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Variational solution to the ground state energy of the Yukawa potential, together with the literature average curve and the fifth-order perturbative ones.
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We study the ground state energy and the critical screening parameter of the Yukawa potential in non-relativistic quantum mechanics. After a short review of the existing literature on these quantities, we apply fifth-order perturbation theory to the calculation of the ground state energy, using the exact solutions of the Coulomb potential together...
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... Fig. 5 we show the result of using (4.4) in (4.3), together with the literature average curve and the curves from both versions of fifth-order perturbation theory (the curves for H 0 and H 0 are practically indistinguishable at the scale of the figure). Note that, for very small µ, the numerical evaluation of the variational curve becomes ...
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Citations
... Some mathematical properties of the RSCP, e.g., the dilatation analyticity and Fourier analysis, have also attracted increasing interest in recent years [1,13]. In view of the extensive application of the conventional screened Coulomb potential in chemical physics [14], atomic and plasma physics [15][16][17][18], and condensed matter systems [19][20][21], the newly proposed RSCP may reveal its potential usefulness in these active areas. ...
... The first iteration of r C min » into the above relation leads to an approximate solution of r C l l e 1 , 18 min » + + ( ) ( ) which means we omit the second-and higher-order terms with respect to l(l + 1)e, which is always a small number compared to large values of C considered here. The substitution of equation (18) into equation (16) produces the minimum value of the effective potential as » - ...
We revisit the energy spectrum of the radial screened Coulomb potential V(r)=−exp(−C/r)/r and analyze the small- and large-parameter (C) asymptotic laws of the system eigenenergies in both the original and C-scaled Schrödinger equations. The discrepancy observed in our previous work [Xu et al (2023) Phys. Lett. A 483, 129 064] with the numerical calculations of Stachura and Hancock [(2021) J. Phys. Commun. 5, 065 004] is resolved by reinterpreting their results as the eigenenergies of two-body systems of equal mass. We derive an improved analytical expression for system energies at small values of C and an approximate asymptotic law for arbitrary l-state energies when C approaches infinity. The behavior of the scaled system energies and wave functions is analyzed in the large-parameter limit. The present study provides solid proof that the radial screened Coulomb potential supports an infinite number of bound states for finite values of screening parameter, which completes the investigation of the energy spectrum of this special potential.
... and criticality (eigenvalue E = 0) occurs for µ = µ c where a numerical analysis yields [28,29] µ c = 1.19αm (72) For us, the spherical SKG equation is now r = |⃗ r| ...
... As our main result, we find that the coloron model can be classically critical. The critical coupling, g 2 c , extracted from [28,29], is astonishingly close to the critical value of the Yukawa coupling in the NJL model. While in the NJL model the critical behavior is O(h) coming from fermion loops, in the bilocal model this is a semiclassical result, and the essential factor of N c comes from the coherent BCS-like enhancement of the fourfermion scattering amplitudes. ...
We give a bilocal field theory description of a composite scalar with an extended binding potential that reduces to the Nambu–Jona-Lasinio (NJL) model in the pointlike limit. This provides a description of the internal dynamics of the bound state and features a static internal wave function, ϕ(r→), in the center-of-mass frame that satisfies a Schrödinger–Klein–Gordon equation with eigenvalues m2. We analyze the “coloron” model (single perturbative massive gluon exchange) which yields a UV completion of the NJL model. This has a BCS-like enhancement of its interaction, ∝Nc the number of colors, and is classically critical with gcritical remarkably close to the NJL quantum critical coupling. Negative eigenvalues for m2 lead to spontaneous symmetry breaking, and the Yukawa coupling of the bound state to constituent fermions is emergent.
... The potentials' screening parameter and eigenvalue pair (α, E ) have been successfully calculated and reported previously, but not near the critical values, especially as E increases towards the continuum. Only a few digits of the critical value of α were calculated until recently [8,9]. Its electronic density at the origin [10] is another quantity that becomes numerically challenging to compute near the critical values. ...
Coupled first-order differential forms of a single-particle Schrödinger equation are presented. These equa- tions are convenient to solve efficiently using the widely available ordinary differential equation solvers. This is particularly true because the solutions to the differential equation are two sets of complemen- tary functions that share simple and consistent mathematical relationships at the boundary and across the domain for a given potential. The differential equations are derived from an integral equation obtained using the Green’s function for the kinetic operator, making them universally applicable to various sys- tems. These equations are applied to the Yukawa potential −e−αr/r to calculate the critical screening parameter α=1.19061242106061770534277710636105 using a standard quadruple precision calcu- lation, which is the most accurate compared to similar calculations in the past that confirm the first 30 significant figures. Also reported is the interesting coincident point with the eigenvalue, α = −E = 0.274 376 862 689 408 994 894 705 268 554 458.
