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We explore some consequences of modifying the usual Heisenberg commutation relations of two simple systems: first, the one-dimensional quantum system given by the infinite square-well potential, and second, the case of a gas of N non-interacting particles in a box of volume V, which permit obtaining analytical solutions. We analyse two possible cases of modified Heisenberg commutation relations: one with a linear and non-linear dependence on the position and another with a linear and quadratic dependence on the momentum. We determine the eigenfunctions, probability densities, and energy eigenvalues for the one-dimensional square well for both deformation cases. For linear and non-linear x deformation dependence, the wave functions and energy levels change substantially when the weight factor associated with the modification term increases. Here, the energy levels are rescaled homogeneously. Instead, for linear and quadratic momentum p deformation dependence, the changes in the energy spectrum depend on the energy level. However, the probability densities are the same as those without any modification. For the non-interacting gas, the position deformation implies that the ideal gas state equation is modified, acquiring the form of a virial expansion in the volume, whereas the internal energy is unchanged. Instead, the ideal gas state equation remains unchanged at the lowest order in β for the momentum modification case. However, the temperature modifies the internal energy at the lowest order in β. Thus, this study indicates that gravity could generate forces on particles by modifying the Heisenberg commutation relations. Therefore, gravitation could be the cause of the other three forces of nature.

We present results, from Monte Carlo (MC) simulations, on polymer systems of freely jointed chains with spherical monomers interacting through the square well potential. Starting from athermal packings of chains of tangent hard spheres, we activate the square well potential under constant volume and temperature corresponding effectively to instantaneous quenching. We investigate how the intensity and range of pair-wise interactions affected the final morphologies by fixing polymer characteristics such as average chain length and tolerance in bond gaps. Due to attraction chains are brought closer together and they form clusters with distinct morphologies. A wide variety of structures is obtained as the model parameters are systematically varied: weak interactions lead to purely amorphous clusters followed by well-ordered ones. The latter include the whole spectrum of crystal morphologies: from virtually perfect hexagonal close packed (HCP) and face centered cubic (FCC) crystals, to random hexagonal close packed layers of single stacking direction of alternating HCP and FCC layers, to structures of mixed HCP/FCC character with multiple stacking directions and defects in the form of twins. Once critical values of interaction are met, fivefold-rich glassy clusters are formed. We discuss the similarities and differences between energy-driven crystal nucleation in thermal polymer systems as opposed to entropy-driven phase transition in athermal polymer packings. We further calculate the local density of each site, its dependence on the distance from the center of the cluster and its correlation with the crystallographic characteristics of the local environment. The short- and long-range conformations of chains are analyzed as a function of the established cluster morphologies.

Access to time delay in a projectile-target scattering is a fundamental tool in understanding their interactions by probing the temporal domain. The present study focuses on computing and analyzing the Eisenbud-Wigner-Smith (EWS) time delay in low energy elastic e−C60 scattering. The investigation is carried out in the framework of a non-relativistic partial wave analysis (PWA) technique. The projectile-target interaction is described in (i) Density Functional Theory (DFT) and (ii) Annular Square Well (ASW) static model, and their final results are compared in details. The impact of polarization on resonant and non-resonant time delay is also investigated.

In a previous paper, we used a classification of the self adjoint extensions, also called self-adjoint determinations, of the differential operator −d2/dx2 in order to obtain the whole list of Supersymmetric (SUSY) partners of those selfadjoint determinations for which the ground state has strictly positive energy. The existence of self adjoint determinations with a ground state of zero or even negative energy is a proved fact. In this paper, we analyze the possibility of constructing SUSY partners for those determinations. We also study those cases for which the ground state has a degeneracy, the study of their SUSY partners should be analyzed separately. So far, we have studied those determinations having an exactly solvable eigenvalue problem. On the present study, we also included some comments in relation to determinations not exactly solvable from this point of view. In addition, the use of self adjoint determinations for which the ground state wave function has nodes (zeroes) produces formal SUSY partners with a finite number of eigenvalues or even with a purely continuous spectrum. We give some worked examples of these situations.

