Bound State Solutions Research Articles

The correspondences between variance and information entropies of a particle confined by a q-deformed hyperbolic potential

The bound state solutions of the Schrödinger equation (SE) under a deformed hyperbolic potential are used in this paper to investigate the correlation between the information content, such as Shannon entropies and Fisher information, and the variance of a quantum system in both momentum and position spaces. The variance was obtained from the expectation moments in conjugated spaces and used to get the uncertainty products, Fisher information products and Shannon entropic sums for different potential parameters. These information measures were observed to vary with the potential parameters and obey their lowest bound inequalities. An increase in the Fisher information leads to a lower uncertainty and less information while decreases in the Fisher information result in higher uncertainties. We proposed a relationship between the Shannon entropies, Fisher information and the variance where an increase in the Fisher information leads to a lower Shannon entropy or a lower spread of the probability distribution and vice versa. The numerical results obtained with the proposed relations agree with the most utilized method in the literature. This current work simplifies the calculation of the Shannon entropies notably for complex potential functions.

Bound state solutions for quasilinear Schrödinger equations with Hardy potential

This paper is concerned with the existence of bound state solutions for quasilinear Schrödinger equations with Hardy potential and Berestycki–Lions type conditions. By using the variational methods, we get the existence results which is a complement to the ones in Hu et al. (2022).

Estimating the magnetic field contributions on thermodynamic functions of diatomic molecules trapped in an isotropic oscillator plus inverse quadratic potential

In this study, the Laplace transform approach have been employed to analyze the bound state solutions of two-dimensional Schrödinger equation with isotropic oscillator plus inverse quadratic potential. The wavefunctions and energy equation were derived. Thereafter, the so-called partition function and the thermodynamic properties were evaluated as a function Larmor frequency in the range 0<ωL<100 for some values of parameter λ. The findings showed that both the partition function and thermodynamic properties of the molecules were modified significantly by magnetic interactions, which invariably may alter the macroscopic behaviour of the system examined.

Complex partition function of 1-dimensional Dirac particle confined in Woods-Saxon interaction

We present the bound state solutions of one-dimensional Schrödinger and Dirac equations for a particle confined in Woods-Saxon potential. In each case the wavefunctions and the energy eigenvalues have been obtained in terms of the Jacobi polynomials by using the Nikiforov-Uvarov method. The complex eigenenergy obtained from the Dirac equation is explicitly shown to contain the nonrelativistic energies plus complex relativistic correction terms The quantum partition functions obtained using the relativistic and the nonrelativistic energies have been used to generate important thermodynamic properties of the system in the high temperature limit. In particular, the complex partition function obtained in the relativistic regime is taken to indicate some important physical effects in the system.

On a class of elliptic equations with critical perturbations in the hyperbolic space

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 ( B N ) , where B N denotes the hyperbolic space, 2 &lt; p &lt; 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 &lt; p &lt; + ∞, if N = 2, λ &lt; ( N − 1 ) 2 4 , and 0 &lt; a ∈ L ∞ ( B N ). We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1. Next, we prove that for a ( x ) ⩾ 1, there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.

Multiple bound states for a class of fractional critical Schrödinger–Poisson systems with critical frequency

In this paper we study the fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u=ϕ|u|2s*−3u+|u|2s*−2u,ε2s(−Δ)sϕ=|u|2s*−1,x∈R3, where s ∈ (0, 1), ɛ &amp;gt; 0 is a small parameter, 2s*=63−2s is the critical Sobolev exponent and V∈L32s(R3) is a nonnegative function which may be zero in some regions of R3, e.g., it is of the critical frequency case. By virtue of a new global compactness lemma, and the Lusternik–Schnirelmann category theory, we relate the number of bound state solutions with the topology of the zero set where V attains its minimum for small values of ɛ.

