The Weyl relations, the harmonic oscillator, the hydrogen atom, the Dirac equation on the lattice are presented with the help of the difference equations and the orthogonal polynomials of discrete variable. This area of research is attracting more interest due to the lattice field theories and the hypothesis of a finite space. 1 Weyl group on finite space We defined the position space of dimens...

A new family of one-dimensional quantum models is proposed in terms of new potentials with a Gaussian asymptotic behavior but approaching to the potential of the harmonic oscillator when x → 0. It is shown that, in the energy basis of the harmonic oscillator, the matrix elements of the Hamiltonian operators of these new models can be derived from generating functionals. Pacs 03.65.Ge The progre...

We present accurate numerical results for the one-dimensional stationary Schrödinger equation in case of three quantum problems: harmonic oscillator, radial a Hydrogen atom, and particle penetration through potential barrier. All them were solved by Numerov method with high accuracy we plot their wave functions using calculations. Furthermore, offer methods solving boundary value problems, eige...

Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L = A +V on H where A is a possibly unbounded self-adjoint operator commuting with J and V a bounded J-self-adjoint operator anti-commuting with J. We establish optimal estimates on the position of the spectrum of L with respect to the spectrum of A and we obtain norm bounds on the operator angles betw...

A free quantum field and its canonical conjugate are equivalent to a family of harmonic oscillators (one oscillator for each plane wave), which is in turn equivalent to a quantum theory of free identical bosons. In this note, I will show how all of this works for the relativistic scalar field φ̂(x) and its conjugate π̂(x). And then I will turn around and show that a quantum theory of any kind of ...

A free quantum field and its canonical conjugate are equivalent to a family of harmonic oscillators (one oscillator for each plane wave), which is in turn equivalent to a quantum theory of free identical bosons. In this note, I will show how all of this works for the relativistic scalar field φ̂(x) and its conjugate π̂(x). And then I will turn around and show that a quantum theory of any kind of ...

In this paper we revisit the quantum damped harmonic oscillator and construct an interesting approximate solution in the operator algebra level. Namely, we first give the general solution to the Lindblad form (equation) and next construct an approximate solution to the full equation by use of it. Our method is a milestone to construct the general solution. In this paper we revisit dynamics of a...

All coboundary Lie bialgebras and their corresponding Poisson–Lie structures are constructed for the oscillator algebra generated by {N,A+, A−,M}. Quantum oscillator algebras are derived from these bialgebras by using the Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both algebra and group levels are obtained, including their universal R–matrices.

A Lie algebra is said to be metric if it admits a symmetric, invariant, and non-degenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of oscillator, smallest solvable non-abelian example. This first step countable series algebras that support invariant Lorentzian forms. Generalizing this situation, paper, we arrive at K-algebras as double ...

The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger, Heisenberg and Weyl-Wigner-Moyal representations of the Lindblad equation are given exp...