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.
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December 2021
ADVANCED TOPICS|
December 01 2021
Williamson theorem in classical, quantum, and statistical physics
F. Nicacio
F. Nicacio
a)
Instituto de Física, Universidade Federal do Rio de Janeiro
, 21941-972, RJ, Brazil
and Universität Wien, NuHAG, Fakultät für Mathematik
, A-1090 Wien, Austria
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a)
Electronic mail: nicacio@if.ufrj.br
Am. J. Phys. 89, 1139–1151 (2021)
Article history
Received:
November 29 2020
Accepted:
August 02 2021
Citation
F. Nicacio; Williamson theorem in classical, quantum, and statistical physics. Am. J. Phys. 1 December 2021; 89 (12): 1139–1151. https://doi.org/10.1119/10.0005944
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