Group approach to the quantization of the Pöschl–Teller dynamics

Abstract

The quantum dynamics of a particle in the Modified Pöschl-Teller potential is derived from the group SL(2, R) by applying a Group Approach to Quantization (GAQ). The explicit form of the Hamiltonian as well as the ladder operators is found in the enveloping algebra of this basic symmetry group. The present algorithm provides a physical realization of the non-unitary, finite-dimensional, irreducible representations of the SL(2, R) group. The non-unitarity manifests itself in that only half of the states are normalizable, in contrast with the representations of SU (2) where all the states are physical.

A local–normal description of vibrational excitations of pyramidal molecules in terms of Morse oscillators

“…; j À 1. The Morse functions are then associated with one branch (in this case to m P 1) of the suð2Þ representations, although a recent work associates a non-compact group to the bound Morse space [45]. The bound solutions (19), however, do not form a complete set of states in the Hilbert space.…”

“…where the interpretation for the operatorsN in terms of tensor couplings is similar to (45). Since the operator N 43=43 may lead to ambiguities, we present its explicit form…”

Symmetries of Non-Linear Systems: Group Approach to Their Quantization

“…In fact, in the GAQ scheme, not only the generators of the original group G can be quantized, but also the entire universal enveloping algebra. This procedure has been explicitly achieved in dealing with the quantum dynamics of a particle in a (modified) Pöschl-Teller potential [25], where the "first-order" (auxiliary) group G used was SL(2, R). We start by parametrizing rotations with a vector ε in the rotation-axis direction and with modulus (ϕ is the rotation angle)…”

The quantum harmonic oscillator on the sphere and the hyperbolic plane

“…At this point we also recall the existence of another family of ''relativistic Hermite polynomials'' obtained by Aldaya et al in Refs. [74][75][76] in the study of the relativistic quantum harmonic oscillator. We also note that the equations (15) and (18) are of hypergeometric type and the relation of the solutions with the hypergeometric series is an interesting problem.…”