Anharmonic Oscillator Potential Research Articles

Supersymmetric Expansion Algorithm and Complete Analytical Solution for the Hulthén and Anharmonic Potentials

Abstract An algorithm for providing analytical solutions to Schrödinger’s equation with non-exactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of the supersymmetric quantum mechanics as extended toward non shape invariant potentials. The complete solution to a given Hamiltonian H0 is obtained from the nodeless states of the Hamiltonian H0 and of a set of supersymmetric partners H1, H2, …, Hr. The nodeless states (dubbed “edge” states) are unique and in general can be ground or excited states. They are solved using the logarithmic expansion which yields an infinite system of coupled first order hierarchical differential equations, converted later into algebraic equations with recurrence relations which can be solved order by order. We formulate the aforementioned scheme, termed to as “Supersymmetric Expansion Algorithm” step by step and apply it to obtain for the first time the complete analytical solutions of the three dimensional Hulthén–, and the one-dimensional anharmonic oscillator potentials.

Rationally-extended Dunkl oscillator on the line

It is shown that the extensions of exactly-solvable quantum mechanical problems connected with the replacement of ordinary derivatives by Dunkl ones and with that of classical orthogonal polynomials by exceptional orthogonal ones can be easily combined. For such a purpose, the example of the Dunkl oscillator on the line is considered and three different types of rationally-extended Dunkl oscillators are constructed. The corresponding wavefunctions are expressed in terms of exceptional orthogonal generalized Hermite polynomials, defined in terms of the three different types of X m -Laguerre exceptional orthogonal polynomials. Furthermore, the extended Dunkl oscillator Hamiltonians are shown to be expressible in terms of some extended Dunkl derivatives and some anharmonic oscillator potentials.

Quantum corrections to the Weyl quantization of the classical time of arrival

A time of arrival (TOA) operator that is conjugate with the system Hamiltonian was constructed by Galapon without canonical quantization in Galapon (J. Math. Phys. 45:3180–3215, 2004). The constructed operator was expressed as an infinite series but only the leading term was investigated which was shown to be equal to the Weyl-quantized TOA-operator for entire analytic potentials. In this paper, we give a full account of the said operator by explicitly solving all the terms in the expansion. We interpret the terms beyond the leading term as the quantum corrections to the Weyl quantization of the classical arrival time. These quantum corrections are expressed as some integrals of the interaction potential and their properties are investigated in detail. In particular, the quantum corrections always vanish for linear systems but nonvanishing for nonlinear systems. Finally, we consider the case of an anharmonic oscillator potential as an example.

Bohr Hamiltonian with sextic potential for γ -rigid prolate nuclei with deformation-dependent mass term

The main aim of the present paper is to solve the eigenvalues problem with the Bohr collective Hamiltonian for $\ensuremath{\gamma}$-rigid nuclei within a model we have elaborated by combining two model approaches: the quantum mechanical formalism, namely, deformation-dependent mass formalism (DDM), and the anharmonic sextic oscillator potential for the variable $\ensuremath{\beta}$ and $\ensuremath{\gamma}=0$. The model developed in this way is conventionally called the sextic and DDM approach. Analytical expressions for energy spectra are conjointly derived by means of quasiexact solvability and a quantum perturbation method. Due to the scaling property of the problem, the energy and $B(E2)$ transition ratios depend on two free parameters apart from an integer number which limits the number of allowed states. Numerical results are given for 35 nuclei---$^{98--108}\mathrm{Ru}, ^{100--102}\mathrm{Mo}, ^{116--130}\mathrm{Xe}, ^{180--196}\mathrm{Pt}, ^{172}\mathrm{Os}, ^{146--150}\mathrm{Nd}, ^{132--134}\mathrm{Ce}, ^{154}\mathrm{Gd}, ^{156}\mathrm{Dy}$, and $^{150--152}\mathrm{Sm}$---revealing a good agreement with experiment. Moreover, as proved for the first time by Bonatsos et al. [D. Bonatsos, P. Georgoudis, D. Lenis, N. Minkov, and C. Quesne, Phys. Lett. B 683, 264 (2010)], the dependence of the mass on the deformation with the sextic potential moderates the increase of the moment of inertia with the deformation, removing an important drawback that has been revealed in the constant mass case [Buganu and Budaca, J. Phys. G: Nucl. Part. Phys. 42, 105106 (2015)]. Additionally, the correlation between the DDM and the minimal length formalism persists for the sextic potential. Finally, the DDM effects on the shape phase transition for the most numerous isotopic chains, namely, Ru, Xe, Nd, and Pt, have been duly investigated.

