Isotropic Oscillator Research Articles

Estimating the magnetic field contributions on thermodynamic functions of diatomic molecules trapped in an isotropic oscillator plus inverse quadratic potential

In this study, the Laplace transform approach have been employed to analyze the bound state solutions of two-dimensional Schrödinger equation with isotropic oscillator plus inverse quadratic potential. The wavefunctions and energy equation were derived. Thereafter, the so-called partition function and the thermodynamic properties were evaluated as a function Larmor frequency in the range 0<ωL<100 for some values of parameter λ. The findings showed that both the partition function and thermodynamic properties of the molecules were modified significantly by magnetic interactions, which invariably may alter the macroscopic behaviour of the system examined.

On the entanglement of co-ordinate and momentum degrees of freedom in noncommutative space

In this paper, we investigate the quantum entanglement induced by phase-space noncommutativity. Both the position–position and momentum–momentum noncommutativity are incorporated to study the entanglement properties of coordinate and momentum degrees of freedom under the shade of oscillators in noncommutative space. Exact solutions for the systems are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp’s shift to the noncommuting degrees of freedom. It is shown that the bipartite Gaussian state for an isotropic oscillator is always separable. To extend our study for the time-dependent system, we allow arbitrary time dependency on parameters. The time-dependent isotropic oscillator is solved with the Lewis–Riesenfeld invariant method. It turns out that even for arbitrary time-dependent scenarios, the separability property does not alter. We extend our study to the anisotropic oscillator, which provides an entangled state even for time-independent parameters. The Wigner quasi-probability distribution is constructed for a bipartite Gaussian state. The noise matrix (covariance matrix) is explicitly studied with the help of Wigner distribution. Simon’s separability criterion (generalized Peres–Horodecki criterion) has been employed to find the unique function of the (mass and frequency) parameters, for which the bipartite states are separable. In particular, we show that the mere inclusion of non-commutativity of phase-space is not sufficient to generate the entanglement, rather anisotropy is important at the same footing. We explore the experimental viability of our result through the computation of extractable work for the current situation.

Molecular Study on the DKP Equation in (1 + 3) Dimensions with Isotropic Oscillator plus Inverse Quadratic Potential in Non-Commutative Space

Molecular Study on the DKP Equation in (1 + 3) Dimensions with Isotropic Oscillator plus Inverse Quadratic Potential in Non-Commutative Space

Tuning the separability in noncommutative space

With the help of the generalized Peres–Horodecki separability criterion (Simon’s condition) for a bipartite Gaussian state, we have studied the separability of the noncommutative (NC) space coordinate degrees of freedom. Non-symplectic nature of the transformation between the usual commutative space and NC space restricts the straightforward use of Simon’s condition in NCS. We have transformed the NCS system to an equivalent Hamiltonian in commutative space through the Bopp shift, which enables the utilization of the separability criterion. To make our study fairly general and to analyze the effect of parameters on the separability of bipartite state in NC-space, we have considered a bilinear Hamiltonian with time-dependent (TD) parameters, along with a TD external interaction, which is linear in field modes. The system is transformed (Sp(4,R)) into canonical form keeping the intrinsic symplectic structure intact. The solution of the TD-Schrödinger equation is obtained with the help of the Lewis–Riesenfeld invariant method (LRIM). Expectation values of the observables (thus the covariance matrix) are constructed from the states obtained from LRIM. It turns out that the existence of the NC parameters in the oscillator determines the separability of the states. In particular, for isotropic oscillators, the separability condition for the bipartite Gaussian states depends on specific values of NC parameters. Moreover, particular anisotropic parameter values for the oscillator may cease the separability. In other words, both the deformation parameters (θ, η) and parameter values of the oscillator (mass, frequency) are important characteristics for the separability of bipartite Gaussian states. Thus tuning the parameter values, one can destroy or recreate the separability of states. With the help of a toy model, we have demonstrated how the tuning of a TD-NC space parameter affects the separability.

Quasi-exactly solvable potentials in Wigner-Dunkl quantum mechanics

It is shown that the Dunkl harmonic oscillator on the line can be generalized to a quasi-exactly solvable one, which is an anharmonic oscillator with known eigenstates for any . It is also proved that the Hamiltonian of the latter can also be rewritten in a simpler way in terms of an extended Dunkl derivative. Furthermore, the Dunkl isotropic oscillator and Dunkl Coulomb potentials in the plane are generalized to quasi-exactly solvable ones. In the former case, potentials with known eigenstates are obtained, whereas, in the latter, sets of potentials associated with a given energy are derived.

