[Paper Review] A new dynamical group approach to the modified Poschl-Teller potential

Shi‐Hai Dong, R. Lemus|ArXiv.org|Oct 27, 2001

Quantum Mechanics and Non-Hermitian Physics7 references6 citations

TL;DR

This paper develops a new dynamical group approach to the modified Pöschl-Teller (MPT) potential by constructing ladder operators directly from the wave functions using the factorization method, identifying them with the $su(2)$ algebra. It establishes an exact quantum-mechanical link between the $su(2)$ vibron model and traditional molecular vibration descriptions, showing that the standard $su(2)$ model corresponds to the dominant $\Delta n = \pm 1$ interaction in coupled MPT oscillators, with extensions possible via higher-order terms in the expansion of $x$ and $p$ in $su(2)$ generators.

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.

Motivation & Objective

• To establish a direct connection between the modified Pöschl-Teller (MPT) potential and the $su(2)$ algebra without auxiliary parameters.
• To construct ladder operators for the MPT potential directly from its wave functions using the factorization method.
• To derive analytical expressions for matrix elements of $\sinh(\alpha x)$ and $\frac{\cosh(\alpha x)}{\alpha}\frac{d}{dx}$ from these ladder operators.
• To expand the coordinate $x$ and momentum $\hat{p}$ in terms of $su(2)$ generators, enabling a quantum-mechanical bridge between the $su(2)$ vibron model and traditional configuration space descriptions.
• To analyze the validity of the $su(2)$ vibron model as an approximation for coupled MPT oscillators, particularly near the dissociation limit.

Proposed method

• The factorization method is applied directly to the MPT wave functions to derive first-order differential ladder operators $\hat{\cal P}_{\pm}$ in terms of the variable $u = \tanh(\alpha x)$, without introducing auxiliary parameters.
• The ladder operators are shown to satisfy the $su(2)$ algebra commutation relations, identifying them as generators of the $su(2)$ Lie algebra.
• Analytical expressions for matrix elements of $\sinh(\alpha x)$ and $\frac{\cosh(\alpha x)}{\alpha}\frac{d}{dx}$ are computed using the derived ladder operators.
• The coordinate $x$ and momentum $\hat{p}$ are expanded in terms of $su(2)$ generators $\hat{b}^\dagger$ and $\hat{b}$ by inverting the expressions for the ladder operators.
• An approximation is introduced where $\hat{c}^\dagger \simeq \hat{b}^\dagger z_n + \hat{b} \zeta_n$, allowing the mapping of MPT operators onto $su(2)$ generators with $\nu = 2q+1$ as a key parameter.
• The method is extended to two coupled MPT oscillators, showing that the standard $su(2)$ vibron interaction arises naturally from the dominant $\Delta n = \pm 1$ coupling when $n \ll n_{\text{max}}$.

Experimental results

Research questions

• RQ1 How can ladder operators for the modified Pöschl-Teller potential be constructed directly from its wave functions without auxiliary parameters?
• RQ2 What is the exact algebraic structure underlying the MPT potential, and how does it relate to the $su(2)$ Lie algebra?
• RQ3 Can the coordinate $x$ and momentum $\hat{p}$ be expressed as expansions in terms of $su(2)$ generators, and what is the physical significance of such an expansion?
• RQ4 To what extent does the standard $su(2)$ vibron model approximate the dynamics of coupled MPT oscillators, particularly in the low- and high-energy regimes?
• RQ5 How can the $su(2)$ vibron model be systematically extended to describe vibrational excitations near the dissociation limit where polyad breaking occurs?

Key findings

• The ladder operators for the MPT potential are constructed directly from the wave functions via the factorization method, and they are shown to satisfy the $su(2)$ algebra commutation relations.
• The matrix elements of $\sinh(\alpha x)$ and $\frac{\cosh(\alpha x)}{\alpha}\frac{d}{dx}$ are derived analytically using the ladder operators, providing exact expressions in terms of $n$ and $\nu$.
• The coordinate $x$ and momentum $\hat{p}$ are expressed as expansions in $su(2)$ generators, with the form $\hat{c}^\dagger \simeq \hat{b}^\dagger z_n + \hat{b} \zeta_n$, where $z_n$ and $\zeta_n$ are functions of $n$ and $\nu$.
• The standard $su(2)$ vibron model is shown to be equivalent to the dominant $\Delta n = \pm 1$ interaction in coupled MPT oscillators, valid when $n \ll n_{\text{max}}$.
• The approximation $\hat{A}(n_1,n_2) \simeq 1$ and $\hat{B}(n_1,n_2) \simeq 0$ holds for low-lying states, confirming the $su(2)$ vibron model as a low-lying approximation to the MPT system.
• The extended $su(2)$ model, incorporating higher-order terms in the expansion of $\hat{c}^\dagger$ and $\hat{c}$, provides a framework for describing vibrational spectra near the dissociation limit where polyad breaking occurs.