... The Yukawa potential is useful as it can be used for describing strong interactions between nucleons, in plasma physics for describing the potential of a charged particle in a weakly non-ideal plasma, in electrolytes and colloids, in solid state physics for describing the effects of a charged particle in a sea of conduction electrons and as well as in quantum mechanics [32]. To show the applicability of this potential, a few of the studies are mentioned here. ...
We have investigated the effect of temperature and geometrical confinement on the behavior of spherical cavities
with Yukawa potential presence inside the cavity within effective mass approximation. Using the Finite-element method, we have calculated the energy eigenvalue with the geometric and temperature effects into consideration with consideration of temperature and position dependent mass and position dependent dielectric constant. The optical transition from the ground to the excited state shows a blueshift in optical absorption coefficient, refractive index changes, second and third harmonic generation with an increase in temperature. The increase in the geometric size of the spherical cavity causes the redshift in the optical resonance peaks.
... In [13], a Keplerian-type parametrization was shown as a solution of the equations of motion for a Yukawa-type potential between two bodies. In fact, the two-body solution for alternative theories yielded a strong constraint for the solar system [14,15], whereas several analyses of Yukawa potential for a two-body system in different contexts were carried out as well [16,17]. The orbit of a single particle moving under Yukawa potential was studied in [18], and precessing ellipse-type orbits were observed. ...
In this manuscript, we review the motion of a two-body celestial system (planet–sun) for a Yukawa-type correction on Newton’s gravitational potential using Hamilton’s formulation. We reexamine the stability using the corresponding linearization Jacobian matrix, and verify that the conditions of Bertrand’s theorem are met for radii ≪1015 m, meaning that bound closed orbits are expected. Applied to the solar system, we present the equation of motion of the planet, then solve it both analytically and numerically. Making use of the analytical expression of the orbit, we estimate the Yukawa strength α and find it to be larger than the nominal value (10−8) adopted in previous studies, in that it is of order (α=10−4−10−5) for the terrestrial planets (Mercury, Venus, earth, Mars, and Pluto) and even larger (α=10−3) for the giant planets (Jupiter, Saturn, Uranus, and Neptune). Taking the inputs (rmin,vmas,e) observed by NASA, we analyse the orbits analytically and numerically for both the estimated and nominal values of α and determine the corresponding trajectories. For each obtained orbit, we recalculate the characterizing parameters (rmin,rmax,a,b,e) and compare their values according to the potential (Newton with/without Yukawa correction) and method (analytical and/or numerical) used. When compared to the observational data, we conclude that the path correction due to Yukawa correction is on the order of up to 80 million km (20 million km) as the maximum deviation occurring for Neptune (Pluto) for a nominal (estimated) value of α.
... It is well known that for a finite screening length there is a finite number of bound states [5,7,12]. Approximate calculations for some of the energy levels exist, using variational methods [4,8,9,[13][14][15], Rayleigh-Schrödinger perturbation theory [5,[16][17][18][19][20], new perturbation schemes [21,22], and other methods [23][24][25][26][27][28][29][30]. The ground state energy has been calculated to very high orders in the expansion on the parameter δ = a 0 /D = M /α g μ, where a 0 = 1/α g μ denotes the Bohr radius and μ stands for the reduced mass of the bounded system. ...
In this work, we present a phenomenological study of the complete analytical solution to the bound eigenstates and eigenvalues of the Yukawa potential obtained previously using the hidden supersymmetry of the system and a systematic expansion of the Yukawa potential in terms of δ = a 0 ∕D, where a 0 is the Bohr radius and D is the screening length. The eigenvalues, ϵnl(δ), are given in the form of Taylor series in δ which can be systematically calculated to the desired order δk. Coulomb l-degeneracy is broken by the screening effects and, for a given n, ϵnl(δ) is larger for higher values of l which causes the crossing of levels for n ≥ 4. The convergence radius of the Taylor series can be enlarged up to the critical values using the Padé approximants technique which allows us to calculate the eigenvalues with high precision in the whole rage of values of δ where bound states exist, and to reach a precise determination of the critical screening lengths, δnl. Eigenstates have a form similar to the solutions of the Coulomb potential, with the associated Laguerre polynomials replaced by new polynomials of order δk with r-dependent coefficients which, in turn, are polynomials in r. In general we find sizable deviations from the Coulomb radial probabilities only for screening lengths close to their critical values. We use these solutions to find the squared absolute value at the origin of the wave function for l = 0, and their derivatives for l = 1, for the lowest states, as functions of δ, which enter the phenomenology of dark matter bound states in dark gauge theories with a light dark mediator.
