Ground State Energy Eigenvalue Research Articles

Analytic ground state energy and wave function of a quantum system in an exponential-form anharmonic interaction potential

An anharmonic oscillator with a perturbed quadratic potential which is coupled with an exponential term is being investigated in this paper by isolating an anharmonic oscillator interaction potential from the actual interaction experienced by the quantum system studied, and using standard perturbative techniques. The anharmonic potential considered is of interest because of its usefulness in the study of non-centrosymmetric materials which have applications in piezoelectricity. The ground state energy eigenvalue and its associated eigenstates were calculated for the quantum system using an analytical approach. Results obtained are compared to those of quantised harmonic oscillator to show the effect of the perturbation.

Analysis of the ground-state energy eigenvalues of fractal quantum potentials

Traditional models in solid-state physics study the motion of electrons in a periodic lattice. Recent developments in the physics of graphene allow scientists to construct two-dimensional structures with fractal geometry and conduct experiments on them. Recently, some theoretical approaches have been developed to study the optical and electrical properties of semiconductor layers with self-similar characteristics. This newly emerged direction in solid-state physics inspired us to focus on studying the quantum mechanical properties of fractal potentials. We first introduce sequences of potential wells converging towards different fractal structures. Then, we calculate the ground-state energy eigenvalues of the time-independent Schrödinger equation for these potential functions using two numerical methods, the Numerov and analytical transfer matrix method, to demonstrate the effect of the potential structure morphology and properties on the behavior of the energy eigenvalues. Ground-state energies for the generalized Cantor set, the Smith–Volterra–Cantor set, a multi-level Cantor set and the Weierstrass function will be calculated and compared. We will deal with the question of how the ground state of these potential functions changes as the fractal generator is applied, and we show that properties such as the Lebesgue measure of the Cantor potentials and the Hausdorff dimension of the Weierstrass function strongly control the convergence of energy eigenvalues.

Nonlinear absorption coefficient and refractive index changes of two-dimensional two-electron quantum dot in rigid confinement

The Hamiltonian and wavefunctions describing two-dimensional (2D) two-electron ZnO quantum dot in rigid confinement are developed. Then the Schrödinger equation is solved analytically and numerically for determining the ground and excited state energies. The ground state energy of 2D two-electron ZnO quantum dot (QD) in rigid confinement is studied using perturbation and variational methods. The obtained result show that our trial wavefunction is good enough to describe the 2D two-electron QD in rigid confinement. The wavefunction describing the ground state is the combination of symmetric spatial wavefunction and antisymmetric spin wavefunction which is a para-state. The ground state energy eigenvalue obtained by variational technique is a little above that of a perturbation technique. Based on this; the trial wavefunction for the excited state is developed. The excited state energy of 2D two-electron ZnO QD in rigid confinement is studied computationally using variational method. The wavefunction describing the excited state is the combination of symmetric spatial wavefunction with antisymmetric spin wavefunction (para-state) or vice versa (ortho-state). The para and ortho-state energies of the first excited state are calculated and their difference is twice of the exchange energy. Based on the obtained energy eigenvalues of the ground and the first excited state at the value of the coupling constant [Formula: see text] [Formula: see text] 1, the third-order nonlinear absorption coefficient and refractive index changes are investigated. The optical transition is only considered between the two lowest para states.

\(Ps^−\) and \(H^+_2\) Ions under Spatial Confinement

The ground state energy eigenvalues of the spatially confined symmetric three-body exotic ions \(Ps^-\) and \(H^+_2\) have been determined under the framework of Rayleigh-Ritz variational method. The spatial confinement is simulated by considering an impenetrable spherical box of varying radius around the ion. The electron-electron correlation in \(Ps^-\) and the protonproton correlation in \(H_+^2\) are properly taken care of by expanding the trial wavefunction in explicitly correlated Hylleraas type basis set. The critical radius and the corresponding critical pressure, at which the respective ions destabilize are also reported.

On the quantum mechanics of bicomplex Hamiltonian system

We investigate the Schrödinger equation in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. We propose an analytical method to solve bicomplex-version of Schrödinger equation corresponding to the systems of Hamiltonians of both hermitian (self-adjoint) and non-hermitian PT symmetric type. In our approach we extend the existing mathematical formulation of quantum system searching for the exact or quasi-exact solution for ground state energy eigenvalues and associated wave functions acting in bicomplex Hilbert space. The model concerning hermitian Hamiltonians is then applied to the problems of two bicomplex valued polynomial oscillators one involving x2 and another of isotonic type. The ground states and associated energy values for both the oscillators are found to be hyperbolic in nature. The model in connection to the unbroken PT symmetric Hamiltonians is then applied to illustrate the problems of complex and bicomplex valued shifted oscillators.

