On the Search for Stationary States in Quantum Mechanics

Communications in Mathematical Physics, 1995

In this paper we prove the existence of stationary solutions of some nonlinear Dirac equations. We do it by using a general variational technique. This enables us to consider nonlinearities which are not necessarily compatible with symmetry reductions.

The Dirac equation is a cornerstone of quantum mechanics that fully describes the behaviour of spin ½ particles. Recently, the energy momentum relationship has been reconsidered such that |E|^2 = |(m0c^ 2)| 2 + |(pc)| 2 has been modified to: |E| 2 = |(m0c^2)|^2-|(pc)|^2 where E is the kinetic energy, moc^2 is the rest mass energy and pc is the wave energy for the spin ½ particle. This has been termed the 'Hamiltonian approach' and with a new starting point, the original Dirac equation has been derived: and the modified covariant form found is where h/2π = c = 1. The behaviour of spin ½ particles is found to be the same as for the original Dirac equation. The Dirac equation will also be expanded by setting the rest energy as a complex number, |(m0c 2)| e^jωt

Chinese Physics Letters, 2007

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrödinger like one. Earlier results are discussed in a unified framework and certain solutions of a large class of potentials are given.

2013 13th Mediterranean Microwave Symposium (MMS), 2013

Dirac equation is one of the fundamental equations of quantum mechanics, but the interpretation of its solutions in terms of spinors is very abstract. It is proposed in this paper to show that some electromagnetic solutions exist in terms of standing waves. We shall exhibit explicit solutions as standing modes of a rectangular electromagnetic cavity. We conclude by proposing a physical interpretation of the new terms which appear in the Dirac solution.

In this paper, we present an exact solution of the Dirac oscillator problem in one dimension in the context of the generalized uncertainty principle (GUP). The solution method presented here depends on the knowledge of the energy eigenvalues of the quantum harmonic oscillator with GUP. The crucial property of harmonic oscillator that the kinetic energy and the potential energy part of the Hamiltonian are of equal weight is used to obtain exact energy spectrum. Our result coincides with the results found in the literature. However, the solution procedure is completely different from others, very handy and alternative one. Moreover, we also remark the super symmetry aspects of the system.

Hadron Physics 2000 - Topics on the Structure and Interaction of Hadronic Systems - Proceedings of the International Workshop, 2001

Exact analytic solutions are found to the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic potentials, with the scalar part dominating, can be chosen to give exact analytic Dirac wave functions.

Calculus of Variations and Partial Differential Equations, 1996

The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic field. We prove the existence of soliton-like solutions of Maxwell-Dirac in (3+1)-Minkowski space-time. The solutions obtained are regular, stationary in time, and localized in space. They are found by a variational method, as critical points of an energy functional. This functional is strongly indefinite and presents a lack of compactness. We also find soliton-like solutions for the Klein-Gordon-Dirac system, arising in the Yukawa model.

A stationary solution of the Dirac equation in the metric of a Reissner-Nordstrom black hole has been found. Only one stationary regular state outside the black hole event horizon and only one stationary regular state below the Cauchy horizon are shown to exist. The normalization integral of the wave functions diverges on both horizons if the black hole is non-extremal. This means that the solution found can be only the asymptotic limit of a nonstationary solution. In contrast, in the case of an extremal black hole, the normalization integral is finite and the stationary regular solution is physically self-consistent. The existence of quantum levels below the Cauchy horizon can affect the final stage of Hawking black hole evaporation and opens up the fundamental possibility of investigating the internal structure of black holes using quantum tunneling between external and internal states.

2023

Quantum Mechanics