... For a given radial potential, there are quite a few upper and lower bounds on the number of bound states; see [7] for a review of some such bounds. These bounds can be translated into conditions on the screening parameter; this is indeed the case for the Yukawa potential [11]. However, these bounds (including the famous Bargmann bound [3]) do not apply with V P as these require the finiteness of a certain integral, which for V P does not converge. ...
... Here, we work in units such that e = 0 =h = m = 1. As in [11] with the Yukawa potential, we take the trial function ...
... For the Yukawa potential, the issue of bound states is an extremely interesting question, given the existence of the critical screening parameter [11]. The problem of counting the number of bound states for this potential has been considered by many, and depends on the dimension. ...
We analyze bound states and other properties of solutions of a radial Schrödinger equation with a new screened Coulomb potential. In particular, we employ hypervirial relations to obtain eigen-energies for a Hydrogen atom with this potential. Additionally, we appeal to a sharp estimate for a modified Bessel function to estimate the ground state energy of such a system. Finally, when the angular quantum number ℓ ≠ 0, we obtain evidence for a critical screening parameter, above which bound states cease to exist.
... It has been used in particle physics, physical chemistry, and in plasma applications because of its simplicity and validity at weak coupling. 28,[52][53][54][55] Quantum statistical potentials have also been considered for finite-T electronic systems since the 1960s, 56,57 and more recently, 58 where the latter authors concluded that such methods treat manybody effects inadequately. ...
We present computationally simple parameter-free pair potentials useful for solids, liquids, and plasmas at arbitrary temperatures. They successfully treat warm-dense matter (WDM) systems like carbon or silicon with complex tetrahedral or other structural bonding features. Density functional theory asserts that only one-body electron densities and one-body ion densities are needed for a complete description of electron–ion systems. Density functional theory (DFT) is used here to reduce both the electron many-body problem and the ion many-body problem to an exact one-body problem, namely, that of the neutral pseudoatom (NPA). We compare the Stillinger–Weber (SW) class of multi-center potentials, the embedded-atom approaches, and N-atom DFT, with the one-atom DFT approach of the NPA to show that many-ion effects are systematically included in this one-center method via one-body exchange-correlation functionals. This computationally highly efficient one-center DFT-NPA approach is contrasted with the usual N-center DFT calculations that are coupled with molecular dynamics simulations to equilibrate the ion distribution. Comparisons are given with the pair-potential parts of the SW, “glue” models, and the corresponding NPA pair-potentials to elucidate how the NPA potentials capture many-center effects using single-center one-body densities.
... As reviewed by Rogers et al. 44 and Edwards et al., 45 for example, the Yukawa potential is also used to model the potential of particle pairs when particle charges are screened or shielded by other charged particles-as one might find in a solid or plasma. In such contexts, a potential of the same form has been called a static screened Coulomb, 44 shielded Coulomb, 46 Debye-H€ uckel, 47 or Thomas-Fermi potential. ...
... Investigation of bound state solutions to Schodinger's equation given an attractive Yukawa potential have a long history, see Ref. 45 for a recent review. Exact analytic solutions are not known, though several analytic conditions on the system have been derived. ...
We present and discuss numerical solutions to a two-body quantum bound state problem closely related to that of the hydrogen atom and the deuterium nucleus. The forces binding the particles of our system are Yukawa forces, which fall off with distance faster than the Coulomb force and arise for non-relativistic particles whose interactions are mediated by massive scalar or vector particles; the Coulomb force arises in the limit of massless mediators such as photons. We use the solutions to explain several features of deuterium. We also present heuristic estimates of ground state energies and energy level degeneracy breaking. Studying this Yukawa two-body system provides insights into hydrogen, deuterium, and two-body bound states, generally. We describe our undergraduate-accessible procedure for numerically solving non-relativistic two-body bound state problems with mathematica in the appendix.
... Here k is the strength of the interaction, is a range of the interaction, r is the separation distance between nucleons. In the case of 0, Yukawa potential tends to Coulomb potential and infinite bound state while in the case of ≠ 0, the Yukawa potential has a finite bound state [8,9,10]. ...
The strength of linear combined short-range potential (Yukawa plus Hulthen) is calculated using two different centrifugal term approximation (equation 2 and equation 3). The calculation shows that the strength of potential calculated using an approximation of equation (3) is greater than equation (2), but the difference between them is negligible. Therefore, both approximation is used to calculate the energy eigenvalue of considering a potential system. Besides also, the functional value goes decreasing with increasing screening parameters and nucleon separation distance. This decreasing functional value shows the nature of potential Yukawa plus Hulthen potential is short-range potential.
