An alternative method of deriving the values of the observables of atomic systems is presented. Rather than using operators and eigenvalues the local variables method uses the continuity equation together with current densities derived from wave functions that are solutions of the Dirac or Pauli equation. The method is applied to atomic hydrogen using the usual language of quantum mechanics rather than that of geometric algebra with which the method is often associated. The picture of the atom that emerges is one in which the electron density as a whole is rotating about a central axis. The results challenge some assumptions of conventional quantum mechanics. Electron spin is shown to be a property of the dynamical motion of the electron and not an intrinsic property of the electron, the ground state of hydrogen is shown to have an orbital angular momentum of ℏ, and excited states are shown to have angular momenta that are different from the eigenvalues of the usual quantum mechanical operators. The uncertainty relations are found not to be applicable to the orthogonal components of the angular momentum. No double electron spin gyromagnetic ratio is required to account for the observed magnetic moments, and the behavior of the atom in a magnetic field is described entirely in kinetic terms.
An alternative method of deriving the values of the observables of atomic systems is presented. Rather than using operators and eigenvalues the local variables method uses the continuity equation together with current densities derived from wave functions that are solutions of the Dirac or Pauli equation. The method is applied to atomic hydrogen using the usual language of quantum mechanics rather than that of geometric algebra with which the method is often associated. The picture of the atom that emerges is one in which the electron density as a whole is rotating about a central axis. The results challenge some assumptions of conventional quantum mechanics. Electron spin is shown to be a property of the dynamical motion of the electron and not an intrinsic property of the electron, the ground state of hydrogen is shown to have an orbital angular momentum of ℏ, and excited states are shown to have angular momenta that are different from the eigenvalues of the usual quantum mechanical operators. The uncertainty relations are found not to be applicable to the orthogonal components of the angular momentum. No double electron spin gyromagnetic ratio is required to account for the observed magnetic moments, and the behavior of the atom in a magnetic field is described entirely in kinetic terms.
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the later cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrodinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the latter cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work, we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrödinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
In Part one of this Paper a hypothesis is forwarded of the electron charge in an atom existing in a distributed form. To check it by methods of electrodynamics and mechanics (without invoking the formalism of quantum mechanics and the concepts of the wave function and of the operators), the potential, kinetic, and total energies were calculated for three states of the hydrogen atom, which were found to agree closely with the available experimental data. The Part two of the Paper offers additional assumptions concerning various scenarios of motion of elements of the distributed electron charge which obey fully the laws of theoretical mechanics. The angular momentum of the ground-state hydrogen atom calculated in the frame of theoretical mechanics is shown to coincide with the spin which is ℏ/2.
A rigorous derivation of the Schrödinger theory from the Pauli (or Dirac) theory implies that the Schrödinger equation describes an electron in an eigenstate of spin. Furthermore, the ground-state kinetic energy is completely determined by the electron spin density. This can be explained by interpreting the spin as an orbital angular momentum, which is necessarily accompanied by a kinetic energy. Thus, the spin is a zero-point angular momentum associated with the zero-point energy of the electron. Since the dispersion in electron momentum is determined by the zero-point energy, the Heisenberg uncertainty relations for an electron can be interpreted as a property of the electronspin motion. The kinetic interpretation of spin and the statistical interpretation of quantum mechanics can be jointly sustained by regarding the electron as a point particle. It follows that stationary electron states are sources of fluctuating electric fields. There is reason to believe that these fluctuating fields are responsible for the Van der Waals force and can be identified with electromagnetic vacuum field fluctuations. The kinetic interpretation of spin then implies that the Van der Waals forces are spin dependent. These ideas are not only consistent with the conventional mathematical formalism of quantum mechanics, they provide a more complete and coherent interpretation of many details in the formalism than does the alternative Copenhagen interpretation. They present some difficulties, however, which, if the kinetic interpretation of spin is correct, probably require some modification of quantum electrodynamics to be resolved.
Abstract This paper is concerned chiefly with the introduction of a co-ordinate operator in the quantum mechanics of the electron. It differs from other work on the same subject in the derivation, of the form of the operator by an analogy with the quantum expression for the angular momentum of the electron. Expressions for velocity and momentum operators are derived, and the relation between them is formally that of the classical theory in the absence of a field of force. By the introduction of reciprocal relations for the co-ordinate and momentum operators a condition is obtained which is applicable when the finite extension of the particle cannot be ignored. This is similar to the condition introduced by Yukawa in the theory of non-local fields. In the course of the development of the work, an explanation is offered of the significance of the fifth co-ordinate introduced in Kaluza’s theory of the gravitational and electromagnetic field and in Klein’s adaptation of it. Further, a new aspect of the question of the quantization of space and time is revealed by the use of co-ordinate operators (Xk), for it appears that the interest lies, not in a property of space-time, but in the quantization of localization of a particle. It should, however, be emphasized that the co-ordinates (xk) are retained with their usual meaning, but it is suggested that they are insufficient as a basis of a mechanical description of the electron. It is by operations upon (Xk) and not upon (xk) that mechanical quantities are obtained. The momentum (uk), like (xk), keeps its usual meaning, but it also is part of an operator (Mk) of wider significance. This is analogous to the case of the angular momentum of the electron where Dirac’s introduction of ‘spin’ shows that the orbital angular momentum is a part of a complete angular momentum operator. The application has been limited to the electron, because the derivation of the form of the operator (Xk) rests upon this analogy. The intention has not been to develop a calculus in which ordinary geometrical co-ordinates are replaced by operators. Finally, in proceeding to a force operator an additional term arises proportional to the derivative of the acceleration with respect to time. This term, which is usually ascribed to the reaction of the field of the electron, occurs here without reference to the self field, and suggests that, in addition to charge and current, a non-Maxwellian quantity must be considered as interacting with the external field.
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.