Energy spectrum design and potential function engineering

Abstract

Starting with an orthogonal polynomial sequence $\{p_n(s){\}}_{n=0}^{\infty }$ that has a discrete spectrum, we design an energy spectrum formula, $E_k=f(s_k)$, where $|{s_k\}$ is the finite or infinite discrete spectrum of the polynomial. Using a recent approach for doing quantum mechanics based, not on potential functions but, on orthogonal energy polynomials, we give a local numerical realization of the potential function associated with the chosen energy spectrum. In this work, we select the three-parameter continuous dual Hahn polynomial as an example. Exact analytic expressions are given for the corresponding bound states energy spectrum, scattering states phase shift, and wavefunctions. However, the potential function is obtained only numerically for a given set of physical parameters.

arXiv comment15 pages, 4 figures (one in color)

References in corpus (5)

• Quantum mechanics without potential function
• Establishing correspondence between the reformulation of quantum mechanics without a potential function and the conventional formulation
• Quantum mechanics with orthogonal polynomials
• Construction of potential functions associated with a given energy spectrum -- An inverse problem. II
• Construction of potential functions associated with a given energy spectrum