Al-Yousef, I.F.; Ekhwan, M.; Bahlouli, H.; Alhaidari, A.D. Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function. Axioms2023, 12, 334.
Al-Yousef, I.F.; Ekhwan, M.; Bahlouli, H.; Alhaidari, A.D. Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function. Axioms 2023, 12, 334.
Al-Yousef, I.F.; Ekhwan, M.; Bahlouli, H.; Alhaidari, A.D. Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function. Axioms2023, 12, 334.
Al-Yousef, I.F.; Ekhwan, M.; Bahlouli, H.; Alhaidari, A.D. Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function. Axioms 2023, 12, 334.
Abstract
We use a recently proposed formulation of quantum mechanics based, not on potential functions but rather, on orthogonal energy polynomials. In this context, the most important building block of a quantum mechanical system, which is the wavefunction at a given energy, is expressed as pointwise convergent series of square integrable functions in configuration space. The expansion coefficients of the series are orthogonal polynomials in the energy; they contain all physical information about the system. No reference is made at all to the usual potential function. We consider, in this new formulation, 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. The objective is to demonstrate the viability of this formulation of quantum mechanics and its power in generating rich energy spectra 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.
Keywords
Energy polynomials; Energy spectrum; harmonic oscillator; orthogonal polynomials; tridiagonal matrix; wavefunction
Subject
Physical Sciences, Mathematical Physics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
