scholarly journals Unitary quantum lattice simulations for Maxwell equations in vacuum and in dielectric media

2020 ◽  
Vol 86 (5) ◽  
Author(s):  
George Vahala ◽  
Linda Vahala ◽  
Min Soe ◽  
Abhay K. Ram
Keyword(s):  
Wave Packet ◽  
Maxwell Equations ◽  
Quantum Computer ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Spinor Representation ◽  
Quantum Lattice ◽  
Dielectric Media ◽  
Uniform Medium ◽  
Lattice Algorithm

Utilizing the similarity between the spinor representation of the Dirac and the Maxwell equations that has been recognized since the early days of relativistic quantum mechanics, a quantum lattice algorithm (QLA) representation of unitary collision-stream operators of Maxwell's equations is derived for both homogeneous and inhomogeneous media. A second-order accurate 4-spinor scheme is developed and tested successfully for two-dimensional (2-D) propagation of a Gaussian pulse in a uniform medium whereas for normal (1-D) incidence of an electromagnetic Gaussian wave packet onto a dielectric interface requires 8-component spinors because of the coupling between the two electromagnetic polarizations. In particular, the well-known phase change, field amplitudes and profile widths are recovered by the QLA asymptotic profiles without the imposition of electromagnetic boundary conditions at the interface. The QLA simulations yield the time-dependent electromagnetic fields as the wave packet enters and straddles the dielectric boundary. QLA involves unitary interleaved non-commuting collision and streaming operators that can be coded onto a quantum computer: the non-commutation being the very reason why one perturbatively recovers the Maxwell equations.

scholarly journals A dynamical time operator in Dirac's relativistic quantum mechanics

2014 ◽  
Vol 29 (06) ◽  
pp. 1450036 ◽  
Author(s):  
M. Bauer
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Phase Velocity ◽  
Relativistic Quantum ◽  
Rate Of Change ◽  
Time Operator ◽  
Relativistic Quantum Mechanics ◽  
Instantaneous Rate ◽  
Expectation Value ◽  
Dynamical Time

A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.

OCTONIC FIRST-ORDER EQUATIONS OF RELATIVISTIC QUANTUM MECHANICS

2009 ◽  
Vol 24 (22) ◽  
pp. 4157-4167 ◽  
Author(s):  
VICTOR L. MIRONOV ◽  
SERGEY V. MIRONOV
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Maxwell Equations ◽  
Relativistic Quantum ◽  
Order Equation ◽  
Second Order ◽  
Relativistic Quantum Mechanics ◽  
Quantum Fields ◽  
First Order ◽  
Spatial Operators

We demonstrate a generalization of relativistic quantum mechanics using eight-component octonic wave function and octonic spatial operators. It is shown that the second-order equation for octonic wave function describing particles with spin 1/2 can be reformulated in the form of a system of first-order equations for quantum fields, which is analogous to the system of Maxwell equations for the electromagnetic field. It is established that for the special types of wave functions the second-order equation can be reduced to the single first-order equation analogous to the Dirac equation. At the same time it is shown that this first-order equation describes particles, which do not have quantum fields.

scholarly journals Relativistic Inversion, Invariance and Inter-Action

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1117
Author(s):  
Martin B. van der Mark ◽  
John G. Williamson
Keyword(s):  
Maxwell Equations ◽  
Relativistic Quantum ◽  
Rest Mass ◽  
Natural Extension ◽  
Basis Set ◽  
Relativistic Quantum Mechanics ◽  
Dimensional Algebra ◽  
Dirac Algebra ◽  
First Order ◽  
Vector Algebra

A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter.

BILINEAR FORM AND SOLITON SOLUTIONS FOR THE COUPLED NONLINEAR KLEIN–GORDON EQUATIONS

2012 ◽  
Vol 26 (15) ◽  
pp. 1250057
Author(s):  
HE LI ◽  
XIANG-HUA MENG ◽  
BO TIAN
Keyword(s):  
Field Theory ◽  
Bilinear Form ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Soliton Solutions ◽  
Relativistic Quantum Mechanics ◽  
Klein Gordon ◽  
Truncated Painlevé Expansion ◽  
Klein Gordon Equation ◽  
Singular Manifolds

With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painlevé expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transformed to a bilinear form. Starting from the bilinear form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.

Stochastic microcausality in relativistic quantum mechanics

1984 ◽  
Vol 14 (9) ◽  
pp. 883-906 ◽  
Author(s):  
D. P. Greenwood ◽  
E. Prugovečki
Keyword(s):  
Quantum Mechanics ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics
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