arXiv subjects

Muhammad Adeel Ajaib

Publications and source records attributed to Muhammad Adeel Ajaib.

14 recordsLinked to original sources

arXiv2025

Dirac Equation and Representation Dependent Scattering Phenomena

We show that spin-flip probabilities emerge in the relativistic regime for scalar potentials, absent in the standard Dirac representation. We examine 1D scattering for the Dirac equation employing an alternate matrix representation introduced by the Author in an earlier study. We demonstrate that the transmission (T) and reflection (R) coefficients can depend on the chosen representation of the Clifford algebra despite the two representations being related by a unitary or non-unitary transformation. We also show that for the non-unitary case quantum interference may arise in scattering phenomena, a testable experimental signature. This representation dependence hints at hidden physics in how spinor components couple to external steps/barriers, even when the relativistic dispersion relation (E^2=p^2+m^2) is the same.

quant-ph↗

arXiv2018

A Hamiltonian for Massless and Zero-Energy States

We present a non-hermitian Hamiltonian which can be employed to explain a condensed matter system with effectively massless and zero energy states. We analyze the 2D tunneling problem and derive the transmission and reflection coefficients for the massless and zero energy states. We find that the transmission coefficient of the massless and zero-energy particles in this case is consistent with non-chiral tunneling of quasiparticles which is in contrast to the Klein tunneling known for electrons described by the Dirac equation. Our analysis predicts that the massless electron can be reflected as a hole-like state whereas this transition does not occur for the zero-energy state. Experimental observations are needed to test whether the presented Hamiltonian can be realized in such a condensed matter system.

cond-mat.mes-hall↗

arXiv2018

Addressing Infinities in the Lévy-Leblond Hamiltonian

We attempt to shed light on the following question: What happens to the negative energy states when we take the non-relativistic limit of the Dirac equation? The Levy-Leblond equation is the non-relativistic limit of the Dirac equation and describes fermions in the non-relativistic limit. The Levy-Leblond equation includes singular matrices and an attempt to write the Hamiltonian appears to show that the negative energy states are "buried" under an infinity. We attempt to isolate the infinite energy states and also present an equivalent way of viewing the Schrodinger dispersion relation. We propose that the Levy-Leblond equation can also be seen as resulting from the contribution of enhanced Lorentz violating terms to the Dirac equation.

quant-ph↗

arXiv2018

Status Update on Selective SUSY GUT Inspired Models

We perform a status analysis of selective supersymmetric GUT models in light of recent constraints from collider and dark matter detection experiments. We find that a significant region of the parameter space of these models is still accessible to these experiments. Amongst the models we analyze, the split family model provides solutions that can explain the observed deviation in anomalous magnetic moment of the muon. Furthermore, there is a notable region of the parameter space of each model which yields the desired relic abundance for neutralino dark matter. We also present the prediction of spin independent and spin dependent neutralino cross sections in these models and find that there is parameter space which can be probed at future experiments searching for dark matter. Our analysis serves as a motivation to continue the search for supersymmetry at various experimental fronts.

hep-ph↗

arXiv2017

Lorentz violation and Condensed Matter Physics

We present heuristic arguments that hint to a possible connection of Lorentz violation with observed phenomenon in condensed matter physics. Various references from condensed matter literature are cited where operators in the Standard Model Extension appear to be enhanced. Furthermore, we consider the Levy-Leblond equation, which is the analogue of Dirac equation in non-relativistic quantum mechanics. We show that we can obtain the Levy-Leblond equation by adding enhanced Lorentz violating operators to the Dirac equation. Based on these observations, we propose that Lorentz violation exhibits itself in non-relativistic quantum mechanics.

hep-th↗

arXiv2017

Introducing Spin in 2D Quantum Tunneling

We study the quantum tunneling of non-relativistic electrons for two dimensional condensed matter systems. We employ the Levy-Leblond equation (which is the analogue of the Dirac equation for non-relativistic fermions) and show that the spin of the particle can be incorporated in the 2D tunneling problem. We derive the transmission and reflection coefficients of spin up and down electrons and show that the sum of these coefficients are consistent with the known results for gapless semiconductors.

