Relativistic quantum mechanics

In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) al

Aspects of two dimensional magnetic Schrödinger operatorsquantum hall systems and magnetic Stark resonances

Christian Ferrari

In this PhD thesis we deal with two mathematical problems arising from quantum mechanics. We consider a spinless non relativistic quantum particle whose configuration space is a two dimensional surface S. We also suppose that the particle feels the effect of an homogeneous magnetic field perpendicular to the surface S. In the first case S = R × SL1, the infinite cylinder of circumference L, corresponding to periodic boundary conditions, while in the second one S = R2. In both cases the particle feels the effect of an additional suitable potential. We are thus left with the study of two specific classes of Schrödinger operators. The operator of the first class generates the dynamics of the particle when it is submitted to an Anderson-type random potential, as well as to a non random potential confining the particle along the cylinder axis in an interval of length L. In this case we describe the spectrum and classify it by the quantum mechanical current carried by the corresponding eigenfunctions. We prove that there are spectral regions in which all the eigenvalues have an order one current with respect to L, and spectral regions where eigenvalues with order one current and eigenvalues with infinitesimal current with respect to L are intermixed. These results are relevant for the theory of the integer quantum Hall effect. The second Schrödinger operator class corresponds to the physical situation where the potential is the sum of a "local" potential and of a potential due to a weak constant electric field F. In this case we show that the resonant states, induced by the electric field, decay exponentially at a rate given by the imaginary part of the eigenvalues of some non self-adjoint operator. Moreover we prove an upper bound on this imaginary part that turns out to be of order exp(-1/F2) as F goes to zero. Therefore the lifetime of the resonant states is at least of order exp(-1/F2).

EPFL2003

Covariant formulation of relativistic mechanics

Miguel Alexandre Ribeiro Correia

Accretion disks surrounding compact objects, and other environmental factors, deviate satellites from geodetic motion. Unfortunately, setting up the equations of motion for such relativistic trajectories is not as simple as in Newtonian mechanics. The principle of general (or Lorentz) covariance and the mass-shell constraint make it difficult to parametrize physically adequate 4-forces. Here, we propose a solution to this old problem. We apply our framework to several conservative and dissipative forces. In particular, we propose covariant formulations for Hooke???s law and the constant force and compute the drag due to gravitational and hard-sphere collisions in dust, gas, and radiation media. We recover and covariantly extend known forces such as Epstein drag, Chandrasekhar???s dynamical friction, and Poynting-Robertson drag. Variable-mass effects are also considered, namely, Hoyle-Lyttleton accretion and the variable-mass rocket. We conclude with two applications: (1) The free-falling spring, where we find that Hooke???s law corrects the deviation equation by an effective anti???de Sitter tidal force and (2) black hole infall with drag. We numerically compute some trajectories on a Schwarzschild background supporting a dustlike accretion disk.

AMER PHYSICAL SOC2022

Electronic Structure of Graphene and Topological Insulators

Jens Christian Johannsen

The study of the electronic band structure of solids is a central task in materials science. Indeed, knowing a material's band structure is key to an understanding of its macroscopic electronic properties. In recent years, a new class of materials has emerged whose unifying feature is an electronic structure that breaks with conventional band theory. Rather than obeying the usual Schrödinger Hamiltonian, the low-energy quasiparticle excitations in the band structure of these materials are better described as chiral, massless Dirac-like carriers, similar in nature to the relativistic quantum particles known from high-energy physics. Prominent members of this class are graphene and topological insulators. Despite being vastly different materials, they share a number of intriguing electronic properties that render them highly attractive for both fundamental and application-oriented studies. The thrust of this thesis work is to investigate the photoexcited Dirac carriers in graphene and in the series of topological insulators (Sb2)m-Sb2Te3 by means of angle-resolved photoemission spectroscopy (ARPES) and XUV laser-based time-resolved ARPES. The aim is to push the research frontier further towards a more complete understanding of the fate of a Dirac carrier when it is brought out of thermodynamic equilibrium by an optical excitation; and what role the substrate material, doping level, laser excitation and the introduction of a band gap play, as well as how one can possibly engineer such a Dirac-like electronic structure. To this end, we take advantage of ARPES' momentum and energy resolving capabilities that can be extended into the time domain by adding femtosecond time-resolution in a pump-probe scheme. By taking the Dirac carriers out of their equilibrium state by a laser pump pulse and watching directly with a second probe pulse how they interact and relax back to equilibrium, a movie of the in situ carrier dynamics is obtained. This dynamical view on the ultrafast scattering processes of photoexcited Dirac carriers in graphene allows us to show a quasi-instant sub 40 femtosecond thermalization to a hot carrier distribution. The carriers cool back to equilibrium on femto- and picosecond timescales via excitations of optical phonons and acoustic phonons assisted by disorder-scattering in supercollisions processes. A method is introduced to accurately extract the hot carriers¿ temperature that is used to quantify the carrier multiplication. Cooling times and carrier multiplication are found to display a remarkable dependence on the doping level of graphene to the extent that a substantial carrier multiplication is possible. The hot carrier dynamics is found to be severely perturbed by the presence of a metal, and also pump-induced electric fields are shown to affect a correct interpretation of TR-ARPES data by masking the intrinsic carrier dynamics. Furthermore, we demonstrate that, in the case of bilayer graphene, the existence of a single-particle band gap in the electronic structure leads to a markedly enhanced carrier lifetime over monolayer graphene, despite the existence of in-gap states. Finally, a complete picture of the evolution of the band structure and Dirac nature of the carriers across a series of compounds, from Sb2Te3 to elemental Sb, is drawn, suggesting a promising strategy for engineering the electronic states.