Morse potential in relativistic contexts from generalized momentum operator: Schottky anomalies, Pekeris approximation and mapping

In this work, we explore a generalization of the Dirac and Klein–Gordon (KG) oscillators, provided with a deformed linear momentum inspired in nonextensive statistics, that gives place to the Morse potential in relativistic contexts by first principles. In the (1 + 1)-dimensional case, the relativistic oscillators are mapped into the quantum Morse potential. Using the Pekeris approximation, in the (3 + 1)-dimensional case, we study the thermodynamics of the S-waves states (l = 0) of the H2, LiH, HCl and CO molecules (in the non-relativistic limit) and of a relativistic electron, where Schottky anomalies (due to the finiteness of the Morse spectrum) and spin contributions to the heat capacity are reported. By revisiting a generalized Pekeris approximation, we provide a mapping from (3 + 1)-dimensional Dirac and KG equations with a spherical potential to an associated one-dimensional Schrödinger-like equation, and we obtain the family of potentials for which this mapping corresponds to a Schrödinger equation with non-minimal coupling.

Evolution of subhalo orbits in a smoothly growing host halo potential

ABSTRACT The orbital parameters of dark matter (DM) subhaloes play an essential role in determining their mass-loss rates and overall spatial distribution within a host halo. Haloes in cosmological simulations grow by a combination of relatively smooth accretion and more violent mergers, and both processes will modify subhalo orbits. To isolate the impact of the smooth growth of the host halo from other relevant mechanisms, we study subhalo orbital evolution using numerical calculations in which subhaloes are modelled as massless particles orbiting in a time-varying spherical potential. We find that the radial action of the subhalo orbit decreases over the first few orbits, indicating that the response to the growth of the host halo is not adiabatic during this phase. The subhalo orbits can shrink by a factor of ∼1.5 in this phase. Subsequently, the radial action is well conserved and orbital contraction slows down. We propose a model accurately describing the orbital evolution. Given these results, we consider the spatial distribution of the population of subhaloes identified in high-resolution cosmological simulations. We find that it is consistent with this population having been accreted at $z \lesssim 3$, indicating that any subhaloes accreted earlier are unresolved in the simulations. We also discuss tidal stripping as a formation scenario for NGC 1052-DF2, an ultra diffuse galaxy significantly lacking DM, and find that its expected DM mass could be consistent with observational constraints if its progenitor was accreted early enough, $z \gtrsim 1.5$, although it should still be a relatively rare object.

CONFINEMENT ENERGY OF QUANTUM DOTS AND THE BRUS EQUATION

A review of the ground state confinement energy term in the Brus equation for the bandgap energy of a spherically shaped semiconductor quantum dot was made within the framework of effective mass approximation. The Schrodinger wave equation for a spherical nanoparticle in an infinite spherical potential well was solved in spherical polar coordinate system. Physical reasons in contrast to mathematical expediency were considered and solution obtained. The result reveals that the shift in the confinement energy is less than that predicted by the Brus equation as was adopted in most literatures.

Controlled-scattering resonances in a spherical potential well

Understanding of collision processes is required in designing robust nano-particles for future applications. This study proposes a technique for controlling scattering resonances by using the tuning of well parameters to impose pre-determined thresholds on resonances and bound states in collision processes. The theoretical concept of scattering in a spherical potential well, at varying depths was adopted. A scan of q from 0 to 5π at incremental steps of q= π/p yields (p x 5)+1 number of state(s), and p-1 state(s) resonate(s) at each bound state.

Quantum Mechanics in Three Dimensions

In this chapter, the mathematical machinery of quantum mechanics is further developed in order to address real-world 3-dimensional physics. 3-dimensional vector notation is used for quantum mechanical operators and the Schrödinger equation is presented in this notation. The density of states of a particle in a box is considered. The angular momentum operators are defined. The eigenfunctions of the Laplacian are found. The Schrödinger equation with a spherical potential is analysed and solved for a Coulomb potential. The spectroscopy of the hydrogen atom is discussed. The spin operators are introduced. The Stern–Gerlach experiment and the Zeeman effect are discussed. The quantum mechanics of identical particles is considered and fundamental particles are shown to behave as either bosons or fermions depending on their spin. The action and the Feynman path integral are shown to offer an alternative approach to quantum mechanics that elucidates the connection between quantum and classical physics.