The Schrödinger Equation with Deng-Fan-Eckart Potential (DFEP): Nikiforov-Uvarov-Functional Analysis (NUFA) Method

In this study, the radial part of the Schrödinger equation with the Deng-Fan-Eckart potential (DFEP) is solved analytically by employing the improved Greene and Aldrich approximation to bypass the centrifugal barrier and using the Nikiforov-Uvarov-Functional Analysis method (NUFA). The energy expression and wave function are obtained respectively. The numerical energy spectra for some diatomic molecules have been studied and compared with the findings of earlier studies and it has been found to be in good agreement. The NUFA method used in this study is easy and very less cumbersome compared to other methods that currently exist and it is recommended that researchers in this area adopt this method. The findings of this study will find direct applications in molecular physics.

Eigensolutions of the N-dimensional Schrödinger equation interacting with Varshni-Hulthén potential model

Analytical solutions of the N-dimensional Schrödinger equation for the newly proposed Varshni-Hulthén potential are obtained within the framework of Nikiforov-Uvarov method by using Greene-Aldrich approximation scheme to the centrifugal barrier. The numerical energy eigenvalues and the corresponding normalized eigenfunctions are obtained in terms of Jacobi polynomials. Special cases of the potential are equally studied and their numerical energy eigenvalues are in agreement with those obtained previously with other methods. However, the behavior of the energy for the ground state and several excited states is illustrated graphically.

Analytical Investigation of the Single-Particle Energy Spectrum in Magic Nuclei of ⁵⁶Ni and ¹¹⁶Sn

The analytical solutions of the radial D-dimensional Schrödinger equation for the Yukawa potential plus spin-orbit and Coulomb interaction terms are presented within the framework of the Nikiforov-Uvarov method by using the Greene-Aldrich approximation scheme to the centrifugal barrier. The energy eigenvalues obtained are employed to calculate the single-energy spectrum of ⁵⁶Ni and ¹¹⁶Sn for distinct quantum states. We have also obtained corresponding normalized wave functions for the magic nuclei manifested in terms of Jacobi polynomials. However, the energy spectrum without Spin-orbit and Coulomb interaction terms precisely matches the quantum mechanical system of the Yukawa potential field at any arbitrary state.

The unified treatment of the non-relativistic bound state solutions, thermodynamic properties and expectation values of exponential-type potentials.

In this paper, we obtained the analytical N-dimensional bound state solutions of exponential-type potential functions using the semi-classical WKB approximation method with an improved approximation to the orbital centrifugal barrier. Furthermore, we derived the partition function with the associated thermodynamic properties. We calculated the expectation values (⟨1/r^2 ⟩ ) via the Hellmann-Feynman theorem. Our results for the energy levels are in excellent agreement compared to the best available results obtained by other methods in the existing literature. The variations of the thermodynamic functions with the temperature parameter (β) agree very well with the other results obtained under different potential energy functions. The expectation values conformed to the ones obtained by another approach in previous work. This work demonstrates the exactness of the leading order WKB approximation.

Attosecond timing of electron emission from a molecular shape resonance

Shape resonances in physics and chemistry arise from the spatial confinement of a particle by a potential barrier. In molecular photoionization, these barriers prevent the electron from escaping instantaneously, so that nuclei may move and modify the potential, thereby affecting the ionization process. By using an attosecond two-color interferometric approach in combination with high spectral resolution, we have captured the changes induced by the nuclear motion on the centrifugal barrier that sustains the well-known shape resonance in valence-ionized N2. We show that despite the nuclear motion altering the bond length by only 2%, which leads to tiny changes in the potential barrier, the corresponding change in the ionization time can be as large as 200 attoseconds. This result poses limits to the concept of instantaneous electronic transitions in molecules, which is at the basis of the Franck-Condon principle of molecular spectroscopy.

Nonrelativistic treatment of hydrogen-like and neutral atoms subjected to the generalized perturbed Yukawa potential with centrifugal barrier in the symmetries of noncommutative quantum mechanics

We have obtained the approximate analytical solutions of the nonrelativistic Hydrogen-like atoms such as [Formula: see text] and [Formula: see text] and neutral atoms such as ([Formula: see text] and [Formula: see text]) atoms with a newly proposed generalized perturbed Yukawa potential with centrifugal barrier (GPYPCB) model using the generalized Bopp’s shift method and standard perturbation theory in the symmetries of noncommutative three-dimensional real space phase (NC: 3D-RSP). By approximating the centrifugal term through the Greene–Aldrich approximation scheme, we have obtained the energy eigenvalues and generalized Hamiltonian operator for all orbital quantum numbers [Formula: see text] in the symmetries of NC: 3D-RSP. The potential is a superposition of the perturbed Yukawa potential and new terms proportional with [Formula: see text]) appear as a result of the effects of noncommutativity properties of space and phase on the perturbed Yukawa potential model. The obtained energy eigenvalues appear as functions of the generalized Gamma function, the discreet atomic quantum numbers [Formula: see text], two infinitesimal parameters [Formula: see text], which are induced by (position–position and phase–phase). In addition, the dimensional parameters [Formula: see text] of perturbed Yukawa potential with centrifugal barrier model in NC: 3D-RSP. Furthermore, we have shown that the corresponding Hamiltonian operator in (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of perturbed Yukawa potential model and the two operators are modified spin–orbit interaction and the modified Zeeman operator for the previous Hydrogenic and neutral atoms.

