© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Axioms

Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function

Ibraheem F. Al-Yousef, Moayad Ekhwan, H. Bahlouli, Abdulaziz D Alhaidari

Published Mar 29, 2023• 1 citation

Abstract

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, Pöschl–Teller, 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.

DOI: 10.3390/axioms12040334

Related papers

Quantum mechanics with orthogonal polynomials

Abdulaziz D Alhaidari

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).

arXiv:1709.07652•Sep 22, 2017•5 citations

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

Abdulaziz D Alhaidari

Abstract We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. 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 quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.

Communications in Theoretical Physics•Dec 30, 2019•19 citations

Reconstructing the Potential Function in a Formulation of Quantum Mechanics Based on Orthogonal Polynomials

Abdulaziz D Alhaidari

Abstract In a recent reformulation of quantum mechanics, the properties of the physical system are derived from orthogonal polynomials that make up the expansion coefficients of the wavefunction in a complete set of square integrable basis. Here, we show how to reconstruct the potential function so that a correspondence with the standard formulation could be established. However, the correspondence places restriction on the kinematics of such problems.

arXiv:1703.08659•Communications in Theoretical Physics•Dec 1, 2017•21 citations

Quantum mechanics without potential function

A. D. Alhaidari, M. E. H. Ismail

In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.

arXiv:1408.4003•Journal of Mathematical Physics•Jul 1, 2015•41 citations

The harmonic oscillator in quantum mechanics: A third way

F. Marsiglio

Courses on undergraduate quantum mechanics usually focus on solutions of the Schrödinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced, only academic examples are used, such as the matrix representation of Dirac’s raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.

arXiv:0806.3051•American Journal of Physics•Feb 13, 2009•46 citations

Chapter 10. The Simple Harmonic Oscillator

The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend

Sciendo eBooks•Sep 1, 2018•0 citations

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, Pöschl–Teller, 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.

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).

Abstract We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. 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 quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.

In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.

Courses on undergraduate quantum mechanics usually focus on solutions of the Schrödinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced, only academic examples are used, such as the matrix representation of Dirac’s raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.

The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend