Energy Levels Of Bound States Research Articles

Impact of electronic correlation on strong laser-induced bound-state transitions

Electron correlation (EC) plays a crucial role in all multi-electron systems and dynamic processes. In this work, we focus on strong laser-induced bound-bound transitions (BBT), which are fundamental to optical absorption measurements. We use the helium atom, the simplest two-electron system, as our test case, utilizing the ab initio code package HeTDSE. We examined the bound state energy levels, transition dipole moments (TDMs), and the dynamics of strong laser-induced BBT, both with and without considering EC. Our results indicate that EC significantly impacts the energy levels of the bound states and the TDMs. These effects collectively influence the transition dynamics of the excited states. Although EC does not alter the quantum transition pathways between resonance states, it generally increases the probability of resonance transitions in most cases. Our findings provide a quantitative description of EC in laser-induced BBT.

Behavior of spin-0 scalar particles in Eddington-inspired Born–Infeld gravity global monopole with harmonic oscillator potential

In this paper, we investigate the behavior of a scalar bosonic field within the backdrop of Eddington-inspired Born–Infeld gravity with a global monopole configuration. Additionally, we consider the presence of both scalar and vector potentials. To introduce the scalar potential, we perform a transformation on the mass term, replacing [Formula: see text] with [Formula: see text]. For the vector potential, we apply a minimal coupling scheme, transforming [Formula: see text] within the relativistic Klein–Gordon equation. We opt for a harmonic oscillator scenario where the scalar and vector potentials are equal, i.e. [Formula: see text]. Through this setup, we analytically solve the wave equation using the confluent Heun equation form. Subsequently, we delve into the quantum system in the absence of any potential, determining the permissible values for bound-state relativistic energy levels and the scalar field’s wave function. Our analysis reveals that the energy levels and wave function are influenced by various parameters present in the eigenvalue expression.

Exact solution of three dimensional schrödinger equation with power function superposition potential.

Finding an analytical solution to the Schrödinger equation with power function superposition potential is essential for the development of quantum theory. For example, the harmonic oscillator potential, Coulomb potential, and Klazer potential are all classed as power function superposition potentials. In this study, the general form of the power function superposition potential was used to decompose the second-order radial Schrödinger equation with this potential into the first-order Ricatti equation. Furthermore, two forms of the power function superposition potential are constructed with an exact analytical solution, and the exact bound-state energy level formula is obtained for these two potentials. Finally, the energy levels of some of the diatomic molecules were determined through calculation. And our results are actually consistent with those obtained by other methods.

Analytical Solutions of the Molecular Kratzer-Feus potential by means of the Nikiforov-Uvarov Method

The analytical methods for solving Schrödinger equation are essential and effective tools with which we can investigate the spectroscopic properties, the electronic structure, and the energetic properties of the diatomic molecules (DMs). Accordingly, in this work, we used the Nikiforov-Uvarov (NU) method to solve the three-dimensional nonrelativistic Schrödinger equation with the molecular Kratzer-Feus (KF) potential and obtain the exact analytical bound state energy eigenvalues as well as their corresponding normalized eigenfunctions. The effective KF diatomic molecular potential well is investigated and represented graphically for several well-known DMs. The bound state energy levels are tabulated numerically for arbitrary values of the vibrational and rotational quantum numbers. The results obtained in this work are found to be in excellent agreement with the already-existing results in the literature.

Study of thermomagnetic properties of the diatomic particle using hyperbolic function position dependent mass under the external hyperbolic magnetic and AB force

In this present work, we study the approximate solution of the Schrodinger equation with hyperbolic function position-dependent mass for a symmetrical Modified Poschl-Teller potential. We consider the system influenced by the external hyperbolic magnetic and Aharonov-Bohm (AB) forces. By using the Laplace transform method, the second-order differential equation of the Schrodinger equation is reduced to a first-order differential equation and so the eigenfunction and energy eigenvalues are obtained. The behavior of the bound state energy levels was demonstrated and analyzed using the computational method for various values of the potential parameter, mass parameter, external magnetic force, and AB force. Moreover, the thermomagnetic properties of the system were analyzed for H2, LiH, and HCl diatomic particles, involving the vibrational mean energy, vibrational free energy, vibrational entropy, specific heat capacity, magnetization, magnetic susceptibility, and persistent current.

