Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters

A D Alhaidari

Table 2. The orthogonal polynomial(s) associated with a given physical system as a function of the structure of its energy spectrum. Note the matching of the spectra of the physical system and that of its corresponding orthogonal polynomial except when the physical system has a mix of a continuous and an infinite discrete energy spectrum then there is a need for two polynomials to represent the system: a continuous one and a discrete one with infinite spectrum.

Energy spectrum Polynomial spectrum Discrete Discrete Continuous Finite Infinite Continuous Finite Infinite ✓ ✗ ✗ ✓ ✗ ✗ ✗ ✗ ✓ ✗ ✗ ✓ ✓ ✓ ✗ ✓ ✓ ✗ ✓ ✗ ✓ ✓ ✗ ✗ ✗ ✗ ✓