[formula omitted]-dimensional Klein–Gordon equation in presence of deformed generalized Deng–Fan with Yukawa potential class: Approximate bound state solutions in relativistic and non-relativistic regimes

In this paper, we examine the bound state solutions of the D-dimensional Klein–Gordon equation for a deformed Deng–Fan potential in generalized form along with Yukawa potential class. The supersymmetric quantum mechanics method and Nikiforov–Uvarov method are employed, utilizing a proper correspondence to the centrifugal potential term. Remarkably, in both methods, we obtain the same analytical expressions for normalized wave functions and energy eigenvalues, expressed in closed form by hypergeometric functions and Jacobi polynomials, for all l and n quantum states. Additionally, we explore the thermodynamic quantities (internal energy, partition function, specific heat, entropy and free energy) in case of both relativistic and non-relativistic regimes for the aforementioned potential. Finally, we demonstrate energy variation against various potential parameters for a few diatomic molecules pictorially.

Approximate bound state solutions of the Dunkl-Schrödinger equation for a hyperbolic double-well interaction

We construct approximate solutions to the stationary, one-dimensional Schrödinger equation for a hyperbolic double-well potential within the Dunkl formalism. Our approximation is applied to an inverse quadratic term contributed by the Dunkl formalism in the effective potential. The solutions we obtain are given in terms of confluent Heun functions. We establish parity of these solutions, discuss their elementary cases, and present an example of a system admitting bound states.

Existence and concentration results for a (p,q)-Laplacian problem with a general critical nonlinearity

The aim of this paper is to study the following (p,q)-Laplacian problem:{−εpΔpv−εqΔqv+V(x)(|v|p−2v+|v|q−2v)=f(v) in RN,v∈W1,p(RN)∩W1,q(RN),v>0 in RN, where ε>0 is a small parameter, N≥3, 1<p<q<N, Δsv:=div(|∇v|s−2∇v), with s∈{p,q}, is the s-Laplacian operator, V:RN→R is a positive continuous potential, and f:R→R is a Berestycki-Lions type nonlinearity with critical growth. By applying suitable variational methods, we construct a localized bound state solution that concentrates around an isolated component of a positive local minimum of V as ε→0.

Determination of the Energy Eigenvalues of the Varshni-Hellmann Potential

In this paper, we solve the bound state problem for the Varshni-Hellmann potential via a useful technique. In our technique, we obtain the bound state solution of the Schrödinger equation for the Varshni-Hellmann potential via ansatz method. We obtain the energy eigenvalues and the corresponding eigenfunctions. Also, the behavior of the energy spectra for both the ground and the excited state of the two body systems is illustrated graphically. The similarity of our results to the accurate numerical values is indicative of the efficiency of our technique.

Normalized bound state solutions of fractional Schrödinger equations with general potential

ABSTRACT In this article, we study a class of fractional Schrödinger equation (1) { ( − Δ ) s u = λu + a ( x ) | u | p − 2 u , ∫ R N | u | 2 d x = c 2 , u ∈ H s ( R N ) , where N>2s, s ∈ ( 0 , 1 ) and 0 $ ]]> p ∈ ( 2 , 2 + 4 s / N ) , c > 0 . a ( x ) ∈ C ( R N , R ) is a positive potential function. By using fixed-point theorem of Brouwer, barycenter function and variational method, we obtain the existence of normalized bound solutions for problem (1).

Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation

Abstract The energy bound-state solutions of the spinless Salpeter equation (SSE) have been obtained under a spin-dependent Cornell potential function via the Wentzel–Kramers–Brillouin approximation. The energy levels were applied to predict the mass spectra for the charmonium, bottomonium, and bottom-charmed mesons. The relativistic corrections for the angular momentum quantum number l &gt; 0 l\gt 0 , total angular momentum quantum numbers j = l , j = l ± 1 j=l,\hspace{.3em}j=l\pm 1 , and the radial quantum numbers n = 1–4 improve the mass spectra. The results agree fairly with experimental data and theoretic results reported in existing works, where the authors utilized different forms of the inter-quark potentials and methods. The deviation of the obtained masses for the charmonium and bottomonium from the observed data yields a total percentage error of 3.32 and 1.11%, respectively. The results indicate that the accuracy of the masses is correlated with the magnitude of masses for the charm and bottom quarks. The SSE together with the phenomenological spin-dependent Cornell potential provides an adequate account of the mass spectroscopy for the heavy mesons and may be used to predict other spectroscopic parameters.