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator −d2/dx2 on L2[−a,a], a>0, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the extensions. There are essentially two big groups of extensions. In one, the ground state has strictly positive energy. On the other, either the ground state has zero or negative energy. In the present paper, we show that each of the extensions belonging to the first group (energy of ground state strictly positive) has an infinite sequence of supersymmetric partners, such that the ℓ-th order partner differs in one energy level from both the (ℓ−1)-th and the (ℓ+1)-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of −d2/dx2 come from a transcendental equation and are all infinite. For the case under our study, we determine the eigenvalues, which are also infinite, all the extensions have a purely discrete spectrum, and their respective eigenfunctions for all of its ℓ-th supersymmetric partners of each extension.

The auto correlation function, S(k) (where k = 4π λsinθ) and its Fourier component g(r) are important quantities characterizing the structure of a liquid. Experimentally these quantities have been determined using neutron or X-ray scattering intensities. Wertheim's solution of the fundamental statistical mechanical equation given by Percus and Yevick for hard spheres mixture is invoked with square well attractive part as a perturbation tail to evaluate the direct correlation function C(k) in momentum space through which structure factor, S(k) of liquid metals were derived. Radial distribution function, g(r) is obtained by Fourier analysis of computed S(k) from which the coordination numbers of ten liquid metals were computed. Surface tension of atomic liquids in terms of diffusion coefficient has been determined under square well interaction. The computed values of structural dependent coordination number, diffusion coefficient and surface tension of these liquid metals are compared with the available experimental results and a good agreement is found between the computed and experimental results

A solution is worked out for a γ-unstable Bohr Hamiltonian with an infinite square well potential having an adjustable step. The evolution of the model’s spectral characteristics are investigated with varying height and width of the additional step, revealing shape coexisting an mixing features. The description of the dynamical shape phase transition in the collective bands of the Te even-even nuclei is offered as a numerical application.

This paper explores how competing interactions in the intermolecular potential of fluids affect their structural transitions. This study employs a versatile potential model with a hard core followed by two constant steps, representing wells or shoulders, analyzed in both one-dimensional (1D) and three-dimensional (3D) systems. Comparing these dimensionalities highlights the effect of confinement on structural transitions. Exact results are derived for 1D systems, while the rational function approximation is used for unconfined 3D fluids. Both scenarios confirm that when the steps are repulsive, the wavelength of the oscillatory decay of the total correlation function evolves with temperature either continuously or discontinuously. In the latter case, a discontinuous oscillation crossover line emerges in the temperature–density plane. For an attractive first step and a repulsive second step, a Fisher–Widom line appears. Although the 1D and 3D results share common features, dimensionality introduces differences: these behaviors occur in distinct temperature ranges, require deeper wells, or become attenuated in 3D. Certain features observed in 1D may vanish in 3D. We conclude that fluids with competing interactions exhibit a rich and intricate pattern of structural transitions, demonstrating the significant influence of dimensionality and interaction features.

We study Andreev reflection in a one-dimensional square-well pair potential. We discuss the history of the model. The current-phase relation is presented as a sum over Matsubara frequencies. How the current arises from bound and continuum levels is found by analytic continuation. We discuss two limiting cases of the square-well potential, the zero-length well and the infinite well. The model is quantitatively valid in some cases but forms the basis for understanding a wide range of problems in inhomogeneous superconductivity and superfluidity.

In this work, the shear viscosity (η) of the liquid alkali metals including Rb, Cs, and Na is determined based on the Stokes–Einstein’s equation by an improved mean spherical approximation theory using an effective square-well potential relation for intermolecular interaction in a wide range of densities and temperatures. In this way, the PVT data and the linear isothermal regularity equation of state are used to determine the radial distribution function at a contact point at any thermodynamic state. The results obtained showed that the calculated values for viscosity increased strongly with increasing density and increased slightly with temperature. A reasonable agreement has been found between the calculated values of η and their available experimental data. Furthermore, a correlation relation between the effective molecular diameter and the temperature was found and the order of magnitude and sign of the coefficients were determined.