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arXiv:1510.02841 · Journal of Mathematical Physics

The number radial coherent states for the generalized MICZ-Kepler problem

M. Salazar-Ramírez, D. Ojeda-Guillén, R. D. Mota

Published Feb 1, 2016• 8 citations
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Abstract

We study the radial part of the McIntosh-Cisneros-Zwanziger (MICZ)-Kepler problem in an algebraic way by using the 𝔰𝔲(1, 1) Lie algebra. We obtain the energy spectrum and the eigenfunctions of this problem from the 𝔰𝔲(1, 1) theory of unitary representations and the tilting transformation to the stationary Schrödinger equation. We construct the physical Perelomov number coherent states for this problem and compute some expectation values. Also, we obtain the time evolution of these coherent states.

DOI
10.1063/1.4940719

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We study the radial part of the McIntosh-Cisneros-Zwanziger (MICZ)-Kepler problem in an algebraic way by using the 𝔰𝔲(1, 1) Lie algebra. We obtain the energy spectrum and the eigenfunctions of this problem from the 𝔰𝔲(1, 1) theory of unitary representations and the tilting transformation to the stationary Schrödinger equation. We construct the physical Perelomov number coherent states for this problem and compute some expectation values. Also, we obtain the time evolution of these coherent states.
We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.
We construct the Perelomov number coherent states for an arbitrary su(1, 1) group operation and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su(1, 1) Lie algebra. By using the tilting transformation we apply our results to obtain the energy spectrum and eigenfunctions of the non-degenerate parametric amplifier. We show that these eigenfunctions are the Perelomov number coherent states of the two-dimensional harmonic oscillator.
We apply the Schrödinger factorization method to the radial second-order equation for the relativistic Kepler-Coulomb problem. From these operators we construct two sets of one-variable radial operators which are realizations for the su(1,1)su(1,1)su(1,1) Lie algebra. We use this algebraic structure to obtain the energy spectrum and the supersymmetric ground state for this system.
Abstract From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.
In a previous paper (2007 J. Phys. A: Math. Theor. 40 11105), we constructed a class of coherent states for a polynomially deformed su (2) algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed su (1, 1) algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su (1, 1) algebra which contains the Barut–Girardello set and the Perelomov set of the SU (1, 1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.