Abstract In the general non-relativity quantum dynamics that has been developed by Dirac and others, the motion of a dynamical system is described by a wave function ψ (ξ1 ... ξn ; t); and one interprets this wave function by postulating that |ψ (ξ1' ξ2' ... ξn' ; t) |2dξ1' ... dξn' is the probability that, at time t ξr' ≤ ξr ≤ ξr' + dξr' (r = 1 ... n). The relativity wave function for a single electron can be interpreted in a similar manner. If ψ(xyzt) be the wave function, and φ the conjugate function, then φψ (xyzt) dx dy dz: is the probability that the electron will be found, at time t, in the volume element dx dy dz.
Abstract In the general non-relativity quantum dynamics that has been developed by Dirac and others, the motion of a dynamical system is described by a wave function ψ (ξ1 ... ξn ; t); and one interprets this wave function by postulating that |ψ (ξ1' ξ2' ... ξn' ; t) |2dξ1' ... dξn' is the probability that, at time t ξr' ≤ ξr ≤ ξr' + dξr' (r = 1 ... n). The relativity wave function for a single electron can be interpreted in a similar manner. If ψ(xyzt) be the wave function, and φ the conjugate function, then φψ (xyzt) dx dy dz: is the probability that the electron will be found, at time t, in the volume element dx dy dz.
Dirac's wave equation of an electron enables one to solve for the wave function of an electron moving in an electromagnetic field. The wave function ψ has 4 components ψ1, ψ2, ψ3, ψ4, and the electromagnetic field is described by a four-vector Aμ, consisting of a scalar potential φ and a vector potential A. In the usual problem, we solve for the wave function ψ when the potential Aμ is given. By doing so, we obtain the wave function of an electron which moves in a given electromagnetic field. It is of some interest to consider the reverse question: given the wave function ψ, what can we say about the electromagnetic potential Aμ, which is connected with ψ by Dirac's equation? Is Aμ uniquely determined, and if not, what is the extent to which it is arbitrary ?
A novel interpretation is given of Dirac's "wave equation for the relativistic electron" as a quantum-mechanical one-particle equation. In this interpretation the electron and the positron are merely the two different "topological spin" states of a single more fundamental particle, not distinct particles in their own right. The new interpretation is backed up by the existence of such bi-particle structures in general relativity, in particular the ring singularity present in any spacelike section of the maximal analytically extended, topologically non-trivial, electromagnetic Kerr-Newman spacetime in the zero-gravity limit (here, "zero-gravity" means the limit G→0, where G is Newton's constant of universal gravitation). This novel interpretation resolves the dilemma that Dirac's wave equation seems to be capable of describing both the electron and the positron in "external" fields in many relevant situations, while the bi-spinorial wave function has only a single position variable in its argument, not two - as it should if it were a two-particle equation! A Dirac equation is formulated for a ring-like bi-particle which interacts with a static point charges located elsewhere in the topologically non-trivial physical space associated with the moving ring bi-particle, the motion being governed by a de-Broglie-Bohm type law extracted from the Dirac equation. As an application, the pertinent general-relativistic zero-gravity Hydrogen problem is studied in the usual Born-Oppenheimer approximation. Its spectral results suggest that the zero-G Kerr-Newman magnetic moment be identified with the so-called "anomalous magnetic moment of the physical electron," not with the Bohr magneton, so that the ring radius is only a tiny fraction of the electron's Compton wave length.
The solution is essentially a complex function which accounts for the matter wave spreading illustrated in Figure 28.3. As the wave function itself cannot be identified with a physical property of the free particle, an indirect interpretation was suggested by Born. He pointed out that its modulus ( tx,Ψ ) ( ),x tΨ is large where the particle becomes localized and small elsewhere, and hence ( ) ( ) 2, ,P x t dx x t dx= Ψ (29.1) is the probability of finding the particle described by ( )tx,Ψ at a point between x and at a time t. Therefore the matter wave spreading is associated with a decrease in the probability density . As the total probability of finding the particle somewhere on the x-axis must be unity, the wave function must be normalized to unity, that is dxx + ( txP , ) ( ) ( ) ( ), , ,P x t dx x t x t dx∞ ∞ ∗ −∞ −∞ = Ψ Ψ∫ ∫ = (29.2) The behaviour of the wave function ( )tx,Ψ is determined by Eq.(28.38), which may be rewritten as ( ) ( ) ( ) t x tx m itxttx ∆∂ Ψ∂+Ψ=∆+Ψ 2 2 , 2 ,, = (29.3) where the expression ( ) ( )/ , ,t x t t x t⎡∂Ψ ∂ ≅ Ψ + ∆ −Ψ ∆⎣ / t⎤⎦ ) , which holds for a small , has been used. In view of Eq.(29.3) the values of the wave function at any time can be derived from the initial value that describes a reference state of the particle.
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.
The Dirac equation consistent with the principles of quantum mechanics and the special theory of relativity, introduces a set of matrices combined with the wave function of a particle in motion to give rise to the relativistic energy-momentum relation. In this paper a new hypothesis, the wave function of a particle in motion is associated with a pair of complementary waves is proposed. This hypothesis gives rise to the same relativistic energy-momentum relation and achieves results identical to those of Dirac. Additionally, both the energy-time and momentum-position uncertainty relations are derived from the complementary wave interpretation. How the complementary wave interpretation of the Dirac equation is related to the time-arrow and the four-vectors are also presented.
Abstract The new quantum mechanics consists of a scheme of equations which are very closely analogous to the equations of classical mechanics, with the fundamental difference that the dynamical variables do not obey the commutative law of multiplication, but satisfy instead the well-known quantum conditions. It follows that one cannot suppose the dynamical variables to be ordinary numbers (c-numbers), but may call them numbers of a special type (q-numbers). The theory shows that these q-numbers can in general be represented by matrices whose elements are c-numbers (functions of a time parameter). When one has performed the calculations with the q-numbers and obtained all the matrices one wants, the question arises how one is to get physical results from the theory, i. e., how can one obtain c-numbers from the theory that one can compare with experimental values ? Hitherto this has been done with the help of a number of special assumptions. In Heisenberg’s original matrix mechanics it was assumed that the elements of the diagonal matrix that represents the energy are the energy levels of the system, and the elements of the matrix that represents the total polarisation, which are periodic functions of the time, determine the frequencies and intensities of the spectral lines in analogy to the classical theory. Schrodinger’s wave representation of the quantum mechanics has provided new ways of obtaining physical results from the theory, based on the assumption that the square of the amplitude of the wave function can in certain cases be interpreted as a probability. From this assumption one can, for instance, work out the probability of a transition being produced in a system (or the number of transitions produced in an assembly of like systems) by an arbitrary external perturbing force, and can thus, by supposing the perturbation to consist of incident radiation, obtain directly Einstein’s B coefficients. Again in Born’s treatment of collision problems it is assumed that the square of the amplitude of the wave function scattered in any direction determines the probability of the colliding electron (or other body) being scattered in that direction.