Two-dimensional Isotropic Oscillator Research Articles

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.

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.

MICZ-Kepler systems in noncommutative space and duality of force laws

In this paper, we analyze the modification of integrable models in the κ-deformed space–time. We show that two-dimensional isotropic oscillator problem, Kepler problem and MICZ-Kepler problem in κ-deformed space–time admit integrals of motion as in the commutative space. We also show that the duality equivalence between κ-deformed Kepler problem and κ-deformed two-dimensional isotropic oscillator explicitly, by deriving Bohlin–Sundman transformation which maps these two systems. These results are valid to all orders in the deformation parameter.

DYNAMICAL SYSTEMS RELATED TO FRACTIONAL HAMILTONIANS ON THE TWO-DIMENSIONAL SPHERE

We consider a class of “fractional” Hamiltonian systems generalizing the classical problem of motion in a central field. Our analysis is based on transforming an integrable Hamiltonian system with two degrees of freedom on the plane into a dynamical system that is defined on the sphere and inherits the integrals of motion of the original system. We show that in the four-dimensional space of structural parameters, there exists a one-dimensional manifold (containing the case of the planar Kepler problem) along which the closedness of the orbits of all finite motions and the third Kepler law are preserved. Similarly, there exists a one-dimensional manifold (containing the case of the two-dimensional isotropic harmonic oscillator) along which the closedness of the orbits and the isochronism of oscillations are preserved. Any deformation of orbits on these manifolds does not violate the hidden symmetry typical of the two-dimensional isotropic oscillator and the planar Kepler problem. We also consider two-dimensional manifolds on which all systems are characterized by the same rotation number for the orbits of all finite motions.

Logarithmically perturbed two dimensional oscillator model of a quantum-dot nanostructure

We present a logarithmically perturbed two-dimensional isotropic oscillator model of a quantum-dot nanostructure. The logarithmic potential has an ionic line charge physical origin. We estimate the magnitude of this line charge and we obtain the matrix elements required by perturbation theory. The measured ground-state energy depression for a GaAs/AlGaAs/InGaAs heterostructure can be quantitatively accounted for by the model (although not uniquely). We include the effects of nonzero angular momentum states, which play a significant role in determining the dot current–voltage curves. We mention the immediate selection rule imposed by a perturbing potential of rotational symmetry, together with further analytic results.

Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom).

Some striking analogies are exhibited between the lossless quantum-mechanical beam splitter and the following two elementary quantum systems: (1) the two-dimensional isotropic oscillator and (2) the two-dimensional hydrogen atom. For example, we show that the unitary transformation connecting the input and output photon statistics at a lossless 50-50 beam splitter is identical to the transformation connecting the polar and parabolic eigenstates of the two-dimensional (2D) hydrogen atom. A similar connection between (Gaussian) laser beams and the eigenstates of the 2D oscillator is pointed out. These analogies are established by identifying the underlying symmetries of these systems and exploiting them by means of group theory.

Isotropic time-dependent coupled oscillators

The problem of a two-dimensional isotropic oscillator coupled through a variable parameter is considered. The problem is reduced to uncoupled form with two different variable frequencies. The solution has been obtained in both the Schr\"odinger and Heisenberg pictures. Applications in quantum optics are indicated.

Projective representations of SU(2)ΛR4

The Mackey little group procedure is used to classify the projective irreducible unitary representations of the group SU(2)ΛR4. Whereas an arbitrary ordinary representation of this group plays the role of both spectral and dynamical group for the three-dimensional hydrogen atom, it is found that one of its projective representations acts as spectral group for the two-dimensional isotropic oscillator. The projective representations of the Lie algebra of SU(2)ΛR4 are treated as ordinary representations of the central extension of this Lie algebra, and shift operator techniques used to obtain the matrix elements of its representatives. The connection with the isotropic oscillator is shown to be related to the realizability of the extended Lie algebra by two-boson creation and annihilation operators.

Two-dimensional harmonic oscillator matrix elements, hyperbolic angular momentum and spherical harmonics

Schwinger's coupled-boson representation of spherical and hyperbolic angular momentum is related to spherical harmonics and is used for the calculation of matrix elements. It is shown that radial matrix elements of the two-dimensional isotropic oscillator and the closely related hydrogenic matrix elements are proportional to 3j-symbols. Matrix elements of exponentials of hyperbolic angular momentum operators which form a formal analogy to the well known rotation matrice dmm'j( theta ) and which are written as tmmjj'( theta ) are derived. As only elementary algebra and some operator formulae are used, the algebraic method supplements analytical and group theoretic methods.