An Invitation to Algebraic Methods in Quantum Theory
Abstract
In this paper, we present various well-studied examples of quantum systems, and discuss how they might be analysed algebraically. We analyse the harmonic oscillator, hydrogenic atom, optical components, and particle spin using multiple algebraic methods, and highlight the central importance of algebraic relationships between operators in determining the physical character of the system. In particular, we show: the ubiquity of the Lie algebra so(2,1)≅su(1,1) as the ``spectrum generating algebra''; the role of su(2) and su(1,1) in determining the action of optical components; that an elementary algebraic approach to particle spin offers novel insights into its nature versus the usual approaches; and how understanding the full (often hidden) symmetry of a problem enables it to be solved immediately without the need to solve differential equations. We hope this work will serve as a gentle introduction to algebraic methods in quantum mechanics for the interested reader --- we therefore signpost active areas of research within the community.
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BibTeX
@misc{bradshaw2026invitation,
title = {{An Invitation to Algebraic Methods in Quantum Theory}},
author = {Peter T. J. Bradshaw and Jonte R. Hance},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32659},
eprint = {2610.32659},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32659},
}
RIS
TY - UNPB
TI - An Invitation to Algebraic Methods in Quantum Theory
AU - Peter T. J. Bradshaw
AU - Jonte R. Hance
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32659
UR - https://arxiv.org/abs/2610.32659
AB - In this paper, we present various well-studied examples of quantum systems, and discuss how they might be analysed algebraically. We analyse the harmonic oscillator, hydrogenic atom, optical components, and particle spin using multiple algebraic methods, and highlight the central importance of algebraic relationships between operators in determining the physical character of the system. In particular, we show: the ubiquity of the Lie algebra so(2,1)≅su(1,1) as the ``spectrum generating algebra''; the role of su(2) and su(1,1) in determining the action of optical components; that an elementary algebraic approach to particle spin offers novel insights into its nature versus the usual approaches; and how understanding the full (often hidden) symmetry of a problem enables it to be solved immediately without the need to solve differential equations. We hope this work will serve as a gentle introduction to algebraic methods in quantum mechanics for the interested reader --- we therefore signpost active areas of research within the community.
ER -