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.
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. 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 three earlier papers the author has developed the theory of the operational wave equation, which was originally suggested by Professor Eddington as the most extensive generalisation possible of the linear wave equation devised by Dirac. The wave function,ψ, plays a very minor rô1e in the development of this theory and, in reality, it is introduced simply to provide an operand which shall be patient of the action of the wave operators, A1, A2, A3, A4. The object of this paper is to show that the wave function may be entirely eliminated from the theory, which then takes the form of a “ matrix mechanics,” i.e. a set of relations between matrices representing the coordinates, the momenta and the spin operators.
Abstract The form of the wave equation for a non-rotating electron suggests that it enters into the theory very much in the same way as the wave equation associated with electromagnetic theory. It would be expected to be derivable from equations of the first order corresponding to Maxwell's equations. It has been suggested that the function Ψ might enter by means of a relation such as s = grad Ψ (1) where s replaces the current four vector of the electromagnetic theory. The difficulty in connection with this procedure is to account for the phenomena associated with electronic rotation. Dirac has shown how to overcome this difficulty and has derived first order equations which can be derived from generalisations of Maxwell's equations. There are certain difficulties with regard to the form of Dirac's results which have been much discussed and some of them have been removed. There are two unsatisfactory points in the treatment of this question. One is the introduction of an operator (h/2πi ∂/∂xα - eϕα) into the equations when it is desired to pass from a non-electromagnetic problem to one in which an electromagnetic field is present. The second difficulty lies in the occurrence of a term in mc. Darwin has pointed out this difficulty and considers that it is due to our inability to calculate electromagnetic mass in the quantum theory.
Abstract The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but satisfy instead certain quantum conditions. One can build up a theory without knowing anything about the dynamical variables except the algebraic laws that they are subject to, and can show that they may be represented by matrices whenever a set of uniformising variables for the dynamical system exists. It may be shown, however (see 3), that there is no set of uniformising variables for a system containing more than one electron, so that the theory cannot progress very far on these lines. A new development of the theory has recently been given by Schrödinger. Starting from the idea that an atomic system cannot be represented by a trajectory, i. e., by a point moving through the co-ordinate space, but must be represented by a wave in this space, Schrödinger obtains from a variation principle a differential equation which the wave function ψ must satisty. This differential equation turns out to be very closely connected with the Hamiltonian equation which specifies the system, namely, if H (qr, Pr - W = 0 is the Hamiltonian equation of the system, where the qr, Pr are canonical variables, then the wave equation for ψ is {H(qr, ih ∂/∂q) - W} ψ = 0.
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 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.