Mathematical Physics
Following the celebrated postulates of quantum mechanics, we write the quantum mechanical wavefunction as a convergent series of suitably selected complete square-integrable basis functions in configuration space. The expansion coefficients of the series are energy orthogonal polynomials that contain all spectral information about the system. We exploit the properties of these polynomials to introduce physical systems with rich and highly nontrivial energy spectra. In this approach, no reference is made at all to the usual potential energy function. We consider, in this new approach, a few representative problems at the level of undergraduate students who took at least two courses in quantum mechanics and are familiar with the basics of orthogonal polynomials. Our aim is to expose students to quantum systems with rich energy spectra that goes beyond the very limited textbook examples of systems with very simple energy spectra (e.g., the harmonic oscillator, Coulomb, Morse, PΓΆschlβTeller, etc.) illustrating the physical significance of these energy polynomials in the description of a quantum system. To assist students, partial solutions are given in an appendix as tables and figures.
Background
I took Classical Mechanics and Quantum Mechanics I with Prof. Bahlouli, where he nurtured my interested in computational physics via Mathematica, and after noticing my expertise, he offered me to join him and the pioneer in the field Prof. Alhaidari. I was responsible for producing and validating computational results that matches the analytical results of the paper. Here is SchrΓΆdingerβs equation in the new language:
\[E{P_n}(E) = a_nP_n(E) + b_{n-1}P_{n-1}(E) + b_nP_{n+1}(E)\]