Coherent states for Smorodinsky-Winternitz potentials

Open Physics ◽  
2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Nuri Ünal
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Wave Packets ◽  
Harmonic Oscillators ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Physical Time ◽  
First Case

AbstractIn this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional harmonic oscillator problem. In the first case, we find the non-spreading wave packets by transforming the system into four oscillators in Cartesian, and also polar, coordinates. In the second case, the coherent states are constructed in Cartesian coordinates by transforming the system into three non-isotropic harmonic oscillators. All of these states evolve in physical-time. We also show that in parametric-time, the second case can be transformed to the first one with vanishing eigenvalues.

Probabilistic aspects of the SU(2)-harmonic oscillator

2000 ◽  
Vol 37 (1) ◽  
pp. 306-314
Author(s):  
Shunlong Luo
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Probability Distributions ◽  
Quantum Probability ◽  
Dimensional Sphere ◽  
Large Radius ◽  
Two Dimensional ◽  
Bargmann Transform ◽  
Gaussian Distributions ◽  
Binomial Distributions

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

Probabilistic aspects of the SU(2)-harmonic oscillator

2000 ◽  
Vol 37 (01) ◽  
pp. 306-314
Author(s):  
Shunlong Luo
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Probability Distributions ◽  
Quantum Probability ◽  
Dimensional Sphere ◽  
Large Radius ◽  
Two Dimensional ◽  
Bargmann Transform ◽  
Gaussian Distributions ◽  
Binomial Distributions

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

ATOMIC COHERENT STATES AS THE EIGENSTATES OF A TWO-DIMENSIONAL ANISOTROPIC HARMONIC OSCILLATOR IN A UNIFORM MAGNETIC FIELD

2009 ◽  
Vol 24 (38) ◽  
pp. 3129-3136 ◽  
Author(s):  
XIANG-GUO MENG ◽  
JI-SUO WANG ◽  
HONG-YI FAN
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Uniform Magnetic Field ◽  
Hermite Polynomial ◽  
Entangled State ◽  
Two Dimensional ◽  
Entangled State Representation ◽  
Single Variable ◽  
State Representation

In the newly constructed entangled state representation embodying quantum entanglement of Einstein, Podolsky and Rosen, the usual wave function of atomic coherent state ∣τ〉 = exp (μJ+-μ*J-)∣j, -j〉 turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2j, where j is the spin value. We then prove that a two-dimensional time-independent anisotropic harmonic oscillator in a uniform magnetic field possesses energy eigenstates which can be classified as the states ∣τ〉 in terms of the spin values j.

scholarly journals NUMERICAL SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION USING RANDOMLY GENERATED FINITE GRIDS AND TWO-DIMENSIONAL FRACTIONAL-ORDER LEGENDRE FUNCTION

2021 ◽  
Author(s):  
Sanaullah Mastoi
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Polar Coordinates ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Uniform Grids

There are various methods to solve the physical life problem involving engineering, scientific and biological systems. It is found that numerical methods are approximate solutions. In this way, randomly generated finite difference grids achieve an approximation with fewer iterations. The idea of randomly generated grids in cartesian coordinates and polar form are compared with the exact, iterative method, uniform grids, and approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions. The most ideal and benchmarking method is the finite difference method over randomly generated grids on Cartesian coordinates, polar coordinates used for numerical solutions. This concept motivates the investigation of the effects of the randomly generated meshes. The two-dimensional equation is solved over randomly generated meshes to test randomly generated grids and the implementation. The feasibility of the numerical solution is analyzed by comparing simulation profiles.

scholarly journals Coherent States for the Isotropic and Anisotropic 2D Harmonic Oscillators

2019 ◽  
Vol 1 (2) ◽  
pp. 260-270 ◽  
Author(s):  
James Moran ◽  
Véronique Hussin
Keyword(s):  
Harmonic Oscillator ◽  
Configuration Space ◽  
Coherent States ◽  
Probability Density Functions ◽  
Harmonic Oscillators ◽  
Uncertainty Relations ◽  
Density Functions ◽  
Ladder Operators ◽  
New States ◽  
2D System

In this paper we introduce a new method for constructing coherent states for 2D harmonic oscillators. In particular, we focus on both the isotropic and commensurate anisotropic instances of the 2D harmonic oscillator. We define a new set of ladder operators for the 2D system as a linear combination of the x and y ladder operators and construct the S U ( 2 ) coherent states, where these are then used as the basis of expansion for Schrödinger-type coherent states of the 2D oscillators. We discuss the uncertainty relations for the new states and study the behaviour of their probability density functions in configuration space.

scholarly journals Two-dimensional non-Hermitian harmonic oscillator: coherent states

2019 ◽  
Vol 94 (11) ◽  
pp. 115215
Author(s):  
Masoumeh Izadparast ◽  
S Habib Mazharimousavi
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Two Dimensional

Even and odd coherent states of supersymmetric harmonic oscillators and their nonclassical properties

2016 ◽  
Vol 30 (07) ◽  
pp. 1650026 ◽  
Author(s):  
Davood Afshar ◽  
Amin Motamedinasab ◽  
Azam Anbaraki ◽  
Mojtaba Jafarpour
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Bell States ◽  
Harmonic Oscillators ◽  
Nonclassical Properties ◽  
Logical Qubits ◽  
Poissonian Statistics

In this paper, we have constructed even and odd superpositions of supercoherent states, similar to the standard even and odd coherent states of the harmonic oscillator. Then, their nonclassical properties, that is, squeezing and entanglement have been studied. We have observed that even supercoherent states show squeezing behavior for some values of parameters involved, while odd supercoherent states do not show squeezing at all. Also sub-Poissonian statistics have been observed for some ranges of the parameters in both states. We have also shown that these states may be considered as logical qubits which reduce to the Bell states at a limit, with concurrence equal to 1.

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