quant-ph↗

arXiv2017

The Hydrogen Atom and the Equivalent Form of Levy-Leblond Equation

We discuss the equivalent form of Levy-Leblond equation [1, 2] such that the nilpotent matrices are two dimensional. We show that this equation can be obtained in the non-relativistic limit of the (2+1) dimensional Dirac equation. Furthermore, we analyze the case with four dimensional matrices and propose a Hamiltonian for the equation in (3+1) dimensions and solve it for a Coulomb potential. We show that the quantized energy levels for the hydrogen atom are obtained and the result is consistent with non-relativistic quantum mechanics.

quant-ph↗

arXiv2016

Non-Relativistic Limit of the Dirac Equation

We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary conditions. Based on these conclusions, we propose that the equation introduced in [1] is the non-relativistic limit of the Dirac equation and more appropriately describes spin 1/2 particles in the non-relativistic limit.

quant-ph↗

arXiv2015

A Fundamental Form of the Schrodinger Equation

We propose a first order equation from which the Schrodinger equation can be derived. Matrices that obey certain properties are introduced for this purpose. We start by constructing the solutions of this equation in 1D and solve the problem of electron scattering from a step potential. We show that the sum of the spin up and down, reflection and transmission coefficients, is equal to the quantum mechanical results for this problem. Furthermore, we present a 3D version of the equation which can be used to derive the Schrodinger equation in 3D.

quant-ph↗

arXiv2014

Sum Over Histories: Discrete Step Interpretation

We study the transition of a particle between two points such that the particle takes discrete spatial steps in this transition. We analyze how the sum over histories interpretation of quantum mechanics can be implemented in this scenario. We show that the Euclidean propagator of a free particle is recovered if the minimum space interval is of the order or greater than the De Broglie's wavelength of the particle. We also describe the statistical ensembles that model this transition. Furthermore, we discuss a possible extension of this model to 2-dimensions which serves as an example to extend it to any number of spatial dimensions.

quant-ph↗

arXiv2013

Numerical Methods and Causality in Physics

We discuss physical implications of the explicit method in numerical analysis. Numerical methods have there own condition for causality, known as the Courant-Friedrichs-Lewy condition. It is proposed that numerical causality merges with physical causality as the grid interval size approaches zero. We discuss the implications of this proposition on the numerical analysis of the wave equation. We also show that, insisting on physical causality, the numerical analysis of Schrodinger's equation implies that the minimum space interval should satisfy $Δx \ge a_0 λ_c$, where $λ_c$ is the reduced Compton wavelength and $a_0$ is a constant of the order unity.

physics.comp-ph↗

arXiv2012

Understanding Lorentz violation with Rashba interaction

Rashba spin orbit interaction is a well studied effect in condensed matter physics and has important applications in spintronics. The Standard Model Extension (SME) includes a CPT-even term with the coefficient H_{μν} which leads to the Rashba interaction term. From the limit available on the coefficient H_{μν} in the SME we derive a limit on the Rashba coupling constant for Lorentz violation. In condensed matter physics the Rashba term is understood as resulting from an asymmetry in the confining potential at the interface of two different types of semiconductors. Based on this interpretation we suggest that a possible way of inducing the H_{μν} term in the SME is with an asymmetry in the potential that confines us to 3 spatial dimensions.

hep-th↗

arXiv2012

Anisotropic to Isotropic Phase Transitions in the Early Universe

We propose that the early Universe was not Lorentz symmetric and that a gradual transition to the Lorentz symmetric phase occurred. An underlying form of the Dirac equation hints to such a transition for fermions. Fermions were coupled to space-time in a non-trivial manner such that they were massless in the Lorentz violating phase. The partition function is used as a transfer matrix to model this transition on a two level thermodynamics system that describes how such a transition might have occurred. The system that models this transition evolves, with temperature, from a state of large to negligible entropy and this is interpreted as describing the transition to a state with Lorentz symmetry. In addition to this, analogy is created with the properties of this system to describe how the fields were massless and how a baryon asymmetry can be generated in this model.

astro-ph.CO↗

arXiv2011

Muons from Neutralino Annihilations in the Sun: Flipped SU(5)

We consider two classes of supersymmetric flipped SU(5) models with gravity mediated supersymmetry breaking such that the thermal neutralino relic abundance provides the observed dark matter density in the universe. We estimate the muon flux induced by neutrinos that arise from neutralino annihilations in the Sun and discuss prospects for detecting this flux in the IceCube/Deep Core experiment. We also provide comparisons with the corresponding fluxes in the constrained minimal supersymmetric standard model and non-universal Higgs models. Regions in the parameter space that can be explored by the IceCube/DeepCore experiment are identified.

hep-ph↗