The problem of a hydrogen atom in a cavity: Oscillator representation solution versus analytic solution

A solution to the problem of a hydrogenic atom in a homogeneous dielectric medium with a concentric spherical cavity using the oscillator representation method (ORM) is presented. The results obtained by the ORM are compared with a known exact analytic solution. The energy levels of the hydrogenic atom in a spherical cavity exhibit a shallow-deep instability as a function of the cavity radius. The sharpness of the transition depends on the value of the dielectric constant of the medium. The results of the ORM agree well with the results obtained by the analytic solution when the shallow-deep transition is not too sharp (i.e., when the dielectric constant is not too large) for all values of the cavity radius. The ORM results in the zeroth order approximation diverge significantly in the region of the shallow-deep transition (i.e., for the values of the radius where the shallow-deep transition occurs) when the dielectric constant is high and as a result the transition is sharp. Even for the sharp transition, the ORM results again agree very well with the analytic results at least for the ground state when a commonly used approximation in the ORM is removed. The ORM methodology for the cavity model presented in this article can potentially be used for two-electron systems in a quantum dot.

Einstein’s E=mc2 Derivable from Heisenberg’s Uncertainty Relations

Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the O ( 2 , 1 ) group. This group has three independent generators. The one-dimensional step-up and step-down operators can be combined into one two-by-two Hermitian matrix which contains three independent operators. If we use a two-variable Heisenberg commutation relation, the two pairs of independent step-up, step-down operators can be combined into a four-by-four block-diagonal Hermitian matrix with six independent parameters. It is then possible to add one off-diagonal two-by-two matrix and its Hermitian conjugate to complete the four-by-four Hermitian matrix. This off-diagonal matrix has four independent generators. There are thus ten independent generators. It is then shown that these ten generators can be linearly combined to the ten generators for Dirac’s two oscillator system leading to the group isomorphic to the de Sitter group O ( 3 , 2 ) , which can then be contracted to the inhomogeneous Lorentz group with four translation generators corresponding to the four-momentum in the Lorentz-covariant world. This Lorentz-covariant four-momentum is known as Einstein’s E = m c 2 .

Howe duality of Higgs – Hahn algebra for 8D harmonic oscillator

In the light of the Howe duality, two different, but isomorphic representations of one algebra as Higgs algebra and Hahn algebra are considered in this article. The first algebra corresponds to the symmetry algebra of a harmonic oscillator on a 2-sphere and a polynomially deformed algebra SU(2), and the second algebra encodes the bispectral properties of corresponding homogeneous orthogonal polynomials and acts as a symmetry algebra for the Hartmann and certain ring-shaped potentials as well as the singular oscillator in two dimensions. The realization of this algebra is shown in explicit form, on the one hand, as the commutant O(4) ⊕ O(4) of subalgebra U(8) in the oscillator representation of universal algebra U (u(8)) and, on the other hand, as the embedding of the discrete version of the Hahn algebra in the double tensor product SU(1,1) ⊗ SU(1,1). These two realizations reflect the fact that SU(1,1) and U(8) form a dual pair in the state space of the harmonic oscillator in eight dimensions. The N-dimensional, N-fold tensor product SU(1,1)⊗N аnd q-generalizations are briefly discussed.

The hardest TC self-energy behavior and radiative corrections in a 331-TC model

The solution of the phenomenological problems of technicolor (TC) models may reside in the different dynamical behaviors of the technifermions self-energy appearing in walking (or quasi-conformal) theories. Motivated by recent results, where it is shown how the boundary conditions (BC) of the anharmonic oscillator representation of the Schwinger–Dyson gap equation (SDE) to [Formula: see text] are directly related with the mass anomalous dimensions, and different BC cause a change in the ultraviolet asymptotic behavior of the self-energies, in this paper, we verify that it is possible to have a hard technifermion self-energy in TC models originated through radiative corrections coming from the interactions mediated by the new massive neutral and charged gauge bosons, [Formula: see text] and [Formula: see text] in the context of a 331-TC model.

Representations of E7, equivalent combinatorics and algebraic varieties

In our earlier work on a new functor from [Formula: see text]-Mod to [Formula: see text]-Mod, we found a one-parameter ([Formula: see text]) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type [Formula: see text] in [Formula: see text] variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector [Formula: see text], we prove that the space forms an irreducible [Formula: see text]-module for any [Formula: see text] if [Formula: see text] is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of [Formula: see text] over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.

Two-Parameter Harmonic Oscillator Expansion Method for Calculating Non-Relativistic Ground State Energy of Coulomb Three-Particle Systems with Two Identical Particles

The variational method in oscillator representation with individual parameters for each Jacobi coordinate is applied to the non-relativistic calculation of the ground state energy of a number of three-particle Coulomb systems, consisting of two identical particles and a different one. The accuracy and convergence rate of the calculations in the constructed oscillator basis are studied up to a total of 28 oscillator quanta. The results are compared with those of the traditional approach using only one such nonlinear variational parameter. The method with individual parameters for Jacobi coordinates is found to possess a number of advantages as compared to the traditional approach.