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arXiv:quant-ph/0110157

A new dynamical group approach to the modified Poschl-Teller potential

Shi‐Hai Dong, R. Lemus

Published Oct 27, 2001• 6 citations
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Abstract

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 $su(2)$ algebra. Analytical expressions for the functions $\sinh(αx)$ and $\frac{\cosh(αx)}α \frac{d}{dx}$ are evaluated from these ladder operators. The expansions of the coordinate $x$ and momentum $\hat p$ in terms of the $su(2)$ generators are presented. This analysis allows to establish an exact quantum-mechanical connection between the $su(2)$ vibron model and the traditional descriptions of molecular vibron.

DOI
10.48550/arxiv.quant-ph/0110157

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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.
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 quantum mechanics of a simplified Pöschl–Teller potential in one dimension is analyzed at an elementary level. The number of bound states of this potential is determined by a parameter l characterizing the depth of the potential energy. The resulting collection of potentials with varying depth, as well as their ground state energies, is found by requiring the ground state wave function to be the hyperbolic secant function raised to power l . A step-by-step operator-based strategy for the construction of the eigenfunctions corresponding to all other bound states and their eigenenergies is discussed. We also show how all these results can be alternatively obtained by means of the Legendre differential equation, familiar from the rigid-rotor and hydrogen atom problems. The connection between all these problems is discussed while highlighting the use of this model for teaching introductory quantum mechanics.
SUSY quantum mechanics can be applied to solve Schrodinger equation for high dimensional system that can be reduced into one dimensional system and represented in lowering and raising operators. Lowering and raising operators can be obtained using relationship between original Hamiltonian equation and the (super) potential equation. In this paper SUSY quantum mechanics is used as a method to obtain the wave function and the energy level of the Modified Poschl Teller potential. The graph of wave function equation and probability density is simulated by using Delphi 7.0 programming language. Finally, the expectation value of quantum mechanics operator could be calculated analytically using integral form or probability density graph resulted by the programming.
This work prolongs recent investigations by Bergeron et al [see 2012 J. Phys. A: Math. Theo. 45 244028] on new SUSYQM coherent states for Pöschl-Teller potentials. It mainly addresses explicit computations of eigenfunctions and spectrum associated to the higher order hierarchic supersymmetric Hamiltonian. Analysis of relevant properties and normal and anti-normal forms is performed and discussed. Coherent states of the hierarchic first order differential operator Am,ν,βA_{m,ν,β}Am,ν,β​ of the Pöschl-Teller Hamiltonian Hν,β(m)H_{ν,β}^{(m)}Hν,β(m)​ and their characteristics are studied.
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.