Ground State Energy Eigenvalues of an Octic Oscillator with Quartic and Sextic Anharmonicities

Ground State Energy Eigenvalues of an Octic Oscillator with Quartic and Sextic Anharmonicities

Dirac Bound States of the Killingbeck Potential Under External Magnetic Fields

Abstract The Killingbeck potential model is used to study the influence of the external magnetic and Aharanov–Bohm (AB) flux fields on the splitting of the Dirac energy levels in a 2+1 dimensions. The ground state energy eigenvalue and its corresponding two spinor components wave functions are investigated in the presence of the spin and pseudo-spin symmetric limit as well as external fields using the wave function ansatz method.

Exact Diagonalization of the Hubbard Model: Ten-electrons on Ten-sites

By exactly diagonalizing the Hubbard model for ten electrons on ten sites in a one-Dimensional (1D) ring, we extend the study of Jafari (2008) to more than two electrons on two sites. We equally show the sparsity patterns of the Hamiltonian matrices for four- and eight-site problems and obtain the ground state energy eigenvalues for ten electrons on ten-sites. The technique we employ will be a good guide to a beginner/programmer.

Linear higher derivative field theories, entanglement entropy and quantum phase transition

Quantum systems described by linear, higher order spatial derivatives of the field exhibit a new kind of phase transition as the coupling strength of the higher order terms varies.This essentially happens because a fundamental change in the ratio of ground state energy eigenvalues, between linear and non-linear regime, takes place which leads to excitation of higher modes. The new phase has inverse relation with area.

Modified semi-classical methods for nonlinear quantum oscillations problems

We develop a modified semi-classical approach to the approximate solution of Schrödinger's equation for certain nonlinear quantum oscillations problems. In our approach, at lowest order, the Hamilton-Jacobi equation of the conventional semi-classical formalism is replaced by an inverted-potential-vanishing-energy variant thereof. With suitable smoothness, convexity and coercivity properties imposed on its potential energy function, we prove, using methods drawn from the calculus of variations together with the (Banach space) implicit function theorem, the existence of a global, smooth “fundamental solution” to this equation. Higher order quantum corrections thereto, for both ground and excited states, can then be computed through the integration of associated systems of linear transport equations, derived from Schrödinger's equation, and formal expansions for the corresponding energy eigenvalues obtained therefrom by imposing the natural demand for smoothness on the (successively computed) quantum corrections to the eigenfunctions. For the special case of linear oscillators our expansions naturally truncate, reproducing the well-known exact solutions for the energy eigenfunctions and eigenvalues. As an explicit application of our methods to computable nonlinear problems, we calculate a number of terms in the corresponding expansions for the one-dimensional anharmonic oscillators of quartic, sectic, octic, and dectic types and compare the results obtained with those of conventional Rayleigh/Schrödinger perturbation theory. To the orders considered (and, conjecturally, to all orders) our eigenvalue expansions agree with those of Rayleigh/Schrödinger theory whereas our wave functions more accurately capture the more-rapid-than-gaussian decay known to hold for the exact solutions to these problems. For the quartic oscillator in particular our results strongly suggest that both the ground state energy eigenvalue expansion and its associated wave function expansion are Borel summable to yield natural candidates for the actual exact ground state solution and its energy. Our techniques for proving the existence of the crucial “fundamental solution” to the relevant (inverted-potential-vanishing-energy) Hamilton-Jacobi equation have the important property of admitting interesting infinite dimensional generalizations. In a project paralleling the present one we shall show how this basic construction can be carried out for the Yang-Mills equations in Minkowski spacetime.

Three-body negative ions under Coulomb interaction

The ground state energy eigenvalues of the symmetric three-body exotic negative ions p+π−π− and p+K−K− have been determined variationally for the first time using an explicitly correlated Hylleraas basis set. Ground state energies of Ps− and p+μ−μ− were also determined to check the accuracy of the present methodology by comparing those results to a few accurate earlier results for these systems.

The solution of the Schrödinger equation for coupled quadratic and quartic potentials in two dimensions

We deal with the difficulties claimed by the author of [Ann. Phys.206, 90 (1991)] while solving the Schrodinger equation for the ground states of two-dimensional anharmonic potentials. It is shown that the ground state energy eigenvalues and eigenfunctions for the coupled quadratic and quartic potentials can be obtained by making some simple assumptions. Expressions for the energy eigenvalues and the eigenfunctions for the first and second excited states of these systems are also obtained.

Ground-state energy eigenvalue calculation of the quantum mechanical well via analytical transfer matrix method

We consider a fundamental quantum mechanical bound-state problem in the form of the quartic-well potential . The analytical transfer matrix method is applied. This yields a quantization condition from which we can calculate the phase contributions and ground-state energy eigenvalues numerically. We also compare the results with those obtained from other typical means popular among physics students, namely the numerical shooting method, perturbation theory and the standard WKB method.

