This paper proposes a probabilistic particle model of single-electrons on graphene. The behavior of a single-electron on graphene is described approximately by the massless Dirac equation. The electron is non-relativistic and given pseudo-spin. The Dirac equation originally describes the motion of a relativistic quantum particle with actual spin. Then, it has seemed difficult that the Nelson's stochastic quantization theory could build a particle model of the electron on graphene since the theory can deal with only non-relativistic quantum particles with spin not being taken into account. In this paper, Nelson's theory is interpreted by using probability density function and probability density current so that it can build a particle model of a single-electron on graphene. Single-electrons on a graphene nano-ribbon and on a graphene sheet in constant magnetic field were modeled as probabilistic particles. The models were described by nonlinear stochastic ordinary differential equations. It has been numerically confirmed that probability distributions of the electron models coincide with distributions derived from the wave functions.
This paper proposes a probabilistic particle model of single-electrons on graphene. The behavior of a single-electron on graphene is described approximately by the massless Dirac equation. The electron is non-relativistic and given pseudo-spin. The Dirac equation originally describes the motion of a relativistic quantum particle with actual spin. Then, it has seemed difficult that the Nelson's stochastic quantization theory could build a particle model of the electron on graphene since the theory can deal with only non-relativistic quantum particles with spin not being taken into account. In this paper, Nelson's theory is interpreted by using probability density function and probability density current so that it can build a particle model of a single-electron on graphene. Single-electrons on a graphene nano-ribbon and on a graphene sheet in constant magnetic field were modeled as probabilistic particles. The models were described by nonlinear stochastic ordinary differential equations. It has been numerically confirmed that probability distributions of the electron models coincide with distributions derived from the wave functions.
We investigate the simulations of the the Schrödinger equation using the onedimensional quantum lattice Boltzmann (QLB) scheme and the irregular behaviour of solution. We isolate error due to approximation of the Schrödinger solution with the non-relativistic limit of the Dirac equation and numerical error in solving the Dirac equation. Detailed analysis of the original scheme showed it to be first order accurate. By discretizing the Dirac equation consistently on both sides we derive a second order accurate QLB scheme with the same evolution algorithm as the original and requiring only a one-time unitary transformation of the initial conditions and final output. We show that initializing the scheme in a way that is consistent with the non-relativistic limit supresses the oscillations around the Schrödinger solution. However, we find the QLB scheme better suited to simulation of relativistic quantum systems governed by the Dirac equation and apply it to the Klein paradox. We reproduce the quantum tunnelling results of previous research and show second order convergence to the theoretical wave packet transmission probability. After identifying and correcting the error in the multidimensional extension of the original QLB scheme that produced asymmetric solutions, we expand our second order QLB scheme to multiple dimensions. Next we use the QLB scheme to simulate Klein tunnelling of massless charge carriers in graphene, compare with theoretical solutions and study the dependence of charge transmission on the incidence angle, wave packet and potential barrier shape. To do this we derive a representation of the Dirac-like equation governing charge carriers in graphene for the one-dimensional QLB scheme, and derive a two-dimensional second order graphene QLB scheme for more accurate simulation of wave packets. We demonstrate charge confinement in a graphene device using a configuration of multiple smooth potential barriers, thereby achieving a high ratio of on/off current with potential application in graphene field effect transistors for logic devices. To allow simulation in magnetic or pseudo-magnetic fields created by deformation of graphene, we expand the scheme to include vector potentials. In addition, we derive QLB schemes for bilayer graphene and the non-linear Dirac equation governing Bose-Einstein condensates in hexagonal optical lattices.
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.
Quantum particles possess both wave and particle natures. When quantum effect devices exploiting the wave nature and the devices described from the view point of particle nature coexist in a system, the following advantages are gained by representing former devices in the same way as we describe the latter devices: First, it is not necessary to compute probability distribution from wave functions and sample device states in accordance with the probability distribution. Secondly, the particle models of the devices are suitable models for conventional circuit simulators. In this paper, simple particle models of an electromagnetic wave detector and an electron lens are constructed by applying Nelson's stochastic quantization theory. The device characteristics obtained from the wave functions and from the behavior of the particle models were almost equal.
Abstract 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 spacetime singularity of the maximal-analytically extended, topologically non-trivial, electromagnetic Kerr–Newman (KN)spacetime in the zero-gravity limit (here, ‘zero-gravity’ means the limit , 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 quantum-mechanical two-particle wave equation. A Dirac equation is formulated for such a ring-like bi-particle which interacts with a static point charge located elsewhere in the topologically non-trivial physical space associated with the moving ring 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 KN 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 reduced Compton wavelength.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Schrödinger equation the wave function is represented by a single complex quantity, in the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant, in contrast to the Dirac equation, which lacks this invariance. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Each of these pseudovectors or pseudospinors corresponds to one of the particles presented in the theory of electrodynamics. Furthermore, each plane wave possesses an additional coefficient in the form of a birth or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors corresponding to particles represented in electrodynamics.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by coordinate and momentum spinors. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Quantum mechanics describes interactions using complex wave functions that cannot be observed directly; only upon measurement do they translate into probabilities expressed as the squares of the magnitudes of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed partial derivatives on coordinates. Section 3 presents a version of quantum theory in spinor coordinate space. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. An equation of motion for the electron is proposed that is relativistically invariant. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Furthermore, each plane wave possesses an additional coefficient in the form of a creation or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations are presented, which can serve as a model of the graviton. Section 5 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor of a charged particle having four indices rather than two. In the case where a particle of matter has a charge, another Riemann tensor is added to Einstein’s equation, in addition to the usual Riemann curvature tensor, which describes the additional curvature created for a charged particle by an electromagnetic field.