SU(1,1) and SU(2) app

Abstract

It is shown that each one of the Lie algebras su(1, 1) and su(2) determine the spectrum of the radial oscillator. States that share the same orbital angular momentum are used to construct the representation spaces of the non-compact Lie group SU (1, 1). In addition, three different forms of obtaining the representation spaces of the compact Lie group SU (2) are introduced, they are based on the accidental degeneracies associated with the spherical symmetry of the system as well as on the selection rules that govern the transitions between different energy levels. In all cases the corresponding generalized coherent states are constructed and the conditions to squeeze the involved quadratures are analyzed.

Position dependent mass Scarf Hamiltonians generated via the Riccati equation

“…For each fixed value of ℓ m i n , the complete space of wave functions splits into the direct sum of the set of subspaces spanned by the corresponding hierarchies. This fact is relevant, eg, in the construction of different families of coherent states and in the generation of the dynamical algebra underlying the corresponding system …”

“…The operators intertwine the wave functions corresponding to the same k ‐hierarchy. Thus, for each fixed value of ℓ m i n , the whole set of wave functions can be decomposed as the direct sum of the subspaces each one spanned by the corresponding k ‐hierarchy …”

“…Thus, for each fixed value of min , the whole set of wave functions n can be decomposed as the direct sum of the subspaces {  k , k = 1, 3 2 , 2· · · } each one spanned by the corresponding k-hierarchy. 41,42…”

Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

“…The uncertainties of the SU(1, 1) variables for these modes are given by with ⟨K 0 ⟩ 0 = K ⟨k, k|K 0 |k, k⟩ K = k [32,50]. Thus…”

“…In this method the second order differential Hamiltonian is factorized, up to an additive constant, as the product of two first order differential operators. These factors can be used to construct the basis elements of the dynamical algebra of the system leading to the group approach of the factorization method (see, [31,32]). …”