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arXiv:quant-ph/0410009 · Journal of Physics A Mathematical and General

Group approach to the quantization of the Pöschl–Teller dynamics

V. Aldaya, Julio Guerrero

Published Jul 20, 2005• 18 citations
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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.

DOI
10.1088/0305-4470/38/31/005

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The quantum dynamics of a particle in the Modified Pöschl-Teller potential is derived from the group SL(2,R)SL(2,R)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)SL(2,R)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.
The properties of the modified Pöschl-Teller (MPT) potential are outlined. The ladder operators are constructed directly from the wave functions without introducing any auxiliary variable. It is shown that these operators are associated to the su(2)su(2)su(2) algebra. Analytical expressions for the functions sinh⁡(αx)\sinh(αx)sinh(αx) and cosh⁡(αx)αddx\frac{\cosh(αx)}α \frac{d}{dx}αcosh(αx)​dxd​ are evaluated from these ladder operators. The expansions of the coordinate xxx and momentum p^\hat pp^​ in terms of the su(2)su(2)su(2) generators are presented. This analysis allows to establish an exact quantum-mechanical connection between the su(2)su(2)su(2) vibron model and the traditional descriptions of molecular vibron.
Based on the concepts of integrability and nonintegrability of a quantum system presented in a previous paper [Zhang, Feng, Yuan, and Wang, Phys. Rev. A 40, 438 (1989)], a realization of the dynamics in the quantum phase space is now presented. For a quantum system with dynamical group scrG and in one of its unitary irreducible-representation carrier spaces ${\mathit{gerh}}_{\mathrm{\ensuremath{\Lambda}}}$, the quantum phase space is a 2${\mathit{M}}_{\mathrm{\ensuremath{\Lambda}}}$-dimensional topological space, where ${\mathit{M}}_{\mathrm{\ensuremath{\Lambda}}}$ is the quantum-dynamical degrees of freedom. This quantum phase space is isomorphic to a coset space scrG/scrH via the unitary exponential mapping of the elementary excitation operator subspace of scrg (algebra of scrG), where scrH (\ensuremath{\subset}scrG) is the maximal stability subgroup of a fixed state in ${\mathit{gerh}}_{\mathrm{\ensuremath{\Lambda}}}$. The phase-space representation of the system is realized on scrG/scrH, and its classical analogy can be obtained naturally. It is also shown that there is consistency between quantum and classical integrability. Finally, a general algorithm for seeking the manifestation of ``quantum chaos'' via the classical analogy is provided. Illustrations of this formulation in several important quantum systems are presented.
In a recent paper (Aldaya V and Guerrero J 2005 J. Phys. A: Math. Gen. 38 6939 ) it is shown that the bound states of the Modified Pöschl–Teller potential should be described by the non-compact dynamical group SU (1, 1) instead of the usual compact group SU (2). Here we prove that SU (2) cannot be the dynamical group for a potential with a finite number of bound states on the basis of the Modified Pöschl–Teller potential and the Morse potential. This suggests that a quantum system with a continuum spectrum and a finite number of bound states should be described, both in the continuum and the discrete spectrum, by a non-compact dynamical group instead of a compact one.