wave is specified by a four-rowed matrix. The theory has been brilliantly successful in accounting for the “duplexity” phenomena of the atom, but has the defect that the wave equations are unsymmetrical and have not the tensor form. In a recent paper an attempt has been made to surmount this difficulty. Eight wave functions are employed instead of Dirac’s four. These are grouped together to form two four-vectors and satisfy tensor equations of the second order. It is shown in 2 of the present paper that these eight wave equations can be reduced, by addition and subtraction, to the four second order equations satisfied by Dirac’s functions, taken twice over; and that, in a sense, the present theory includes Dirac’s.
Abstract It is well known that de Broglie’s wave mechanics is based on the analogy between the mechanical principle of least action and the optical principle of least time. Schrodinger’s original theory of the hydrogen atom was likewise based on a variational principle, and the method has been used extensively in subsequent developments of the subject. Very recently Darwin has given the form of the principle appropriate to Dirac’s theory and has shown that expressions for the current vector and electric density can be deduced from it. In Dirac’s work the electron wave is specified by a four-rowed matrix. The theory has been brilliantly successful in accounting for the “duplexity” phenomena of the atom, but has the defect that the wave equations are unsymmetrical and have not the tensor form. In a recent paper an attempt has been made to surmount this difficulty. Eight wave functions are employed instead of Dirac’s four. These are grouped together to form two four-vectors and satisfy tensor equations of the second order. It is shown in 2 of the present paper that these eight wave equations can be reduced, by addition and subtraction, to the four second order equations satisfied by Dirac’s functions, taken twice over; and that, in a sense, the present theory includes Dirac’s.
In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.
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.
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.
Investigations into the nature of the principle of least action have shown that there is an intrinsic relationship between geometrical and topological methods and the variational principle in classical mechanics. In this work, we follow and extend this kind of mathematical analysis into the domain of quantum mechanics. First, we show that the identification of the momentum of a quantum particle with the de Broglie wavelength in 2-dimensional space would lead to an interesting feature; namely the action principle S=0 would be satisfied not only by the stationary path, corresponding to the classical motion, but also by any path. Thereupon the Bohr quantum condition possesses a topological character in the sense that the principal quantum number is identified with the winding number, which is used to represent the fundamental group of paths. We extend our discussions into 3-dimensional space and show that the charge of a particle also possesses a topological character and is quantised and classified by the homotopy group of closed surfaces. We then discuss the possibility to extend our discussions into spaces with higher dimensions and show that there exist physical quantities that can be quantised by the higher homotopy groups. Finally we note that if Einstein’s field equations of general relativity are derived from Hilbert’s action through the principle of least action then for the case of n=2 the field equations are satisfied by any metric if the energy-momentum tensor is identified with the metric tensor, similar to the case when the momentum of a particle is identified with the curvature of the particle’s path.
The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schroedinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton's equations also give the Schroedinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order derivatives. For a time-independent system the Principle of Least Action reduces to the energy variational principle. For time-dependent systems the Principle of Least Action gives time-dependent approximations. Using a Hartree product trial wave function for a time-dependent many-boson system, we apply the Principle of Least Action to obtain the Gross-Pitaevskii equation that describes a Bose-Einstein condensate.
The paper gives an account of the author's work on a new form of quantum theory. \textsection{}1. The Hamiltonian analogy between mechanics and optics. \textsection{}2. The analogy is to be extended to include real "physical" or "undulatory" mechanics instead of mere geometrical mechanics. \textsection{}3. The significance of wave-length; macro-mechanical and micro-mechanical problems. \textsection{}4. The wave-equation and its application to the hydrogen atom. \textsection{}5. The intrinsic reason for the appearance of discrete characteristic frequencies. \textsection{}6. Other problems; intensity of emitted light. \textsection{}7. The wave-equation derived from a Hamiltonian variation-principle; generalization to an arbitrary conservative system. \textsection{}8. The wave-function physically means and determines a continuous distribution of electricity in space, the fluctuations of which determine the radiation by the laws of ordinary electrodynamics. \textsection{}9. Non-conservative systems. Theory of dispersion and scattering and of the "transitions" between the "stationary states." \textsection{}10. The question of relativity and the action of a magnetic field. Incompleteness of that part of the theory.