New bound-state solutions and statistical properties of the IMK-GIQYP and IMSK-IQYP models in 3D-NRNCPS symmetries

Within the framework of three-dimensional non-relativistic noncommutative quantum phase-space (3D-NRNCPS) symmetries, we study the three-dimensional deformed Schrödinger equation (3D-DSE) using the improved modified Kratzer plus generalized inverse quadratic Yukawa potential (IMK-GIQYP) and the improved modified screened Kratzer plus inversely quadratic Yukawa potential (IMSK-IQYP) models. For this consideration, the well-known generalized Bopp’s shifts method and standard perturbation theory are used to solve the DSE in the 3D-NRNCPS regime. For the homogeneous (H2, N2 and I2) and heterogeneous (CO, CH and NO) diatomic molecules, the new non-relativistic energy equation and eigenfunction for the IMK-GIQYP and the IMSK-IQYP models in the presence of deformation phase-space are obtained to be sensitive to the atomic quantum numbers ([Formula: see text] and m), the mixed potential depths ([Formula: see text] and V) and ([Formula: see text] and [Formula: see text]), the screening parameters ([Formula: see text] and [Formula: see text]), and non-commutativity parameters ([Formula: see text] and [Formula: see text]) for the IMK-GIQYP and the IMSK-IQYP, respectively. We investigate the newly obtained bound state eigenvalues of the DSE in 3D-NRNCPS symmetries using the IMK-GIQYP and the IMSK-IQYP, with appropriate adjustments made to the improved modified Kratzer potential, improved modified screened Kratzer potential, improved generalized inverse quadratic Yukawa potential model and improved inversely quadratic Yukawa potential model. Additionally, in 3D-NRNCPS symmetries, the thermal properties of the IMK-GIQYP and the IMSK-IQYP, including their partition function, mean energy, free energy, specific heat and entropy, are thoroughly examined. Significant areas, including atomic and molecular physics, find many uses for this study.

Approximate bound state solutions of the fractional Schr\"{o}dinger equation under the spin-spin-dependent Cornell potential

In this work, the approximate bound state solutions of the fractional Schr\"{o}dinger equation under a spin-spin-dependent Cornell potential are obtained via the convectional Nikiforov-Uvarov approach. The energy spectra are applied to obtain the mass spectra of the heavy mesons such as bottomonium, charmonium and bottom-charm. The masses for the singlet and triplet spin numbers increase as the quantum numbers increase. The fractional Schr\"{o}dinger equation improves the mass spectra compared to other theoretic masses. The bottomonium masses agree with the experimental results where percentage errors for fractional parameters of $\beta =1,\alpha =0.97$ and $\beta =1,\alpha =0.50$ were found to be 0.67\% and 0.49\% respectively. The respective percentage errors of 1.97\% and 1.62\% for fractional parameters of $\beta =1,\alpha =0.97$ and $\beta =1,\alpha =0.50$ were obtained for charmonium meson. The results indicate that the potential curves coupled with the fractional parameters account for the short-range gluon exchange between the quark-antiquark interactions and the linear confinement phenomena which is associated with the quantum chromo-dynamic and phenomenological potential models in particle and high-energy physics.

Energy spectrum and applications of Eckart plus Hellmann potential in hyperspherical coordinates

We investigate the approximate bound-state solutions of the N-dimensional Schrödinger equation with a combination of Eckart and Hellmann potentials via the asymptotic iteration approach. By the use of the Greene-Aldrich approximation, which is valid for small values of the screening parameter, we establish the N-dimensional energy spectrum and the N-dimensional radial wave function in approximate analytic form. In hyperspherical coordinates, the normalized radial wave function is expressed in terms of hypergeometric and Jacobi polynomials. To double-check the energy spectrum in hyperspherical coordinates, we utilize polynomial solutions in view of the asymptotic iteration approach. The expressions for the expectation values of inverse position, square of inverse position, kinetic energy, and square of momentum are derived in hyperspherical coordinates by using the Hellmann-Feynmann theorem. We also present the analytically calculated energy eigenvalues for Eckart plus Hellmann potential. We deduce special forms of the relevant potential, such as Eckart, Hellmann, and Hulthén potentials.

