Williamson theorem in classical, quantum, and statistical physics

In this work, we present (and encourage the use of) the Williamson theorem and its consequences in several contexts in physics. We demonstrate this theorem using only basic concepts of linear algebra and symplectic matrices. As an immediate application in the context of small oscillations, we show t...

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Bibliographic Details
Published in:American journal of physics 2021-12, Vol.89 (12), p.1139-1151
Main Author: Nicacio, F.
Format: Article
Language:English
Subjects:
Mathematical functions
Physics
Quantum dots
Quantum physics
Theorems
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Summary:In this work, we present (and encourage the use of) the Williamson theorem and its consequences in several contexts in physics. We demonstrate this theorem using only basic concepts of linear algebra and symplectic matrices. As an immediate application in the context of small oscillations, we show that applying this theorem reveals the normal-mode coordinates and frequencies of the system in the Hamiltonian scenario. A modest introduction of the symplectic formalism in quantum mechanics is presented, using the theorem to study quantum normal modes and canonical distributions of thermodynamically stable systems described by quadratic Hamiltonians. As a last example, a more advanced topic concerning uncertainty relations is developed to show once more its utility in a distinct and modern perspective.
ISSN:0002-9505
1943-2909
DOI:10.1119/10.0005944