Dynamical symmetries of the anisotropic oscillator

It is well known that the Hamiltonian of an n-dimensional isotropic oscillator admits an SU(n) symmetry, making the system maximally superintegrable. However, the dynamical symmetries of the anisotropic oscillator are much more subtle. We introduce a novel set of canonical transformations that map an n-dimensional anisotropic oscillator to the corresponding isotropic problem. Consequently, the anisotropic oscillator is found to possess the same number of conserved quantities as the isotropic oscillator, making it maximally superintegrable too. The first integrals are explicitly calculated in the case of a two-dimensional anisotropic oscillator and remarkably, they admit closed-form expressions.

Rational extensions of the Dunkl oscillator in the plane and exceptional orthogonal polynomials

It is shown that rational extensions of the isotropic Dunkl oscillator in the plane can be obtained by adding some terms either to the radial equation or to the angular one obtained in the polar coordinates approach. In the former case, the isotropic harmonic oscillator is replaced by an isotropic anharmonic one, whose wave functions are expressed in terms of [Formula: see text]-Laguerre exceptional orthogonal polynomials. In the latter, it becomes an anisotropic potential, whose explicit form has been found in the simplest case associated with [Formula: see text]-Jacobi exceptional orthogonal polynomials.

Virtual excitations and quantum correlations in ultra-strongly coupled harmonic oscillators under intrinsic decoherence

We study the effect of Milburn intrinsic decoherence in a system of ultra-strongly coupled quantum harmonic oscillators, putting emphasis on the virtual excitations of the ground state and their interconnection with quantum correlations like entanglement and steering. For isotropic (equal frequency) oscillators we derive an analytic expression for the common steady state value of virtual excitations and then show analytically and numerically that the influence of intrinsic decoherence can be delayed by increasing the ultra-strong coupling. When increasing the anisotropy in the oscillator frequencies, we numerically observe an asymmetric redistribution of virtual excitations among the oscillators, which leads to an asymmetry in quantum steering. Furthermore, virtual excitations are increased, causing also the strengthening of entanglement and steering immediately after their generation. For large anisotropy values we observe quantum synchronization between the oscillators, with both excitations and quantum correlations being significantly enhanced. When increasing the ultra-strong coupling in the case of large anisotropy, a considerable reinforcement of quantum correlations is observed, over long simulation times and in spite of the presence of decoherence. This is the most important result of this work, since it suggests how to properly design the experimental parameters of the system in order to delay the effect of intrinsic decoherence.

Time-Dependent 4D Quantum Harmonic Oscillator and Reacting Hydrogen Atom

With the help of low-dimensional reference equations (ordinary differential equations) and the corresponding coordinate transformations, the non-stationary 4D quantum oscillator in an external field is reduced to an autonomous form. The latter, in particular, reflects the existence of a new type of dynamical symmetry that reduces the equation of motion of a non-stationary oscillator to an autonomous form that does not change with time. By imposing an additional constraint on the wave function of the isotropic oscillator, we have obtained the total wave functions of the reacting hydrogen atom in two different cases: (a) when the non-stationary frequency has two asymptotic values and there is no external field; and (b) when, in addition to the non-stationary frequency, an external force acts on the hydrogen atom. The transition S-matrix elements of various elementary atomic–molecular processes are constructed. The probabilities of quantum transitions of the hydrogen atom to others, including new bound states, are studied in detail, taking into account the influence of external forces.

Fabrication of submillimeter-sized spherical self-oscillating gels and control of their isotropic volumetric oscillatory behaviors.

In this study, we established a fabrication method and analyzed the volumetric self-oscillatory behaviors of submillimeter-sized spherical self-oscillating gels. We validated that the manufactured submillimeter-sized spherical self-oscillating gels exhibited isotropic volumetric oscillations during the Belousov-Zhabotinsky (BZ) reaction. In addition, we experimentally elucidated that the volumetric self-oscillatory behaviors (i.e., period and amplitude) and the oscillatory profiles depended on the following parameters: (1) the molar composition of N-(3-aminopropyl)methacrylamide hydrochloride (NAPMAm) in the gels and (2) the concentration of Ru(bpy)3-NHS solution containing an active ester group on conjugation. These clarified relationships imply that controlling the amount of Ru(bpy)3 in the gel network could influence the gel volumetric oscillation during the BZ reaction. These results of submillimeter-sized and spherical self-oscillating gels bridge knowledge gaps in the current field because the gels with corresponding sizes and shapes have not been systematically explored yet. Therefore, our study could be a cornerstone for diverse applications of (self-powered) gels in various scales and shapes, including soft actuators exhibiting life-like functions.