Theoretical study on potential energy surface and bound states of the Kr-CNCN complex: Compared with the Kr-NCCN system

The first ab initio potential energy surface of the Kr-CNCN system is investigated using the single double excitation coupled-cluster theory with noniterative treatment of triple excitation. The mixed basis sets are used with aug-cc-pVTz for the C and N atoms and aug-cc-pVTz-pp for the Kr atom, including an additional midbond function of 3s3p2d1f1g. The computed single point energies at 585 configurations are fitted to an analytical potential model. The potential energy surface of the Kr-CNCN complex has a nearly T-shaped global minimum with the energy of −286.6 cm−1 located at R = 6.88 a0, θ = 89.3°. The interaction of the linear CNCNKr geometry is stronger than that of the linear KrCNCN structure. The whole potential energy surface of the Kr-CNCN presents angular anisotropy. The bound state energies up to J = 9 and the wave functions are calculated by analytically solving the Schrödinger equation based on the ab initio potential energy surface. The calculated fundamental frequencies of bending vibration and stretching vibration of the Kr-CNCN are 24.0 cm−1 and 32.7 cm−1, respectively. The transition frequencies and the spectroscopic constants of the Kr-CNCN system are predicted, and the relevant results are compared with the Kr-NCCN complex. The differences in the PES and bound state energy levels between the Kr-CNCN and Kr-NCCN complexes are clarified. These results will help to further explore the formation causes of CN radical in the interstellar medium and provide a theoretical basis for future experimental spectra research of the Kr-CNCN system.

Phase space path integral for modified pöschl-teller potential

The phase space path integral for a particle inside the modified Pöschl-Teller potential hole (MPT) is exactly evaluated by using the delta functional technique. The bound-state energy-levels and the corresponding suitably normalized wave functions are derived.

Density profile of a semi-infinite one-dimensional Bose gas and bound states of the impurity

We study the effect of the boundary on a system of weakly interacting bosons in one dimension. It strongly influences the boson density which is completely suppressed at the boundary position. Away from it, the density is depleted over the distances on the order of the healing length at the mean-field level. Quantum fluctuations modify the density profile considerably. The local density approaches the average one as an inverse square of the distance from the boundary. We calculate an analytic expression for the density profile at arbitrary separations from the boundary. We then consider the problem of localization of a foreign quantum particle (impurity) in the potential created by the inhomogeneous boson density. At the mean-field level, we find exact results for the energy spectrum of the bound states, the corresponding wave functions, and the condition for interaction-induced localization. The quantum contribution to the boson density gives rise to small corrections of the bound state energy levels. However, it is fundamentally important for the existence of a long-range Casimir-like interaction between the impurity and the boundary.

Two-dimensional hydrogen-like atom in a weak magnetic field

.We consider a non-relativistic two-dimensional (2D) hydrogen-like atom in a weak, static, uniform magnetic field perpendicular to the atomic plane. Within the framework of the Rayleigh-Schrödinger perturbation theory, using the Sturmian expansion of the generalized radial Coulomb Green function, we derive explicit analytical expressions for corrections to an arbitrary planar hydrogenic bound-state energy level, up to the fourth order in the strength of the perturbing magnetic field. In the case of the ground state, we correct an expression for the fourth-order correction to energy available in the literature.

A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

Abstract We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term ~x −1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x −2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

A conditionally integrable Schrödinger potential of a bi-confluent Heun class

We present a bi-confluent Heun potential for the Schrödinger equation involving inverse fractional powers and a repulsive centrifugal-barrier term the strength of which is fixed to a constant. This is an infinite potential well defined on the positive half-axis. Each of the fundamental solutions for this conditionally integrable potential is written as an irreducible linear combination of two Hermite functions of a shifted and scaled argument. We present the general solution of the problem, derive the exact energy spectrum equation and construct a highly accurate approximation for the bound-state energy levels.

