Ring-shaped Oscillator Potential Research Articles

Analytical eigenstate solutions of Schrödinger equation with noncentral generalized oscillator potential by extended Nikiforov-Uvarov method

Exact eigenstate solutions of Schrödinger equation for a generalized oscillator system that includes, as special cases; ring shaped oscillator potential and isotropic harmonic oscillator potential are examined in an analytical treatment by using extended Nikiforov Uvarov method. Radial and angular parts of the Schrödinger equation for the relevant potential can be systematically solved in the same manner. Moreover the relation between the eigenfunction solutions for this potential and Heun polynomials are presented.

Dirac Equation in the Presence of Hartmann and Ring-Shaped Oscillator Potentials

The importance of the energy spectrum of bound states and their restrictions in quantum mechanics due to the different methods have been used for calculating and determining the limit of them. Comparison of Schrödinger-like equation obtained by Dirac equation with the nonrelativistic solvable models is the most efficient method. By this technique, the exact relativistic solutions of Dirac equation for Hartmann and Ring-Shaped Oscillator Potentials are accessible, when the scalar potential is equal to the vector potential. Using solvable nonrelativistic quantum mechanics systems as a basic model and considering the physical conditions provide the changes in the restrictions of relativistic parameters based on the nonrelativistic definitions of parameters.

Spin–orbit interaction for the double ring-shaped oscillator

The spin–orbit interactions (SOI) for the single and double ring-shaped oscillator potentials are studied as an energy correction to the Schrödinger equation. We find that the degeneracy for the energy levels with angular quantum number m=0 keeps invariant in the case of the SOI. The degeneracy is still 2 for single ring-shaped potential and 4 for double ring-shaped potential. However, for the energy levels with angular quantum number m≠0 the degeneracy is reduced from original 4 for the single ring-shaped potential and 8 for the double ring-shaped potential to 2. That is, their energy levels in the case of the SOI are split to 2 (single) and 4 (double) sublevels. There exists an accidental degeneracy for the cases |m|=2,3,4,…. We note that around the critical value b0, the energy levels are reversed. We also discuss some special cases for η=2,3,4,5,6,…, and the b=0,c>0. It should be pointed out that the parameter b0 is relevant for the angular part parameter b in the single and double ring-shaped potentials and it makes the energy levels changed from positive to negative, but the parameter c corresponds to the angular part parameter in double ring-shaped potential and the η is related to it. This model can be useful for investigations of axial symmetric subjects like the ring-shaped molecules or related problems and may also be easily extended to a many-electron theory.

Energy spectra of Hartmann and ring-shaped oscillator potentials using the quantum Hamilton–Jacobi formalism

In the present work, we apply the exact quantization condition, introduced within the framework of Padgett and Leacock's quantum Hamilton–Jacobi formalism, to angular and radial quantum action variables in the context of the Hartmann and the ring-shaped oscillator potentials which are separable and non-central. The energy spectra of the two systems are exactly obtained.

Exact solutions of the Klein—Gordon equation with ring-shaped oscillator potential by using the Laplace integral transform

We present exact solutions for the Klein—Gordon equation with a ring-shaped oscillator potential. The energy eigenvalues and the normalized wave functions are obtained for a particle in the presence of non-central oscillator potential. The angular functions are expressed in terms of the hypergeometric functions. The radial eigenfunctions have been obtained by using the Laplace integral transform. By means of the Laplace transform method, which is efficient and simple, the radial Klein—Gordon equation is reduced to a first-order differential equation.

Non-central potentials, exact solutions and Laplace transform approach

Exact bound state solutions and the corresponding wave functions of the Schrödinger equation for some non-central potentials including Makarov potential, modified-Kratzer plus a ring-shaped potential, double ring-shaped Kratzer potential, modified non-central potential and ring-shaped non-spherical oscillator potential are obtained by using the Laplace transform approach. The energy spectrums of the Hartmann potential, modified-Kratzer potential and ring-shaped oscillator potential are also briefly studied as special cases. It is seen that our analytical results for all these potentials are consistent with those obtained by other works. We also give some numerical results obtained for the modified non-central potential for different values of the related quantum numbers.

Relativistic solutions for double ring-shaped oscillator potential via asymptotic iteration method

The three dimensional Klein–Gordon equation is solved with a double ring-shaped oscillator for the case of the equal vector and scalar potential, , by using the asymptotic iteration method which is very efficient, systematic and practical. The bound states energy eigenvalues and corresponding eigenfunctions are presented for a particle bound in a potential of a double ring-shaped oscillator.

Bound states of Klein–Gordon equation for double ring-shaped oscillator scalar and vector potentials

In spherical polar coordinates, double ring-shaped oscillator potentials have supersymmetry and shape invariance for θ and r coordinates. Exact bound state solutions of Klein–Gordon equation with equal double ring-shaped oscillator scalar and vector potentials are obtained. The normalized angular wavefunction expressed in terms of Jacobi polynomials and the normalized radial wavefunction expressed in terms of the Laguerre polynomials are presented. Energy spectrum equations are obtained.

The supersymmetric WKB approximation of ring-shaped oscillator potential in r and dimensions with spherical polar coordinates

According to calculation of the energy spectrum of ring-shape oscillator potential by using the supersymmetric WKB approximation, it is shown that the energy spectrum of some noncentral separable potentials can be exactly obtained in r and θ dimensions by above method.

Supersymmetry and Shape Invariance of Hartmann Potential and Ring-Shaped Oscillator Potential in the r and θ Dimensions of Spherical Polar Coordinates

This article shows that in spherical polar coordinates, some noncentral separable potentials have supersymmetry and shape invariance in the r and θ dimensions, we choose Hartmann potential and ring-shaped oscillator as two important examples, thus in principle the energy eigenvalues and energy eigenfunctions of such the potentials in the r and θ dimensions can be obtained by the method of supersymmetric quantum mechanics. Here we use an alternative method to get the required results.