Lie symmetries, quantisation and c-isochronous nonlinear oscillators

In a series of papers Calogero and Graffi [F. Calogero, S. Graffi, On the quantisation of a nonlinear Hamiltonian oscillator, Phys. Lett. A 313 (2003) 356–362] and Calogero [F. Calogero, On the quantisation of two other nonlinear harmonic oscillators, Phys. Lett. A 319 (2003) 240–245; F. Calogero, On the quantisation of yet another two nonlinear harmonic oscillators, J. Nonlinear Math. Phys. 11 (2004) 1–6] treated the quantisation of several one-degree-of-freedom Hamiltonians containing a parameter, c. Two of these systems possess the Lie algebra sl ( 2 , R ) characteristic of the Ermakov–Pinney problem and are related to the Hamiltonian of that problem by an autonomous canonical transformation. Calogero found that the ground-state energy eigenvalues of the corresponding three Schrödinger equations differed when the standard quantisation procedures were used. We examine three simpler c-isochronous oscillators to determine if the method of quantisation is responsible for this unexpected result. We propose a quantisation scheme based on the preservation of the algebraic properties of the Lie point symmetries of the kinetic energy. We find that this criterion removes the dependence of the ground-state eigenvalue on the parameter c and that in fact the eigenvalues are the same for the three systems. Similarly for the Ermakov–Pinney problem and the two derivate models of Calogero we find consistency of ground-state eigenvalues.

Ground state and resonance state ofPs−in plasmas with various Debye lengths

We have made an investigation on the resonances for positronium negative ion ${\mathrm{Ps}}^{\ensuremath{-}}$ in various model plasma environments. The $2{s}^{2}\phantom{\rule{0.2em}{0ex}}^{1}S^{e}$ autoionization resonance state in ${\mathrm{Ps}}^{\ensuremath{-}}$ ion is determined by calculating the density of resonance states using the stabilization method. We have also performed accurate variational calculations to obtain ground-state energy eigenvalues of ${\mathrm{Ps}}^{\ensuremath{-}}$ for various Debye lengths. A screened Coulomb potential obtained from the Debye model is used to represent the interaction between the charged particles. A correlated wave function has been used to represent the correlation effect between the three charge particles. The calculated resonance energies and widths for various Debye parameters ranging from infinity to a small value along with the ground-state energies are reported.

Quantum Phase Transitions and Bipartite Entanglement

We develop a general theory of the relation between quantum phase transitions (QPTs) characterized by nonanalyticities in the energy and bipartite entanglement. We derive a functional relation between the matrix elements of two-particle reduced density matrices and the eigenvalues of general two-body Hamiltonians of d-level systems. The ground state energy eigenvalue and its derivatives, whose nonanalyticity characterizes a QPT, are directly tied to bipartite entanglement measures. We show that first-order QPTs are signaled by density matrix elements themselves and second-order QPTs by the first derivative of density matrix elements. Our general conclusions are illustrated via several quantum spin models.

A Single-Band Constant-Confining-Potential Model for Self-Assembled InAs/GaAs Quantum Dots

ABSTRACTA simple and versatile numerical method for electronic structure calculations in InAs pyramidal dots is presented, and its predictions compared with both theoretical and experimental data. The calculated ground state energy eigenvalues agree well with those of more sophisticated treatments which take into account band mixing and the microscopic effects of the strain distribution. The number of electron bound states predicted is in excellent agreement with very recent calculations for strained quantum dots performed in the framework of the 8-band k · p theory. Very good agreement is obtained with both the number and the energy of the peaks in several experimental photoluminescence spectra. Furthermore our ca agreement with that deduced from capacitance and photoluminescence measurements.

An underrated entanglement: Riccati and Schrödinger equations

The solution of the time-independent Schrödinger equation is fractionated by solving a corresponding Riccati equation. Simple particular solutions of the Riccati equation are used to construct exact ground state energy eigenvalues and eigenfunctions of otherwise complex problems without solving for the complete energy spectrum. In some instances, a partial series of the complete spectrum is also obtained with little extra effort. A procedure for obtaining the complete energy spectrum from the Riccati equation is also presented, and the entire method is illustrated by detailed analysis of several examples.

Variational approach to the spinless relativistic Coulomb problem.

By application of a straightforward variational procedure we derive a simple, analytic upper bound on the ground-state energy eigenvalue of a semirelativistic Hamiltonian for (one or two) spinless particles which experience some Coulomb-type interaction.

Quasiexactly solvable problems in the path-integral formalism.

Quasiexactly solvable problems are treated via the path-integral approach. The generalized Duru-Kleinert formalism involving a combination of a point-canonical transformation and a new-time transformation is applied to power-law and Morse-like quasiexactly solvable problems, and it is shown that these can be grouped into two families. A truncated kernel hypothesis is advanced, whereby the exactly solvable parts of the kernels of each problem within a given family can be transformed into one another. As an application, the ground-state energy eigenvalue and wave function of the N=0 generalized harmonic oscillator are derived from those of the N=0 generalized Coulomb problem.