Point-particle effective field theory III: relativistic fermions and the Dirac equation

We formulate point-particle effective field theory (PPEFT) for relativistic spin-half fermions interacting with a massive, charged finite-sized source using a first-quantized effective field theory for the heavy compact object and a second-quantized language for the lighter fermion with which it interacts. This description shows how to determine the near-source boundary condition for the Dirac field in terms of the relevant physical properties of the source, and reduces to the standard choices in the limit of a point source. Using a first-quantized effective description is appropriate when the compact object is sufficiently heavy, and is simpler than (though equivalent to) the effective theory that treats the compact source in a second-quantized way. As an application we use the PPEFT to parameterize the leading energy shift for the bound energy levels due to finite-sized source effects in a model-independent way, allowing these effects to be fit in precision measurements. Besides capturing finite-source-size effects, the PPEFT treatment also efficiently captures how other short-distance source interactions can shift bound-state energy levels, such as due to vacuum polarization (through the Uehling potential) or strong interactions for Coulomb bound states of hadrons, or any hypothetical new short-range forces sourced by nuclei.

Path integral solution for a Klein–Gordon particle in vector and scalar deformed radial Rosen–Morse-type potentials

The problem of a Klein–Gordon particle moving in equal vector and scalar Rosen–Morse-type potentials is solved in the framework of Feynman’s path integral approach. Explicit path integration leads to a closed form for the radial Green’s function associated with different shapes of the potentials. For $$q\le -1$$ , and $$\frac{1}{2\alpha }\ln \left| q\right|<r<+\infty$$ , the energy equation and the corresponding wave functions are deduced for the l states using an appropriate approximation to the centrifugal potential term. When $$-1<q<0$$ or $$q>0$$ , it is shown that the quantization conditions for the bound state energy levels $$E_{n_{r}}$$ are transcendental equations which can be solved numerically. Three special cases such as the standard radial Manning–Rosen potential $$(\left| q\right| =1)$$ , the standard radial Rosen–Morse potential $$(V_{2}\rightarrow -V_{2},q=1)$$ and the radial Eckart potential $$(V_{1}\rightarrow -V_{1},q=1)$$ are also briefly discussed.

Solutions of the bi-confluent Heun equation in terms of the Hermite functions

We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the restrictions that are imposed on the involved parameters in order that the series terminates thus resulting in closed-form finite-sum solutions of the bi-confluent Heun equation. A physical application of the closed-form solutions is discussed. We present the five six-parametric potentials for which the general solution of the one-dimensional Schrödinger equation is written in terms of the bi-confluent Heun functions and further identify a particular conditionally integrable potential for which the involved bi-confluent Heun function admits a four-term finite-sum expansion in terms of the Hermite functions. This is an infinite well defined on a half-axis. We present the explicit solution of the one-dimensional Schrödinger equation for this potential and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and construct an accurate approximation for the bound-state energy levels.

Quasi exactly solvable extension of Calogero model associated with exceptional orthogonal polynomials

By using the technique of supersymmetric quantum mechanics, we study a quasi exactly solvable extension of the N-particle rational Calogero model with harmonic confining interaction. Such quasi exactly solvable many particle system, whose effective potential in the radial direction yields a supersymmetric partner of the radial harmonic oscillator, is constructed by including new long-range interactions to the rational Calogero model. An infinite number of bound state energy levels are obtained for this system under certain conditions. We also calculate the corresponding bound state wave functions in terms of the recently discovered exceptional orthogonal Laguerre polynomials.

Scattering States of l-Wave Schrödinger Equation with Modified Rosen—Morse Potential**Supported by the National Natural Science Foundation of China under Grant No. 11405128, and Natural Science Basic Research Plan in Shaanxi Province of China under Grant No. 15JK2093

Within a Pekeris-type approximation to the centrifugal term, we examine the approximately analytical scattering state solutions of the l-wave Schrödinger equation with the modified Rosen—Morse potential. The calculation formula of phase shifts is derived, and the corresponding bound state energy levels are also obtained from the poles of the scattering amplitude.

