Quantum $SU(2,2)$-Harmonic Oscillator

We study the isotropic and anisotropic Hamiltonian of two coupled harmonic oscillators from an algebraic approach of the $SU(1,1)$ and $SU(2)$ groups. In order to obtain the energy spectrum and eigenfunctions of this problem, we write its Hamiltonian in terms of the boson generators of the $SU(1,1)$ and $SU(2)$ groups. We use the one boson and two boson realizations of the $su(1,1)$ Lie algebra, and the one boson realization of the $su(2)$ Lie algebra to apply three tilting t...

The linearized Kepler problem is considered, as obtained from the Kustaanheimo-Stiefel (K-S)transformation, both for negative and positive energies. The symmetry group for the Kepler problem turns out to be SU(2,2). For negative energies, the Hamiltonian of Kepler problem can be realized as the sum of the energies of four harmonic oscillator with the same frequency, with a certain constrain. For positive energies, it can be realized as the sum of the energies of four repulsiv...

Operator angle-action variables are studied in the frame of the SU(2) algebra, and their eigenstates and coherent states are discussed. The quantum mechanical addition of action-angle variables is shown to lead to a novel non commutative Hopf algebra. The group contraction is used to make the connection with the harmonic oscillator.

We show that the $(2+1)$-dimensional Dirac-Moshinsky oscillator coupled to an external magnetic field can be treated algebraically with the $SU(1,1)$ group theory and its group basis. We use the $su(1,1)$ irreducible representation theory to find the energy spectrum and the eigenfunctions. Also, with the $su(1,1)$ group basis we construct the relativistic coherent states in a closed form for this problem.

We solve the Gauss law of SU(2) lattice gauge theory using the harmonic oscillator prepotential formulation. We construct a generating function of a manifestly gauge invariant and orthonormal basis in the physical Hilbert space of (d+1) dimensional SU(2) lattice gauge theory. The resulting orthonormal physical states are given in closed form. The generalization to SU(N) gauge group is discussed.

We study a supersymmetric 2-dimensional harmonic oscillator which carries a representation of the general graded Lie algebra GL(2$\vert$1), formulate it on the superspace, and discuss its physical spectrum.

In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schr\"odinger equation of the model. The approach can be generalized to the $D$-dimens...

In this paper we present a harmonic oscillator realization of the most degenerate discrete series representations of the SU(2,1) group and the deformation quantization of the coset space $D=SU(2,1)/U(2)$ with the method of coherent state quantization.

States which minimize the Schr\"odinger--Robertson uncertainty relation are constructed as eigenstates of an operator which is a element of the $h(1) \oplus \su(2)$ algebra. The relations with supercoherent and supersqueezed states of the supersymmetric harmonic oscillator are given. Moreover, we are able to compute gneneral Hamiltonians which behave like the harmonic oscillator Hamiltonian or are related to the Jaynes--Cummings Hamiltonian.

We show that the elementary modes of the planar harmonic oscillator can be quantised in the framework of quantum mechanics based on pseudo-hermitian hamiltonians. These quantised modes are demonstrated to act as dynamical structures behind a new Jordan - Schwinger realization of the SU(1,1) algebra. This analysis complements the conventional Jordan - Schwinger construction of the SU(2) algebra based on hermitian hamiltonians of a doublet of oscillators.