Comment on “Effects of (1+2)-dimensional circularly symmetric and static traversable wormhole with cosmic string on oscillator field”

We analyze the mathematical results and physical conclusions put forward in a recent paper about “the relativistic quantum dynamics of an oscillator field described by the Klein–Gordon equation in a circularly symmetric and static (1+2)-dimensional traversable wormhole spacetime with cosmic strings”. We show that the existence of allowed oscillator frequencies conjectured by the author is a mere artifact of the truncation of a power series with the purpose of obtaining bound-state solutions in terms of polynomials. Our theoretical analysis suggest that the eigenvalues are continuous, increasing functions of the oscillator frequency, conclusion that is verified by a straightforward numerical calculation.

The calculation of ro-vibrational energies, franck-condon factors and r-centroids for Yttrium/Scandium oxides in D-dimension

The spectroscopic data as ro-vibrational energy values, Franck-Condon Factors and r-centroids are obtained for Yttrium/Scandium oxides with (B 2Σ+, X 2Σ+) different electronic states. Ro-vibrational energy values are calculated by bound state solution of the Schrödinger equation for the general molecular potential by applying Pekeris-type approximation to the centrifugal term in D-dimensional. Using these solutions, the relation giving the bound state energy eigenvalues and the corresponding wave functions expression are derived. The Franck-Condon factors and r-centroids values are determined by using Fraser-Jarmain method for the B 2Σ+ → X 2Σ+ electronic transitions in both molecules. All the results obtained are given numerically in the tables. The obtained values are compared with the previously get data in the literature and it is seen that the results are consistent.

Solutions of Schrödinger equation based on Modified Exponential Screened Plus Yukawa Potential (MESPYP) within the frame of parametric Nikiforov-Uvarov method

In this paper, we deduced an approximate bound state solutions of Schrӧdinger equation based on Modified Exponential Screened Plus Yukawa Potential (MESPYP), with the help Greene–Aldrich Approximation to determine the centrifugal term. The analytical solution was obtained based on Nikiforov–Uvarov (NU) method. The bound state solutions of five diatomic molecules which are mercury hydride (HgH), zinc hydride (ZnH), cadmium hydride (CdH), hydrogen bromide (HBr) and hydrogen fluoride (HF) molecules were also calculated numerically. The estimated energy eigenvalues for these molecules were deduced using the resulting energy eigenequation and total unnormalized wavefunction is expressed in term of Jacobi polynomial. The numerical solution shows that the bound state solutions increase with increase in the quantum state.

Topological defects on the energy spectra of diatomic molecules embedded with shifted Deng–Fan oscillator potential under AB flux field

AbstractA study on the effect of point like global monopole topological defects on the energy eigenvalues of the diatomic molecules , , , embedded with Shifted Deng–Fan Oscillator Potential under the influence of Aharonov–Bohm flux field has been made here. Asymptotic Iteration Method (AIM) is used to find out the bound state solutions for arbitrary states by solving the non‐relativistic Schrödinger equation. A Pekeris‐type approximation has been used to approximate the centrifugal barrier term. It is observed that, energy levels of the diatomic molecules is significantly affected by the global effects of the point like global monopole, flux field and effective potential field.

Existence of a positive bound state solution for logarithmic Schrödinger equation

We investigate the existence of bound state solutions for the logarithmic Schrödinger equation−Δu+V(x)u=ulog⁡u2inRN,N≥1, where V(x)∈C(RN) has a limit at infinity. In the case when the ground state is not attained, we construct a higher critical value for the variational functional. The classical variational methods cannot be applied directly to deal with the above problem due to nonsmoothness. To recover the smoothness, we use the superlinear power-law perturbation of the logarithmic nonlinearity.