Relativistic treatment of the Hellmann-generalized morse potential

We present the relativistic treatment of the Hellmann-generalized Morse potential using Nikiforov-Uvarov(NU) method. The relativistic equations(Klein-Gordon and Dirac equation) were solved using the conventional NU method. In order to overcome the centrifugal barrier, we employed the well-known Greene and Aldrich approximation scheme. The corresponding normalized eigenfunctions was also obtained in each case. It was shown that in the non-relativistic limits, both energy equations obtained by solving Klein-Gordon and Dirac equations, and wavefunctions reduced to the non-relativisitc energy Equation. The bound state energy eigenvalues for N2, CO, NO, CH and HCl diatomic molecules were computed for various vibrational and rotational quantum numbers. It was found that our results agree with those in literature.

Disc kinematics and stability in high-mass star formation

Context. In the disc-mediated accretion scenario for the formation of the most massive stars, high densities and high accretion rates could induce gravitational instabilities in the disc, forcing it to fragment and produce companion objects. Aims. We investigate the effects of inclination and spatial resolution on the observable kinematics and stability of discs in high-mass star formation. Methods. We studied a high-resolution 3D radiation-hydrodynamic simulation that leads to the fragmentation of a massive disc. Using RADMC-3D we produced 1.3 mm continuum and CH3CN line cubes at different inclinations. The model was set to different distances, and synthetic observations were created for ALMA at ~80 mas resolution and NOEMA at ~0.4′′. Results. The synthetic ALMA observations resolve all fragments and their kinematics well. The synthetic NOEMA observations at 800 pc with linear resolution of ~300 au are able to resolve the fragments, while at 2000 pc with linear resolution of ~800 au only a single structure slightly elongated towards the brightest fragment is observed. The position–velocity (PV) plots show the differential rotation of material best in the edge-on views. A discontinuity is seen at a radius of ~250 au, corresponding to the position of the centrifugal barrier. As the observations become less resolved, the inner high-velocity components of the disc become blended with the envelope and the PV plots resemble rigid-body-like rotation. Protostellar mass estimates from PV plots of poorly resolved observations are therefore overestimated. We fit the emission of CH3CN (12K−11K) lines and produce maps of gas temperature with values in the range of 100–300 K. Studying the Toomre stability of the discs, we find low Q values below the critical value for stability against gravitational collapse at the positions of the fragments and in the arms connecting the fragments for the resolved observations. For the poorly resolved observations we find low Q values in the outskirts of the disc. Therefore, although we could not resolve any of the fragments, we are able to predict that the disc is unstable and fragmenting. This conclusion is valid regardless of our knowledge about the inclination of the disc. Conclusions. These synthetic observations reveal the potential and limitations of studying discs in high-mass star formation with current (millimetre) interferometers. While the extremely high spatial resolution of ALMA reveals objects in extraordinary detail, rotational structures and instabilities within accretion discs can also be identified in poorly resolved observations.

Breaking the centrifugal barrier to giant planet contraction by magnetic disc braking

ABSTRACT During the runaway phase of their formation, gas giants fill their gravitational spheres of influence out to Bondi or Hill radii. When runaway ends, planets shrink several orders of magnitude in radius until they are comparable in size to present-day Jupiter; in 1D models, the contraction occurs on the Kelvin–Helmholtz time-scale tKH, which is initially a few thousand years. However, if angular momentum is conserved, contraction cannot complete, as planets are inevitably spun up to their breakup periods Pbreak. We consider how a circumplanetary disc (CPD) can de-spin a primordially magnetized gas giant and remove the centrifugal barrier, provided the disc is hot enough to couple to the magnetic field, a condition that is easier to satisfy at later times. By inferring the planet’s magnetic field from its convective cooling luminosity, we show that magnetic spin-down times are shorter than contraction times throughout post-runaway contraction: tmag/tKH ∼ (Pbreak/tKH)1/21 ≲ 1. Planets can spin-down until they corotate with the CPD’s magnetospheric truncation radius, at a period Pmax/Pbreak ∼ (tKH/Pbreak)1/7. By the time the disc disperses, Pmax/Pbreak ∼ 20–30; further contraction at fixed angular momentum can spin planets back up to ∼10Pbreak, potentially explaining observed rotation periods of giant planets and brown dwarfs.

Theoretical investigation of the isomerization pathways of diazenes: torsion vs. inversion

Ab initio classical trajectory simulations show that diazenes isomerize via out-of-plane torsion and not in-plane inversion due to a centrifugal barrier.