Perturbing free motions on hyperspheres without degeneracy lift

Abstract

We consider quantum motion on S 3 perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, ℓ, the ordinary orbital angular momentum value, and another, continuous parameter, b, through which an external scale is introduced. We argue that a loss of the geometric hyper-spherical so(4) symmetry of the free motion occurs that leaves intact the unperturbed hydrogen-like degeneracy patterns characterizing the spectrum under discussion. The argument is based on the observation that the expansions of the Scarf I wave functions for fixed ℓ-values in the basis of properly identified so(4) representation functions are power series in the perturbation parameter, b, in which carrier spaces of dimensionality (K + 1) 2 with K varying as K ∈ [ℓ, N − 1], and N being the principal quantum number of the Scarf I potential problem, contribute up to the order O b N −1−K . Nonetheless, the degeneracy patterns can still be interpreted as a consequence of an effective so(4) symmetry, i.e. a symmetry realized at the level of the dynamic of the system, in so far as from the perspective of the eigenvalue problem, the Scarf I results are equivalently obtained from a Hamiltonian with matrix elements of polynomials in a properly identified so(4) Casimir operator. The scheme applies to any dimension d. Symmetry and degeneracy: Introductory remarksSymmetry and degeneracy are two concepts which one traditionally associates with the basics of the quantum mechanics teachings. One may think of the degeneracy with respect to the magnetic quantum number, m, of a quantum level of a given angular momentum ℓ which is (2ℓ + 1)-fold, and typical for the states bound within all central potentials. A more advanced example would be the degeneracy of the states within the levels describing the quantum motion within the Coulomb potential of an electron without spin, which is N 2 -fold with N standing for the principal quantum number of the Coulomb potential problem. In the first case, the degeneracy is due to the rotational invariance of the three-dimensional position space, which requires conservation of angular momentum, and demands the total wave functions of the central potentials to be simultaneously eigenfunctions of L 2 , and L z , with L standing for the angular momentum pseudo-vector, and L z for its z-component. As long as L 2 acts as the Casimir invariant of the so(3) algebra, the degeneracy with respect to the magnetic quantum number (the L z eigenvalues m ∈ [−ℓ, +ℓ]) is attributed to the rotational invariance of the Hamiltonian. The second case is bit more involved in so far as in order to explain the larger N 2 -fold degeneracy, one needs to invoke the higher so(4) symmetry algebra underlying the Coulomb potential problem by accounting for the constancy of the Runge-Lenz vector next to that of angular momentum [1]. 1

Modelling duality between bound and resonant meson spectra by means of free quantum motions on the de Sitter space-time dS4

“…To be specific, we are interested in the possibility to formulate effective (one-color) chromo-statics on S 3 . In parallel to electrostatics, one may start approaching this goal by considering the fundamental solutions to the S 3 Laplace operator in (17). It is not difficult to cross check that the (− q 4π tan χ) function, where q stands for a color charge, is such a solution, ∆ S 3 (χ, θ, ϕ) tan χ = 0.…”

“…If the B(R) parameter were to be in some way related to the dilaton mass, then the latter could throughout leave the conformal symmetry intact. In more technical terms, it can be shown that the Hamiltonian in (65) is intertwined with the χ dependent part of the Lapalcian in (17). The parameter values fitting the data are listed in the Table 1.…”

“…As a pair of potentials allowing for the construction of the aforementioned duality we encounter the trigonometric Scarf well,V Sc (χ), earlier considered in [17], and the Pöschl-Teller barrier, V PT (ρ) [18],…”

Parity-violating effects in an exotic perturbation of the rigid rotator

“…In order to get an insight into the symmetry properties of the perturbed Hamiltonian for this case one has to follow the standard procedure and find the complete set of operators with which it is commuting. In calculating the commutators of these operators, a clue on a possible Lie algebra might be obtained [19].…”

“…Alternatively, such equations can also be handled by the so called factorization method underlying modern super-symmetric quantum mechanics (see [7] for details), or by employing the recently elaborated Nikiforov-Uvarov method [15]. The solutions to (19) are given, among others, in [12]. Upon equipping the wave functions with quantum numbers according to,…”