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Chinese Physics Letters

Transport Properties of the Universal Quantum Equation

Arbab I. Arbab

Published Mar 1, 2012• 4 citations
Publisher

Abstract

The universal quantum equation (UQE) is found to describe the transport properties of the quantum particles. This equation describes a wave equation interacting with constant scalar and vector potentials propagating in spacetime. A new transformation that sends the Schrödinger equation with a potential energy to Dirac's equation is proposed. The Cattaneo telegraph equation as well as a one-dimensional UQE are compatible with our recently proposed generalized continuity equations. Furthermore, a new wave equation resulted from the invariance of the UQE under the post-Galilean transformations is derived. This equation is found to govern a Klein—Gordon's particle interacting with a photon-like vector field (ether) whose magnitude is proportional to the particle's mass.

DOI
10.1088/0256-307x/29/3/030304

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The universal quantum equation (UQE) is found to describe the transport properties of the quantum particles. This equation describes a wave equation interacting with constant scalar and vector potentials propagating in spacetime. A new transformation that sends the Schrödinger equation with a potential energy to Dirac's equation is proposed. The Cattaneo telegraph equation as well as a one-dimensional UQE are compatible with our recently proposed generalized continuity equations. Furthermore, a new wave equation resulted from the invariance of the UQE under the post-Galilean transformations is derived. This equation is found to govern a Klein—Gordon's particle interacting with a photon-like vector field (ether) whose magnitude is proportional to the particle's mass.
A new formulation of quantum mechanics based on differential commutator brackets is developed. We have found a wave equation representing the fermionic particle. In this formalism, the continuity equation mixes the Klein-Gordon and Schrodinger probability density while keeping the Klein-Gordon and Schrodinger current unaltered. We have found time and space transformations under which Dirac’s equation is invariant. The invariance of Maxwell’s equations under these transformations shows that the electric and magnetic fields of a moving charged particle are perpendicular to the velocity of the propagating particle. This formulation agrees with the quaternionic formulation recently developed by Arbab.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.
The quaterionic formulation of quantum mechanics yields the unified quantum wave equation (UQWEs). From these equations, Dirac, Klein - Gordon and Schrodinger equations can be derived. While the UQWEs represent a matter wave (de Broglie), the Maxwell equations represent a transverse wave (field). Owing to UQWEs, the spin-0 and spin-1/2 particle are described by a wavepacket consisting of waves traveling to the left and to the right with speed of light. UQWEs show that spin-0 and spin-1/2 are in continuous states of creation and annihilation that are compatible with Heisenberg uncertainty relation. The creation - annihilation process is a result of the time translation property of the particle wavefunction. These are E′=E−im0c2E'=E-im_0c^2E′=E−im0​c2 and E′=E±m0c2E'=E\pm m_0c^2E′=E±m0​c2, for Klein-Gordon' and Dirac' particles, respectively. It is found that ℏm0c2\frac{\hbar}{m_0c^2}m0​c2ℏ​ is the period of the creation -annihilation process.
Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential where this particle is confined, the vector potential of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.
A wave equation complying with Galilean transformations is proposed for electromagnetic and matter waves. A fundamental feature entirely different from the principle of relativity is that the wave equation is referred specifically to the proposed local-ether frame. For electromagnetic wave, the local-ether wave equation accounts for a wide variety of experiments of propagation or interference. For matter wave, the wave equation leads to modifications of Schrodinger's equation which in turn lead to a unified quantum theory of electromagnetic and gravitational forces in conjunction with the identity of inertial and gravitational mass. Further, it leads to modifications of the Lorentz force law and of Maxwell's equations. Moreover, the wave equation leads to dispersion of matter wave, from which the speed-dependences in the mass of a particle and in the wavelength, angular frequency, and quantum energy of a matter wave are derived. The consequences of the wave equation are in accord with a wide variety of experiments which are commonly ascribed to the special relativity, the general relativity, the Lorentz mass-variation law, or to the de Broglie matter wavelength. Thereby, the local-ether wave equation unifies quantum mechanics, electromagnetics, and gravitation.