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Journal of Physics A Mathematical and General

Inconsistencies in the description of a quantum system with a finite number of bound states by a compact dynamical group

Julio Guerrero, V. Aldaya

Published Apr 19, 2006• 6 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/39/18/l03

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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.
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.
Abstract In this research work, an exact normalised bound eigenstate and energy eigenvalue of the Hulthen potential, for the zero angular momentum within the framework of the non-relativistic time independent Schrödinger wave equation were obtained using the Nikiforov–Uvarov method. Also, their self-adjoint differential creation and annihilation (ladder) operators were derived, andfrom the commutation relations their Casimir operator was defined. These results showed that the non-compact S U ( 1 , 1 ) group is the dynamical group for this exact bound state, described by the representation D + ( i ) with a spectrum bounded below.