The Clifford-Yang algebra, noncommutative Clifford phase spaces and the deformed quantum oscillator

We construct the novel Clifford-Yang algebra which is an extension of the Yang algebra in noncommutative phase spaces. The Clifford-Yang algebra allows us to write down the commutators of the [Formula: see text] polyvector-valued coordinates and momenta which are compatible with the Jacobi identities, the Weyl–Heisenberg algebra, and paves the way for a formulation of quantum mechanics (QM) in noncommutative Clifford spaces. We continue with a detailed study of the isotropic 3D quantum oscillator in noncommutative spaces and find the energy eigenvalues and eigenfunctions. These findings differ considerably from the ordinary quantum oscillator in commutative spaces. We find that QM in noncommutative spaces leads to very different solutions, eigenvalues, and uncertainty relations than ordinary QM in commutative spaces. The generalization of QM to noncommutative Clifford (phase) spaces is attained via the Clifford-Yang algebra. The operators are now given by the generalized angular momentum operators involving polyvector coordinates and momenta. The eigenfunctions (wave functions) are now more complicated functions of the polyvector coordinates. We conclude with some important remarks.

Investigating Some Diatomic Molecules Bounded by the Two-Dimensional Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

We investigate the nonrelativistic magnetic effect on the energy spectra, expectation values of some quantum mechanical observables, and diamagnetic susceptibility for some diatomic molecules bounded by the isotropic oscillator plus inverse quadratic potential. The energy eigenvalues and normalized wave functions are obtained via the parametric Nikiforov-Uvarov method. The expectation values square of the position r 2 , square of the momentum p 2 , kinetic energy T , and potential energy V are obtained by applying the Hellmann-Feynman theorem, and an expression for the diamagnetic susceptibility X is also derived. Using the spectroscopic data, the low rotational and low vibrational energy spectra, expectation values, and diamagnetic susceptibility X for a set of diatomic molecules (I2, H2, CO, and HCl) for arbitrary values, Larmor frequencies are calculated. The computed energy spectra, expectation values, and diamagnetic susceptibility X were found to be more influenced by the external magnetic field strength and inverse quadratic potential strength g than the vibrational frequencies and the masses of the selected molecules.

Finite series representation for the bound states of a spiked isotropic oscillator with inverse-quartic singularity

In this paper, we use the tridiagonal representation approach to obtain a finite series solution of the radial Schrödinger equation in three dimensions for a spiked oscillator with inverse quartic singularity.

Analytical approach to the mean-return-time phase of isotropic stochastic oscillators.

One notion of phase for stochastic oscillators is based on the mean return-time (MRT): a set of points represents a certain phase if the mean time to return from any point in this set to this set after one rotation is equal to the mean rotation period of the oscillator (irrespective of the starting point). For this so far only algorithmically defined phase, we derive here analytical expressions for the important class of isotropic stochastic oscillators. This allows us to evaluate cases from the literature explicitly and to study the behavior of the MRT phase in the limits of strong noise. We also use the same formalism to show that lines of constant return time variance (instead of constant mean return time) can be defined, and that they in general differ from the MRT isochrons.

A noncommutative spacetime realization of quantum black holes, Regge trajectories and holography

It is shown that the radial spectrum associated with a fuzzy sphere in a noncommutative phase space characterized by the Yang algebra, leads exactly to a Regge-like spectrum GMl2=l=1,2,3,…, for all positive values of l, and which is consistent with the extremal quantum Kerr black hole solution that occurs when the outer and inner horizon radius coincide r+=r−=GM. The condition GMl2=l is tantamount to the mass-angular momentum relation Ml2=lMp2 implying a mass-squared quantization in multiples of the Planck-mass-squared. Another important feature (also pointed out by Tanaka) is the holographic nature of these results that are based in recasting the Yang algebra associated with an 8D noncommuting phase space, involving xμ,pν,μ,ν=0,1,2,3, in terms of the undeformed realizations of the Lorentz algebra generators JAB corresponding to a 6D-spacetime, and associated to a 12D-phase-space with coordinates XA,PA;A=0,1,2,…,5. We finalize with a discussion of the noncommuting 3D isotropic and Born oscillators. Finding solutions to these oscillators merit investigation because they introduce explicit dynamics to the quantum black holes. We hope that the findings in this work, relating the Regge-like spectrum l=GM2 and the quantized area of black hole horizons in Planck bits, via the Yang algebra in Noncommutative phase spaces, will help us elucidate some of the impending issues pertaining the black hole information paradox and the role that string theory and quantum information will play in its resolution.

