Quantum mechanics with orthogonal polynomials

TL;DR

This paper introduces a novel formulation of quantum mechanics using orthogonal polynomials, expanding wavefunctions over a basis where coefficients are polynomials in energy, leading to more exactly solvable problems.

Contribution

It develops a new approach that enlarges the class of exactly solvable quantum systems by leveraging orthogonal polynomial properties in the wavefunction expansion.

Findings

Increased class of exactly solvable quantum problems

Wavefunctions expanded over orthogonal polynomial basis

Derivation of physical system properties from polynomial characteristics

Abstract

We present a formulation of quantum mechanics based on orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis in configuration space where the expansion coefficients are orthogonal polynomials in the energy. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable non-relativistic quantum mechanical problems becomes larger than in the conventional formulation (see, for example, Table 1 in the text).