https://doi.org/10.1140/epjs/s11734-025-01764-z
Regular Article
Why is it so difficult to generalize Heisenberg’s matrix mechanics from the harmonic oscillator to other exactly solvable problems?
Department of Physics, Georgetown University, 37th and O Sts. NW, 20057, Washington, DC, USA
a
james.freericks@georgetown.edu
Received:
18
December
2024
Accepted:
24
June
2025
Published online:
8
July
2025
In 1925, Heisenberg, and later Born and Jordan, developed matrix mechanics as the first viable modern theory for quantum mechanics. While this approach worked beautifully for the simple harmonic oscillator and for angular momentum states, it was not able to solve other problems, which ultimately lost favor to the wavefunction approach of Schrödinger. In this article, I discuss why this was the case. In particular, I discuss how one can use similar techniques to what Heisenberg, Born and Jordan did to find the ground state and ground-state energy of all exactly solvable problems by properly factorizing the Hamiltonian into a positive semidefinite operator form. However, only by introducing an abstract vector space and abstract operators can one find all eigenvalues and eigenstates of these systems. I describe how one might have proceeded to do this using the matrix mechanics formalism. Along the way, I will describe, where the approach is limited and how one can advance past those limitations.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