Non-linear ladder operators and coherent states for the 2:1 oscillator

The 2:1 two-dimensional anisotropic quantum harmonic oscillator is considered and new sets of states are defined by means of normal-ordering non-linear operators through the use of non-commutative binomial theorems as well as solving recurrence relations. The states generated are good candidates for the natural generalisation of the coherent states of the two-dimensional isotropic oscillator. The two-dimensional non-linear generalised ladder operators lead to several chains of states which are connected in a non trivial way. The uncertainty relations of the defining chain of states are calculated and it is found that they admit a resolution of the identity and the spatial distribution of the wavefunction produces Lissajous figures in correspondence with the classical 2:1 oscillator.

Conformal generation of an exotic rotationally invariant harmonic oscillator

An exotic rotationally invariant harmonic oscillator (ERIHO) is constructed by applying a non-unitary isotropic conformal bridge transformation (CBT) to a free planar particle. It is described by the isotropic harmonic oscillator Hamiltonian supplemented by a Zeeman type term with a real coupling constant $g$. The model reveals the Euclidean ($|g|<1$) and Minkowskian ($|g|>1$) phases separated by the phases $g=+1$ and $g=-1$ of the Landau problem in the symmetric gauge with opposite orientation of the magnetic field. A hidden symmetry emerges in the system at rational values of $g$. Its generators, together with the Hamiltonian and angular momentum produce non-linearly deformed $\mathfrak{u}(2)$ and $\mathfrak{gl}(2,{\mathbb R})$ algebras in the cases of $0<|g|<1$ and $\infty>|g|>1$, which transmute one into another under the inversion $g\rightarrow -1/g$. Similarly, the true, $\mathfrak{u}(2)$, and extended conformal, $\mathfrak{gl}(2,{\mathbb R})$, symmetries of the isotropic Euclidean oscillator ($g=0$) interchange their roles in the isotropic Minkowskian oscillator ($|g|=\infty$), while two copies of the $\mathfrak{gl}(2,{\mathbb R})$ algebra of analogous symmetries mutually transmute in Landau phases. We show that the ERIHO system is transformed by a peculiar unitary transformation into the anisotropic harmonic oscillator generated, in turn, by anisotropic CBT. The relationship between the ERIHO and the subcritical phases of the harmonically extended Landau problem, as well as with a plane isotropic harmonic oscillator in a uniformly rotating reference frame, is established.

Adding potentials to superintegrable systems with symmetry

In previous work, we have considered Hamiltonians associated with three-dimensional conformally flat spaces, possessing two-, three- and four-dimensional isometry algebras. Previously, our Hamiltonians have represented free motion, but here we consider the problem of adding potential functions in the presence of symmetry. Separable potentials in the three-dimensional space reduce to 3 or 4 parameter potentials for Darboux–Koenigs Hamiltonians. Other three-dimensional coordinate systems reveal connections between Darboux–Koenigs and other well-known super-integrable Hamiltonians, such as the Kepler problem and isotropic oscillator.

Symmetries and integrals of motion of a superintegrable deformed oscillator

The symmetry structure of twodimensional nonlinear isotropic oscillator, introduced in Ballesteros et al. (2008), is discussed. It is shown that it possesses three independent integrals of motion which can be chosen in such a way that they span SU(2), E(2) or SU(1,1) algebras, depending on the value of total energy. They generate the infinitesimal canonical symmetry transformations; integrability of the latter is analyzed. The results are then generalized to the case of arbitrary number of degrees of freedom.

Superintegrability of three-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems

We study four particular three-dimensional natural Hamiltonian systems defined in conformally Euclidean spaces. We prove their superintegrability and we obtain, in the four cases, the maximal number of functionally independent integrals of motion. The two first systems are related to the three-dimensional isotropic oscillator and the superintegrability is quadratic. The third system is obtained as a continuous deformation of an oscillator with ratio of frequencies 1:1:2 and with three additional nonlinear terms of the form k 2/x 2, k 3/y 2 and k 4/z 2, and the fourth system is obtained as a deformation of the Kepler Hamiltonian also with these three particular nonlinear terms. These third and fourth systems are superintegrable but with higher-order constants of motion. The four systems depend on a real parameter in such a way that they are continuous functions of the parameter (in a certain domain of the parameter) and in the limit of such parameter going to zero the Euclidean dynamics is recovered.