Arbitrary <Emphasis Type="Italic">l</Emphasis>-state solutions of the Schrödinger equation for the Eckart potential with an improved approximation of the centrifugal term

Applying an improved approximation scheme to the centrifugal term, the approximate analytical solutions of the Schrödinger equation for the Eckart potential are presented. Bound state energy eigenvalues and the corresponding eigenfunctions are obtained in closed forms for the arbitrary radial and angular momentum quantum numbers, and different values of the screening parameter. The results are compared with those obtained by the other approximate and numerical methods. It is shown that the present method is systematic, more efficient and accurate.     

Oscillator Representation for Pseudoharmonic Potential

Oscillator representation method is applied to obtain the bound state energy eigenvalues and the corresponding eigenfunctions of the n-dimensional Schrödinger equation for the pseudoharmonic potential with arbitrary angular momentum.     

Spectral singularities and zero energy bound states

Single particle scattering around zero energy is re-analysed in view of recent experiments with ultra-cold atoms, nano-structures and nuclei far from the stability valley. For non-zero orbital angular momentum the low energy scattering cross section exhibits dramatic changes depending on the occurrence of either a near resonance or a bound state or the situation in between, that is a bound state at zero energy. Such state is singular in that it has an infinite scattering length, behaves for the eigenvalues but not for the eigenfunctions as an exceptional point and has no pole in the scattering function. These results should be observable whenever the interaction or scattering length can be controlled.     

Specifying Angular Momentum and Center of Mass for Vacuum Initial Data Sets

We show that it is possible to perturb arbitrary vacuum asymptotically flat spacetimes to new ones having exactly the same energy and linear momentum, but with center of mass and angular momentum equal to any preassigned values measured with respect to a fixed affine frame at infinity. This is in contrast to the axisymmetric situation where a bound on the angular momentum by the mass has been shown to hold for black hole solutions. Our construction involves changing the solution at the linear level in a shell near infinity, and perturbing to impose the vacuum constraint equations. The procedure involves the perturbation correction of an approximate solution which is given explicitly.     

Bound states and band structure—A unified treatment through the quantum Hamilton-Jacobi approach

We analyze the Scarf potential, which exhibits both discrete energy bound states and energy bands, through the quantum Hamilton-Jacobi approach. The singularity structure and the boundary conditions in the above approach, naturally isolate the bound and periodic states, once the problem is mapped to the zero energy sector of another quasi-exactly solvable quantum problem. The energy eigenvalues are obtained without having to solve for the corresponding eigenfunctions explicitly. We also demonstrate how to find the eigenfunctions through this method.     

On the effects on a Landau-type system for an atom with no permanent electric dipole moment due to a Coulomb-type potential

We analyse the bound states for a Landau-type system for an atom with no permanent electric dipole moment subject to a Coulomb-type potential. By comparing the energy levels for bound states of the system with the Landau quantization for an atom with no permanent electric dipole moment (Furtado et al., 2006), we show that the energy levels of the Landau-type system are modified, where the degeneracy of the energy levels is broken. Another quantum effect investigated is a dependence of the angular frequency of the system on the quantum numbers associated with the radial modes and the angular momentum. As examples, we obtain the angular frequency and the energy levels associated with the ground state and the first excited state of the system.     

On a relativistic particle and a relativistic position-dependent mass particle subject to the Klein–Gordon oscillator and the Coulomb potential

The relativistic quantum dynamics of an electrically charged particle subject to the Klein–Gordon oscillator and the Coulomb potential is investigated. By searching for relativistic bound states, a particular quantum effect can be observed: a dependence of the angular frequency of the Klein–Gordon oscillator on the quantum numbers of the system. The meaning of this behaviour of the angular frequency is that only some specific values of the angular frequency of the Klein–Gordon oscillator are permitted in order to obtain bound state solutions. As an example, we obtain both the angular frequency and the energy level associated with the ground state of the relativistic system. Further, we analyse the behaviour of a relativistic position-dependent mass particle subject to the Klein–Gordon oscillator and the Coulomb potential.     

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

We present a non-variational approach to the solution of the quantum three-body problem, based on the decomposition of the three-body Laplacian operator through the use of its intrinsic symmetries. With the judicious choice of angular momentum eigenfunctions, a clean separation of spatial rotation from kinematic rotation is achieved, leading to a finite set of coupled PDEs in terms of the canonical variables. Numerical implementation of this approach to the three-body Coulomb problem is shown to yield accurate ground state eigenvalues and wavefunctions, together with those of low-lying excited states. We present results on some typical three-body systems. In particular, the eigenvalues and wavefunctions of the even-parity state of the negative hydrogen ion are detailed for the first time. The issue of computational efficiency is also briefly discussed.     

Exact Solutions of Klein-Gordon Equation with Exponential Scalar and Vector Potentials

We obtain the exact analytical solution of the Klein-Gordon equation for the exponential vector and scalar potentials by using the asymptotic iteration method. For the scalar potential greater than the vector potential case, the exact bound state energy eigenvalues and corresponding eigenfunctions are presented. The bound state eigenfunction solutions are obtained in terms of the confluent hypergeometric functions.     

Semi-Relativistic Two-Body States of Spinless Particles with a Scalar-Type Interaction Potential

A semi-relativistic quantum approximation for mutual scalar interaction potentials is outlined and discussed.Equations are consistent with two-body Dirac equations for bound states of zero total angular momentum. Two-body effects near the non-relativistic limit for a linear scalar potential is studied in some detail.     