Effects of Intense Laser Field on Electronic and Optical Properties of Harmonic and Variable Degree Anharmonic Oscillators

In this paper, we calculated the electronic and optical properties of the harmonic oscillator and single and double anharmonic oscillators, including higher-order anharmonic terms such as the quartic and sextic under the non-resonant intense laser field. Calculations are made within the effective mass and parabolic band approximations. We have used the diagonalization method by choosing a wave function based on the trigonometric orthonormal functions to find eigenvalues and eigenfunctions of the electron confined within the harmonic and anharmonic oscillator potentials under the non-resonant intense laser field. A two-level approach in the density matrix expansion is used to calculate the linear and third-order nonlinear optical absorption coefficients. Our results show that the electronic and optical properties of the structures we focus on can be adjusted to obtain a suitable response to specific studies or aims by changing the structural parameters such as width, depth, coupling between the wells, and applied field intensity.

Extended study on the application of the sextic potential in the frame of X(3)-sextic

The main aim of the present paper is to extensively study the γ-rigid Bohr Hamiltonian with anharmonic sextic oscillator potential for the variable β and γ = 0. For the corresponding spectral problem, a finite number of eigenvalues are explicitly found, by algebraic means, the so-called quasi-exact solvability (QES). The evolution of the spectral and electromagnetic properties by considering higher exact solvability orders is investigated, especially the approximate degeneracy of the ground and first two β bands at the critical point of the shape phase transition from a harmonic to an anharmonic prolate β-soft, as well as the shape evolution within an isotopic chain. The numerical results are given for 39 nuclei, namely, 98–108Ru, 100–102Mo, 116–130Xe, 180–196Pt, 172Os, 146–150Nd, 132–134Ce, 152–154Gd, 154–156Dy, 150–152Sm, 190Hg and 222Ra. Across this study, it seems that the higher QES order improves our results by decreasing the root mean square, mostly for deformed nuclei. The nuclei 100,104Ru, 118,120,126,128Xe, 148Nd and 172Os fall exactly at the critical point.

Conformable fractional Bohr Hamiltonian with Bonatsos and double-well sextic potentials

Using the conformable fractional calculus, a new formulation of the Bohr Hamiltonian is introduced. The conformable fractional energy spectra of free- and two- parameters anharmonic oscillator potentials are investigated. The energy eigenvalues and wave functions are calculated utilizing the finite-difference discretization method. It is proved that the conformable fractional spectra of the free-parameter Bonatsos potentials, close completely the gaps between the classical spectra of the vibrational dynamical symmetry, the models, and the critical point symmetry. The ground effective sextic potential, which generates both the ground state and the excited states is considered to have two degenerate minima. In this case, the conformable fractional spectra of sextic potentials show a change, as a function of barrier height, from -unstable energy level sequence to the spectrum of model and simultaneously provide new features. The shape coexistence phenomena in the ground band states are identified. The energy spectrum and shape coexistence with mixing phenomena in 96Mo nucleus are discussed in the framework of the conformable fractional Bohr Hamiltonian.

Perturbative approach for strong and weakly coupled time-dependent for non-Hermitian quantum systems

We propose a perturbative approach to determine the time-dependent Dyson map and the metric operator associated with time-dependent non-Hermitian Hamiltonians. We apply the method to a pair of explicitly time-dependent two dimensional harmonic oscillators that are weakly coupled to each other in a PT-symmetric fashion and to the strongly coupled explicitly time-dependent negative quartic anharmonic oscillator potential. We demonstrate that once the perturbative Ansatz is set up the coupled differential equations resulting order by order may be solved recursively in a constructive manner, thus bypassing the need for making any guess for the Dyson map or the metric operator. Exploring the ambiguities in the solutions of the order by order differential equations naturally leads to a whole set of inequivalent solutions for the Dyson maps and metric operators implying different physical behaviour as demonstrated for the expectation values of the time-dependent energy operator.

