Non-spherical Oscillator Research Articles

Bound states of Dirac equation using the proper quantization rule

Using the proper quantization rule, we investigate the exact solution of Dirac equation for Hartmann and the ring-shaped non-spherical harmonic oscillator potentials under the condition of equal scalar and vector potentials. By considering the proper quantization condition within angular and radial variables, the exact relativistic energy spectra are obtained for each system. Then by the mean of suitable changes of variables, the corresponding spinor wave-functions are constructed where the normalization constants are exactly calculated. We also derived the non-relativistic limit of energy spectra.

Effect of confining potential on information entropy measures in hydrogen atom: extensive and non-extensive entropy

A spherically confined hydrogen atom has been considered using two different confined potentials such as modified Kratzer and non-spherical oscillator potentials. First, the Schrodinger equation in $$ r $$ -space has been solved and the exact analytical wave functions and energy levels have been obtained. Then, the Renyi entropy and Shannon entropy in $$ r $$ -space have been calculated. In addition, we have numerically calculated the wave functions in $$ p $$ -space by performing Fourier transform. The calculations have been done for different quantum numbers $$ n $$ and $$ l $$ and also quantum size $$ r_{0} $$ . The Bialynicki–Birula–Mycielski inequality is also tested. It is found that this inequality depends on non-extensive parameter $$ q $$ .

Exact Solutions of Schrödinger Equation with Improved Ring-Shaped Non-Spherical Harmonic Oscillator and Coulomb Potential

We propose improved ring shaped like potential of the form, V(r, θ) = V(r) + (ħ2/2Mr2)[(β sin2 θ + γ cos2 θ + λ) / sin θ cos θ]2 and its exact solutions are presented via the Nikiforov–Uvarov method. The angle dependent part V(θ) = (ħ2 / 2 Mr2)[(β sin2 θ + γ cos2 θ + λ) / sin θ cos θ]2, which is reported for the first time embodied the novel angle dependent (NAD) potential and harmonic novel angle dependent potential (HNAD) as special cases. We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.

Tridiagonal representation with pseudospin symmetry for a noncentral electric dipole and a ring-shaped anharmonic oscillator potential

The concepts of pseudospin symmetry in atomic nuclei and spin symmetry in anti-nucleon are reviewed. The exploration for understanding the origin of pseudospin symmetry and its breaking mechanism, and the empirical data supporting the pseudospin symmetry are introduced. A noncentral anharmonic oscillatory potential model is proposed, in which a noncentral electric dipole and a double ring-shaped component are included. The pseudospin symmetry for this potential model is investigated by working on a complete square integrable basis that supports a tridiagonal matrix representation of the Dirac wave operator. Then, solving the Dirac equation is translated into finding solutions of the recursion relation for the expansion coefficients of the wavefunction. The angular/radial wavefunction is written in terms of the Jacobi/Laguerre polynomials. The discrete spectrum of the bound states is obtained by diagonalization of the radial recursion relation, and the property of energy equation is discussed for showing the exact pseudospin symmetry. Several particular cases obtained by setting the parameters of the potential model to appropriate values are analyzed, and the energy equations are reduced to that of the anharmonic oscillator and that of the ring-shaped non-spherical harmonic oscillator, respectively. Finally, it is pointed out that the exact spin symmetry exists also in this potential model.

Linear and nonlinear intersubband optical absorption coefficients and refractive index changes with the ring-shaped non-spherical oscillator potential

Linear and nonlinear intersubband optical absorption coefficients and refractive index changes with the ring-shaped non-spherical oscillator potential (RNOP) are theoretically investigated within the framework of the compact-density-matrix approach and iterative method. The energy levels and the wave functions of an electron in the RNOP are obtained by using the effective mass approximation. It is found that the optical absorption coefficients and refractive index changes are strongly affected not only by the dimensionless parameters γ and β, but also by the confinement frequency ω0.

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.

Solving Dirac Equation with New Ring-Shaped Non-Spherical Harmonic Oscillator Potential

A new ring-shaped non-harmonic oscillator potential is proposed. The precise hound solution of Dirac equation with the potential is gained when the scalar potential is equal to the vector potential. The angular equation and radial equation are obtained through the variable separation method. The results indicate that the normalized angle wave function can be expressed with the generalized associated-Legendre polynomial, and the normalized radial wave function can be expressed with confluent hypergeometric function. And then the precise energy spectrum equations are obtained. The ground state and several low excited states of the system are solved. And those results are compared with the non-relativistic effect energy level in Phys. Lett. A 340 (2005) 94. The positive energy states of system are discussed and the conclusions are made properly.