Bound states resulting from interaction of the non-relativistic particles with the multiparameter potential

In this study, we present the analytical solutions of bound states for the Schrodinger equation with the multiparameter potential containing the different types of physical potentials via the asymptotic iteration method by applying the Pekeristype approximation to the centrifugal potential. For any n and l(states) quantum numbers, we derive the relation that gives the energy eigenvalues for the bound states numerically and the corresponding normalized eigenfunctions. We also plot some graphics in order to investigate effects of the multiparameter potential parameters on the energy eigenvalues.Furthermore, we compare our results with the ones obtained in previous works and it is seen that our numerical results are in good agreement with the literature.     

Approximate analytical solutions and mean energies of stationary Schrödinger equation for general molecular potential

The Schrödinger equation is solved with general molecular potential via the improved quantization rule. Expression for bound state energy eigenvalues, radial eigenfunctions, mean kinetic energy, and potential energy are obtained in compact form. In modeling the centrifugal term of the effective potential, a Pekeris-like approximation scheme is applied. Also, we use the Hellmann-Feynman theorem to derive the relation for expectation values. Bound state energy eigenvalues, wave functions and meanenergies of Woods-Saxon potential, Morse potential, Möbius squared and Tietz-Hua oscillators are deduced from the general molecular potential. In addition, we use our equations to compute the bound state energy eigenvalues and expectation values for four diatomic molecules viz. H₂, CO, HF, and O₂. Results obtained are in perfect agreement with the data available from the literature for the potentials and molecules. Studies also show that as the vibrational quantum number increases, the mean kinetic energy for the system in a Tietz-Hua potential increases slowly to a threshold value and then decreases. But in a Morse potential, the mean kinetic energy increases linearly with vibrational quantum number increasing.     

Nyström Method for the Coulomb and Screened Coulomb Potentials

In this paper, we show that the simple Nyström method can yield very accurate eigenvalues and eigenfunctions not only for large principal quantum number but also for large angular momentum quantum number. We demonstrate that the furcation phenomenon emerging in the calculated eigenfunctions can be regarded as an indicator for the bad behavior of the integral equation and the unreliability of the obtained results.     

Hyper-radial adiabatic expansion for a muonic molecule dtμ

By using the hyper-radius, adiabatic potential energy curves with correct asymptotic energies are obtained for the Coulomb three-body problem. The bound state energies of the muonic molecules dtμ with total angular momentum J=0 calculated adopting the three lowest adiabatic potential energy curves are −318.72 and −34.36 eV for vibrational quantum numbers ν=0 and 1, respectively.     

Exact Solution of the Pseudoharmonic Oscillator in the Space of Constant Positive Curvature

We present analytically the exact solution of the radial Schrödinger equation with the pseudoharmonic oscillator potential in constant positive curvature representation. Exact bound state eigenfunctions and eigenvalues obtained using factorization method. Finally, energy eigenvalues obtained here compared with the results of the theoretical methods in the limit of flat space.     

Bound state solution of the Klein-Gordon equation for vector and scalar Hellmann plus modified Kratzer potentials

In this study, the analytical solutions of the Klein–Gordon equation for any l states of the scalar and vector Hellmann plus modified Kratzer potential are derived by using an approximation method to the centrifugal potential term. The analytical expressions for eigenvalues and corresponding normalized eigenfunctions of the spin-zero particle have been estimated by using the parametric Nikiforov-Uvarov method. The solution for the radial part of the Klein-Gordon equation is formulated in terms of the generalized Jacobi polynomials. The energy state equation and the wave function for special cases are in good agreement with the previous literature. In addition, we have measured the numerical results of the energy eigenvalues and also the trend of the eigenvalues concerning of different potential parameters have been plotted. Furthermore, it was shown that the energy levels E and quantum numbers n and l are inversely proportional to each other.     

Fermionic particles with position-dependent mass in the presence of inversely quadratic Yukawa potential and tensor interaction

Approximate solutions of the Dirac equation with position-dependent mass are presented for the inversely quadratic Yukawa potential and Coulomb-like tensor interaction by using the asymptotic iteration method. The energy eigenvalues and the corresponding normalized eigenfunctions are obtained in the case of position-dependent mass and arbitrary spin-orbit quantum number k state and approximation on the spin-orbit coupling term.     

Unified treatment of the bound states of the Schioberg and the Eckart potentials using Feynman path integral approach

We obtain analytical expressions for the energy eigenvalues of both the Schioberg and Eckart potentials using an approximation of the centrifugal term.In order to determine the e-states solutions,we use the Feynman path integral approach to quantum mechanics.We show that by performing nonlinear space-time transformations in the radial path integral,we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system.The explicit expression of bound state energy is obtained and the associated eigenfunctions are given in terms of hypergeometric functions.We show that the Eckart potential can be derived from the Schioberg potential.The obtained results are compared to those produced by other methods and are found to be consistent.     

Exact Solutions to Three-Dimensional Schrödinger Equation with an Exponentially Position-Dependent Mass

For an exponentially position-dependent mass, we obtain the exact solutions of the three-dimensional Schrödinger equation by using coordinate transformation method for the reference problems with Coulomb potential, Kratzer potential, and spherically square potential well of infinite depth, respectively. The explicit expressions for the energy eigenvalues and the corresponding eigenfunctions of the three systems are presented.