Coexistence and fluctuations phenomena with Davidson-like potentials in quadrupole–octupole deformed nuclei

The energy spectra of three types of two-dimensional potentials (we will call them ‘Davidson-like potentials’ (DLPs)), characterized by four minima separated by barriers, are investigated. The predictions for spectra and wave functions are obtained by using the nine-point finite-difference method. For these potentials, with the existence of a single configuration, a transition of spectra, as a function of barrier height, is covered from tunneling splitting modes to fluctuations phenomena, with equal peaks wave functions, crossing to the spectra of purely anharmonic oscillator potentials (AOPs). A different type of phase transition occurs when two (or more) configurations coexist. With the change of the parameters, a transition of spectra is covered from coexistence phenomena of two distinct quantum tunneling modes to fluctuations phenomena, with unequal peaks wave functions, crossing to the spectra of purely AOPs. Using DLPs, a particular application of the coherent quadrupole–octupole model to describe energy bands with alternating parity of the nuclei 100Mo, 146,148Nd, 148,150Sm, 220Ra, 220,222Rn, and 220,222Th is presented. The global parametrization of the model for the selected nuclei is achieved.

Critical potentials and fluctuations phenomena with quartic, sextic, and octic anharmonic oscillator potentials

This is an investigation of the energy levels of three classes of two-dimensional anharmonic oscillator potentials characterized by free-, one-, and two- parameters with quartic, sextic, and octic anharmonicity. The wave functions and energy eigenvalues are found through the discretization using the nine-point finite-difference method. It is proved that the free-parameter potentials provide a “bridge” between the two-dimensional harmonic oscillator potential and two-dimensional infinite square well potential. However, the one- and two- parameter potentials show a transition from tunneling splitting mode to the spectra of free-parameter potentials crossing to the spectrum of infinite square well potential and in the same time present new features. One can distinguish cases of fluctuations phenomena in the ground state and excited states. We obtained, for the first time, a ground state with a four-peak structure. The quantum tunneling between the four minima of one- and two- parameter potentials depends on the characteristics of the separating barrier. These phase transitions are not of the ordinary thermodynamic category, but truly they are phase transitions in the structure of the low-lying energy spectra. The concept of critical potentials is associated with the potentials at the phase-transition points. The coherent quadrupole-octupole motion model, using these potentials, is applied to describe yrast and non-yrast energy sequences with alternating parity in several even-even nuclei from different regions, namely, in 100Mo, 146Ba, 226Ra, and 226Th.

Development and refinement of the Variational Method based on Polynomial Solutions of Schrödinger Equation

Abstract The variational method is known as a powerful and preferred technique to find both analytical and numerical solutions for numerous forms of anharmonic oscillator potentials. In the present study, we considered certain conditions for the choice of the trial wave function. The current form of the trial wave function is based on the possible polynomial solutions of the Schrödinger equation. The advantage of our modified variational method is its ability to reduce the calculation steps and hence computation time. Also, we compared the results provided by our modified method with the results obtained by different methods in general but particularly Numerov method for the same problem.

Application of quantum self-frictional nonperturbative theory for the study of atomic anharmonic oscillator potentials and their arbitrary derivatives

The self-frictional (SF) nonperturbative theory, introduced by one of the authors, is used for the evaluation of the $$V^{{(p_{l}^{ * } )}}$$ and $$V^{(\alpha^{ * })}$$ atomic anharmonic oscillator potentials and their derivatives, where $$p_{l}^{*} = 2l + 2 - \alpha^{ * }$$ and $$\alpha^{ * }$$ represent the integer $$(\alpha^{ * } = \alpha ,\; - \infty < \alpha \le 2$$ ) or non-integer ( $$\alpha^{ * } \ne \alpha ,\,\,\, - \infty < \alpha^{ * } < 3$$ ) SF quantum numbers. This study is based on the use of complete sets of $$\text{L}_{{}}^{{\left( {p_{l}^{*}} \right)}}$$ and $$\text{L}_{{}}^{{\left( {\alpha^{ * }} \right)}}$$ SF polynomials. The dependence of the potentials and their derivatives from the nucleus distances is investigated. All of the obtained results are valid for the arbitrary values of quantum numbers, scaling parameters and SF quantum numbers.

A family of analytically solvable Schrödinger equations related by Levi-Civita transformation

Some two-dimensional problems in non-relativistic quantum mechanics can connect to each other by certain spatial transformations such as Levi-Civita transformation. This property allows forming a series of two-dimensional problems into an interrelated family. Starting from two related problems namely Coulomb plus harmonic oscillator and sextic double-well anharmonic oscillator potentials, such family is constructed via repeatedly applying Levi-Civita transformations. Obviously, this family contains various of exactly analytically solvable problems. The quasi-exact solution for each unknown member of this family is also obtained and systematically investigated.

Anharmonic oscillator potentials in the prolate γ-rigid regime of the collective geometrical model

Based on higher order JWKB approximation, an analytical formula is derived for the energy spectrum of the prolate γ-rigid Bohr-Mottelson Hamiltonian with an oscillator potential and a quartic or sextic anharmonicity in β shape variable. The model is found to provide good description for near vibrational nuclei situated in vicinity of a closed neutron shell.

Bohr Hamiltonian with a potential having spherical and deformed minima at the same depth

A solution for the Bohr-Mottelson Hamiltonian with an anharmonic oscillator potential of sixth order, obtained through a diagonalization in a basis of Bessel functions, is presented. The potential is consid- ered to have simultaneously spherical and deformed minima of the same depth separated by a barrier (a local maximum). This particular choice is appropriate to describe the critical point of the nuclear phase transition from a spherical vibrator to an axial rotor. Up to a scale factor, which can be cancelled by a corresponding normalization, the energy spectra and the electromagnetic E2 transition probabilities depend only on a single free parameter related to the height of the barrier. Investigations of the numerical data revealed that the model represents a good tool to describe this critical point.

Sextic and decatic anharmonic oscillator potentials: Polynomial solutions

We seek the possible polynomial solutions of the Schrödinger equation for the sextic and decatic potentials. Under certain conditions on the parameters of the potentials, we show that these potentials are exactly solvable. We evaluate the first four eigenstates for both potentials. We derive general expressions of the energy levels, for high energy levels, eigenvalues are a function of potentials' parameters and the eigenfunction's zeros.

Optomechanical self-oscillations in an anharmonic potential: engineering a nonclassical steady state

We study self-oscillations of an optomechanical system, where coherent mechanical oscillations are induced by a driven optical or microwave cavity, for the case of an anharmonic mechanical oscillator potential. A semiclassical analytical model is developed to characterize the limit cycle for large mechanical amplitudes corresponding to a weak nonlinearity. As a result, we predict conditions to achieve subpoissonian phonon statistics in the steady state, indicating classically forbidden behavior. We compare with numerical simulations and find very good agreement. Our model is quite general and can be applied to other physical systems such as trapped ions or superconducting circuits.

Airy function approach and Numerov method to study the anharmonic oscillator potentials V(x) = Ax2α + Bx2

The numerical solutions of the time independent Schrödinger equation of different one-dimensional potentials forms are sometime achieved by the asymptotic iteration method. Its importance appears, for example, on its efficiency to describe vibrational system in quantum mechanics. In this paper, the Airy function approach and the Numerov method have been used and presented to study the oscillator anharmonic potential V(x) = Ax2α + Bx2, (A&amp;gt;0, B&amp;lt;0), with (α = 2) for quadratic, (α =3) for sextic and (α =4) for octic anharmonic oscillators. The Airy function approach is based on the replacement of the real potential V(x) by a piecewise-linear potential v(x), while, the Numerov method is based on the discretization of the wave function on the x-axis. The first energies levels have been calculated and the wave functions for the sextic system have been evaluated. These specific values are unlimited by the magnitude of A, B and α. It’s found that the obtained results are in good agreement with the previous results obtained by the asymptotic iteration method for α =3.

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.

Construction of the Barut-Girardello type of coherent states for Pöschl-Teller potential

The Pöschl-Teller (PT) potential occupies a privileged place among the anharmonic oscillator potentials due to its applications from quantum mechanics to diatomic molecules. For this potential, a polynomial su(1, 1) algebra has been constructed previously. So far, the coherent states (CSs) associated with this algebra have never appeared. In this paper, we construct the coherent states of the Barut-Girardello coherent states (BG-CSs) type for the PT potentials, which have received less attention in the scientific literature. We obtain these CSs and demonstrate that they fulfil all conditions required by the coherent state. The Mandel parameter for the pure BG-CSs and Husimi’s and P-quasi distributions (for the mixed-thermal states) is also presented. Finally, the exponential form of the BG-CSs for the PT potential has been presented and enabled us to build Perelomov type CSs for the PT potential. We point out that the BG-CSs and the Perelomov type coherent states (PCSs) are related via Laplace transform.