A single short pulse produced by the excited state of Na atom exposed to an infrared laser

The time-dependent Schrodinger equation of alkali metal Na atom in an infrared laser field is solved numerically by using the pseudo-spectral method. In the calculation, an accurate model potential of Na atom is used. The bound state energy levels, which are consistent with experimental data, are obtained with the potential, so that we can study the characteristics of high-order harmonic generation for emission of the exited stated of Na atom. Our results show that the high-order generation spectrum of emission of 4s, and 5s excited states of Na atom is super-continuum in the over-barrier ionization regime. By superposed certain orders harmonics below threshold, a single pulse can be obtained with the central frequency from high frequency of visible light to the ultraviolet band. Through the calculated ionization probability of Na atom and the time-frequency analysis by wavelet transform of the superposed harmonics, it reveals that the emission process of low-order harmonic generation in over-barrier ionization regime is different from in the tunnel ionization regime.

Curvature-induced bound states and coherent electron transport on the surface of a truncated cone

We study the curvature-induced bound states and the coherent transport properties for a particle constrained to move on a truncated cone-like surface. With longitudinal hard wall boundary condition, the probability densities and spectra energy shifts are calculated, and are found to be obviously affected by the surface curvature. The bound-state energy levels and energy differences decrease as increasing the vertex angle or the ratio of axial length to bottom radius of the truncated cone. In a two-dimensional (2D) GaAs substrate with this geometric structure, an estimation of the ground-state energy shift of ballistic transport electrons induced by the geometric potential (GP) is addressed, which shows that the fraction of the ground-state energy shift resulting from the surface curvature is unnegligible under some region of geometric parameters. Furthermore, we model a truncated cone-like junction joining two cylinders with different radii, and investigate the effect of the GP on the transmission properties by numerically solving the open-boundary 2D Schrödinger equation with GP on the junction surface. It is shown that the oscillatory behavior of the transmission coefficient as a function of the injection energy is more pronounced when steeper GP wells appear at the two ends of the junction. Moreover, at specific injection energy, the transmission coefficient is oscillating with the ratio of the cylinder radii at incoming and outgoing sides.

Proton Radius Puzzle in Hamiltonian Dynamics

Relativistic lepton-proton bound-state eigenvalue equations for Hamiltonians derived from quantum field theory using second-order renormalization group procedure for effective particles, are reducible to two-body Schroedinger eigenvalue equations with the effective Coulomb potential that exhibits a tiny sensitivity to the characteristic momentum-scale of the bound system. The scale dependence is shown to be relevant to the theoretical interpretation of precisely measured lepton-proton bound-state energy levels in terms of a 4 percent difference between the proton radii in muon-proton and electron-proton bound states.

An investigation of the Casimir energy for a fermion coupled to the sine-Gordon soliton with parity decomposition

We consider a fermion chirally coupled to a prescribed pseudoscalar field in the form of the soliton of the sine-Gordon model and calculate and investigate the Casimir energy and all of the relevant quantities, i.e. the spectrum of the states and the phase shifts, for each parity channel, separately. We present and use a simple prescription to construct the simultaneous eigenstates of the Hamiltonian and parity in the continua from the scattering states. We also use a prescription we had introduced earlier to calculate unique expressions for the phase shifts and check their consistency with both the weak and the strong forms of the Levinson theorem. In the graphs of the total and parity decomposed Casimir energies as a function of the parameters of the pseudoscalar field distinctive deformations appear whenever a fermionic bound state energy level with definite parity crosses the line of zero energy. However, the latter graphs show considerable sensitivity to the fine details of the shape of the background field which cannot be seen from the graph of the total Casimir energy. Finally, we consider a system consisting of a valence fermion in the ground state and find that the most energetically favorable configuration is the one with a soliton of winding number one, and this conclusion does not hold for each parity, separately.