Pseudospin symmetry for a new ring-shaped non-spherical harmonic oscillator potential

A new ring-shaped non-spherical harmonic oscillator potential is proposed, which consists of a generalized non-harmonic oscillator potential plus an angle-dependent potential, The pseudospin symmetry for a spin- particle moving in this potential is investigated by solving the Dirac equation with an equal mixture of scalar and vector potentials with opposite signs. The normalized spinor wave function and energy equation are obtained, and the algebraic property of the energy equation and some particular cases are also discussed.

Pseudospin Symmetry for a Ring-Shaped Non-spherical Harmonic Oscillator Potential

The pseudospin symmetry for a ring-shaped non-spherical harmonic oscillator potential is investigated by solving the Dirac equation with equal mixture of scalar and vector potentials with opposite signs. The normalized spinor wave function and energy equation are obtained, the algebraic property of the energy equation and some particular cases are also discussed.

Quesne-like ring-shaped spherical harmonic oscillator potential and pseudospin symmetry

A Quesne-like ring-shaped spherical harmonic oscillator potential is put foword and studied for spin 1/2 particles based on the Dirac equation， the Dirac Hamiltonian contains a scalar and a vector Quesne-like ring-shaped harmonic oscillator potentials. Setting Σ＝S(r)+V(r)=0,we obtain the bound state solutions and eigenenergies with the two-component approach. The result shows the pseudospin symmetry exists in the Quesne-like ring-shaped harmonic oscillator potential. The general properties of both the ring-shaped spherical harmonic oscillator potential and the ring-shaped non-spherical harmonic oscillator potential are discussed.

The relativistic bound states for a new ring-shaped harmonic oscillator

In this paper a new ring-shaped harmonic oscillator for spin 1/2 particles is studied, and the corresponding eigenfunctions and eigenenergies are obtained by solving the Dirac equation with equal mixture of vector and scalar potentials. Several particular cases such as the ring-shaped non-spherical harmonic oscillator, the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.

The position dependent band gap in Kane-type quantum dots

The energy spectrum of electrons in narrow band gap semiconductor nanocrystals which have position dependent band gap in an external non-uniform electric field which compensate the position dependence of the band edge of the valence band potential is studied theoretically taking into account the non-parabolicity of electrons in dispersion laws. The exact solutions of the Kane equations with strong spin–orbital interaction are determined via the band-gap changes as a function of position. The band edge gap potential is taken as the ring-shaped non-spherical oscillator potential

Pseudospin symmetry and the relativistic ring-shaped non-spherical harmonic oscillator

A generalized relativistic harmonic oscillator for spin 1/2 particles is studied. The eigenfunctions and eigenenergies are obtained for the ring-shaped non-spherical harmonic oscillator by solving Dirac equation with equal mixture of vector and scalar potentials in opposite signs, for which pseudospin symmetry is exact. Several particular cases such as the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.

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

Solving the Klein–Gordon equation and Dirac equation with ring-shaped non-spherical oscillator gives the exact bound state wavefunction and energy equation and the relations between non-relativistic Schrodinger equation, Klein–Gordon equation and Dirac equation with equal scalar and vector potentials.

An algebraic approach to the ring-shaped non-spherical oscillator

The eigenfunctions and eigenvalues of the Schrödinger equation with a ring-shaped non-spherical oscillator are obtained. A realization of the ladder operators for the radial wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1,1) group. The closed analytical expressions for the matrix elements of different functions ρ and ρ d dρ with ρ= r 2 are evaluated.

Bound states of Klein-Gordon equation for ring-shaped non-spherical harmonic oscillator potentials

By using the ordinary method of variable separation,the bound states of Klein-Gordon equation of the ring-shaped non-spherical harmonic oscillator with equal scalar and vector potentials are solved. The normalized angular wave function expressed in terms of the universal associatedLegendre polynomial and the normalized radial wave function expressed in terms of the confluent hyper-geometric function are presented. The exact energy spectrum equations are obtained.

Explicit expressions and recurrence formulas of the radial average values for a kind of nonharmonic oscillator model potentials

In this paper, two recurrence formulas are derived for the radial average values of a kind of non-harmonic oscillator model potentials; these non-harmonic oscillator model potentials are ring-shaped non-spherical oscillator, non-spherical oscillator, and ring-shaped oscillator. Explicit expressions for their radial average values are given.

Exact solution of a kind of n-dimensions non-spherical harmonic oscillator p otential

In this paper,the Schrdinger equation with a non-spherical harmonic oscillator potential in n-dimensions is solved.The exact energy equation and the normalized wave function are obtained.The general formulas of matrix elements of op erator rs are presented.